Paul Erdos
The Art of Counting
Selected Writings
edited by Joel Spencer
The MIT Press
Cambridge, Massachusetts, and London, England

Copyright © 1973 by
The Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording, or by any information storage
and retrieval system, without permission in writing from the publisher.
This book was printed on 50$ white Semline Publisher's offset
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and bound in Columbia Milbank Vellum (MBV-4368)
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Library of Congress Cataloging in Publication Data
Erdos, P
Paul Erdos: the art of counting.
(Mathematicians of our time, v. 5)
A commemorative collection of the author's papers.
1. Mathematics—Collected works. I. Spencer,
Joel H., ed. II. Title. III. Series.
QA3.E69 510'.8 73-4275
ISBN 0-262-19116-4

To the memory of my parents,
Ann and Louis Erdos

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Paul Erdos

Dedication
[This volume is being published to coincide with the sixtieth birthday of Paul Erdos.
Richard Rado, one of the world's foremost mathematicians, and a collaborator with
Paul Erd'ds over many years, has graciously contributed this commentary on the
mathematics, and life style, of Paul Erd'ds.—ed.]
Paul Erdos sixty? Never! Those who have been fortunate enough to have
known him over a longish stretch of time will testify that he is simply a fixed
point in the evolving universe of mathematics, unchanging in his penetrating
interest in everything around him, in the sharing of mathematical news, and
in his selfless squandering of ideas. A most fruitful activity, and one he takes
very seriously indeed, is his encouragement of child prodigies. He will go to
any length to arrange a talk with them, and his pleasure at discovering a new
star knows no bounds. Himself the son of two mathematicians, his genius
became apparent at a very early age, and so he surely feels a close affinity
with exceptionally gifted children.
His mode of life is that of a Wandering Scholar. Just as such scholars used
to roam freely all over Europe in the Middle Ages, so Erdos is constantly on
the move across the five continents, the veritable peripatetic mathematician.
He is steadfast in his determination never to settle down anywhere. His loyalty
is to the Queen of Sciences and not to any particular place or institution. Over
many years and into her very old age, his mother accompanied him on his
travels. Just as naturally as others breathe, so he listens to mathematics or
lectures himself, using an idiosyncratic vocabulary which has become widely
known as Erdosian.
Paul Erdos is assuredly no narrow specialist and his contributions to
mathematical discovery form a very wide spectrum. No catalogue of his
research topics will do justice to the extent or quality of his investigations; all
the same, a few of these topics might be mentioned. Erdos has worked in
almost every domain of the theory of numbers, in set theory and
combinatorics, in the theory of designs, graph theory, probability and its applications
(to number theory, group theory, and generally to the study of random
structures), in real analysis, the theory of infinite series, and the theory of
interpolation. His insight into mathematical situations is phenomenal and
innumerable problems which seemed much too ferocious to tackle have been
tamed by him to the point where a common or garden mathematician may
venture to approach them without trepidation.
DEDICATION

This volume is presented to him as a birthday gift from his uncountable
friends all over the world. It will also, we hope, provide an opportunity for
fellow mathematicians to see, from a small but representative selection of his
papers, the breadth of Paul Erdos's interest, to savour his unique style, and to
catch a glimpse of some of the astonishing results with which he has enriched
mathematics. Long may he continue with his own inimitable mode of creative
work, to the benefit of our science and the delight of its practitioners!
Richard Rado
The University,
Reading, England
viii DEDICATION

Contents
Bracketed numbers refer to the list of papers of Paul Erdos given at the back
of this volume. An alphabetical list of the papers of this volume is also given
there.
Foreword by P. Turan
xvii
A Combinatorial Problem in Geometry:
Reminiscences by Gy. Szekeres
xix
Introductory Commentary by J. Spencer
xxiii
Part I
Papers of Special Interest
l
Chapter 1. An Early Gem
3
Commentary
[18] (with Gy. Szekeres) A combinatorial problem in geometry 5
Chapter 2. Problems
13
Commentary
[346] Some unsolved problems (partial paper) 15
[419] Some applications of probability to graph theory and
combinatorial problems 23
[548] Some unsolved problems in graph theory and combinatorial
analysis 27
[575] Problems and results in combinatorial analysis 40
CONTENTS

Part II
Graph Theory
53
Chapter 3. Representation of Graphs
55
Commentary
[344] (with T. Gallai) On the minimal number of vertices
representing the edges of a graph 57
[426] (with L. Moser) On the representation of directed graphs as
unions of orderings 79
[464] (with A. W. Goodman and L. Posa) The representation of a
graph by set intersections 87
Chapter 4. Coloring of Graphs
95
Commentary
[386] On circuits and subgraphs of chromatic graphs 97
[487] Some remarks on chromatic graphs 103
[518] (with D. J. Kleitman) On coloring graphs to maximize the
proportion of multicolored k-edges 107
[525] Problems and results in chromatic graph theory 113
[i] (with J. Spencer) Imbalances in ^-colorations 122
Chapter 5. Extremal Graph Theory
129
Commentary
[360] On a theorem of Rademacher-Turan 131
[366] Uber ein Extremalproblem in der Graphentheorie 137
[378] Remarks on a paper of Posa 143
[389] On the number of complete subgraphs contained in certain
graphs 145
x CONTENTS

On the number of triangles contained in certain graphs 151
Extremal problems in graph theory 155
(with A. Hajnal and J. W. Moon) A problem in graph theory 163
(with A. Hajnal) On complete topological subgraphs of certain
graphs 167
On extremal problems of graphs and generalized graphs 174
On an extremal problem in graph theory 182
A problem on independent r-tuples 186
On the construction of certain graphs 189
(with M. Simonovits) A limit theorem in graph theory 194
(with A. Renyi and V. T. Sos) On a problem of graph theory 201
Some recent results on extremal problems in graph theory 222
Extremal problems in graph theory 229
On some new inequalities concerning extremal properties of
graphs 235
(with L. Moser) An extremal problem in graph theory 240
(with M. Simonovits) Some extremal problems in graph theory 246
(with V. T. Sos) Some remarks on Ramsey's and Turan's
theorem 260
Chapter 6. Circuits
271
Commentary
[301] (with T. Gallai) On the maximal paths and circuits of graphs 273
[367] (with L. Posa) On the maximal number of disjoint circuits of
a graph 293
[392] (with G. Dirac) On the maximal number of independent
circuits in a graph 303
[406] On the structure of linear graphs 319
[440] (with L. Posa) On independent circuits contained in a graph 324
xi CONTENTS

Chapter 7. Assorted Graph Theory
331
Commentary
[401] (with A. Renyi) Asymmetric graphs 333
[410] (with P. Kelly) The minimal regular graph containing a given
graph 354
[459] (with F. Harary and W. T. Tutte) On the dimension of a
graph 356
[480] On cliques in graphs 361
[520] Uber die in Graphen enthaltenen saturierten planaren
Graphen 363
Part III
Combinatorial Analysis
369
Chapter 8. Ramsey's Theorem
371
Commentary
[124] Some remarks on the theory of graphs 373
[160] (with R. Rado) A combinatorial theorem 376
[185] (with R. Rado) Combinatorial theorems on classifications of
subsets of a given set 383
[296] Graph theory and probability. I 406
[339] Graph theory and probability. II 411
[376] (with C. A. Rogers) The construction of certain graphs 418
[407] (with J. W. Moon) On subgraphs of the complete bipartite
graph 424
[iii] Applications of probabilistic methods to graph theory 429
[iv] (with R. J. McEliece and H. Taylor) Ramsey bounds for
graph products 434
xii CONTENTS

Chapter 9. Property B
437
Commentary
[390] On a combinatorial problem. I 439
[430] On a combinatorial problem. II 445
[527] On a combinatorial problem. Ill 448
Chapter 10. Systems of Sets
453
Commentary
[108] On a lemma of Littlewood and Offord 455
[312] (with R. Rado) Intersection theorems for systems of sets 460
[357] (with Chao Ko and R. Rado) Intersection theorems for
systems of finite sets 466
[539] On a lemma of Hajnal- Folk man 474
[542] (with J. Komlos) On a problem of Moser 479
[564] (with D. J. Kleitman) Extremal problems among subsets of
a set 482
[569] (with D. J. Kleitman) On collections of subsets containing no
4-member Boolean algebra 507
Chapter 11. Block Designs
511
Commentary
[243] (with A. Renyi) On some combinatorial problems 513
[412] (with H. Hanani) On a limit theorem in combinatorial
analysis 521
Chapter 12. Tournaments
525
Commentary
[399] On a problem in graph theory 527
[424] (with L. Moser) A problem on tournaments 531
[450] (with J. W. Moon) On sets of consistent arcs in a tournament 537
CONTENTS

Chapter 13. Information Theory
541
Commentary
[405] (with A. Renyi) On two problems of information theory 543
Part IV
Miscellany
557
Chapter 14. Random Objects
559
Commentary
[309] (with A. Renyi) On random graphs. I 561
[v] (with A. Renyi) On the evolution of random graphs 569
[324] (with A. Renyi) On the evolution of random graphs 574
[337] (with A. Renyi) On the strength of connectedness of a random
graph 618
[414] (with A. Renyi) On random matrices. I 625
[517] (with A. Renyi) On random matrices. II 632
Chapter 15. Latin Squares
639
Commentary
[120] (with I. Kaplansky) The asymptotic number of Latin
rectangles 641
[314] (with S. Chowla and E. G. Straus) On the maximal number
of pairwise orthogonal Latin squares of a given order 648
[413] (with A. Ginzburg) On a combinatorial problem in Latin
squares 653
Chapter 16. Geometry
659
Commentary
[106] (with N. H. Anning) Integral distances 661
xiv CONTENTS

[121] On sets of distances of n points 664
[175] (with P. Bateman) Geometrical extrema suggested by a lemma
of Besicovitch 667
[325] On sets of distances of n points in Euclidean space 676
[vi] (with Gy. Szekeres) On some extremum problems in
elementary geometry 680
[576] (with G. Purdy) Some extremal problems in geometry 690
Bibliography of Paul Erdos
697
Alphabetical Listing of Papers in this Volume
739
CONTENTS

Foreword
I am glad to have an opportunity to write a foreword to The Art of Counting.
I first met Paul Erdos in 1930 at the University of Budapest when he was a
freshman. This was already two or three years after our first joint work
appeared. We were both devoted solvers of the problems of the Kozepiskolai
A[atematikai Lapok, a monthly periodical for high school students interested in
mathematics. The best solutions of the problems were printed in the periodical
two months later with the names of all who solved the problem. The names of
the students who submitted the best solutions were given special prominence.
It was, of course, a special pride of the students to see their names as best
solvers. On one problem, our names were not only the only ones given special
prominence—but we were the only contributors having solved the problem.
Neither of us thought at the time that this would not be the last publication
with our joint names.
So I have known Paul Erdos long enough to be a bit personal. We met
daily at the university, made excursions practically every Sunday with a group
of other fellow students, steady members of which were (among others)
Gy. Szekeres and T. Gallai. The main subject of conversations was
mathematics; since Gallai and Erdos attended the graph theoretical lectures of
D. Konig, graph theory was discussed often. The first graph-theoretical result
of Erdos, an extension of Menger's theorem to infinite graphs, arose as early
as 1931; it was an answer to a question of Konig a few weeks after Konig
raised it in his course. This was published in Konig's classical book in 1936
(probably nowhere else).
In this period, Erdos was greatly influenced by Sperner's Theorem on
the maximal number of subsets of a finite set none of which are contained in
another. A decisive moment in his combinatorial studies was the rediscovery
of Ramsey's Theorem in 1934. (See [18] and Szekeres' Reminiscences.—ed.)
Erdos' main interest in the thirties was number theory but gradually
more and more combinatorial moments occurred even in these works. As a
clear sign of this I quote from a paper of his entitled "On sequences of integers
no one of which divides the product of two others and on some related
problems" which appeared in 1938 ([50]): "The argument was really based upon
the following theorem for graphs. Let 2k points be given. We split them into
two classes each containing k of them. The points of two classes are connected
by segments such that the segments form no closed quadrilateral. Then the
number of segments is less than 3P;2." No doubt he was not far from the
discovery of extremal graph theory before 1938.
xvii FOREWORD

The combinatorial touch is characteristic of Erdos' whole oeuvre; so the
papers reproduced in this volume cannot give a complete picture of his
combinatorial work and in particular not on the several aspects of combinatorics
which he used in dealing with problems in other parts of mathematics. Another
difficulty is the complete description of his effect on the whole development of
combinatorics. The new genre of papers created by him, the "problem-
papers" which are reproduced in this book, give, with all their richness, only
an indication of his influence.
Paul Erdos is really a prophet, a missionary, of combinatorics. Traveling
frequently, lecturing at many places on appealing combinatorial problems
which may be understood without learning a mass of definitions, Paul Erdos
has acquired an always growing army oi new followers. His prophetical
activity is facilitated by the fact that he has not to entertain the uninitiated with
hopes for other-wordly rewards: the intellectual pleasures offered by dealing
with combinatorial problems can be felt and understood even after working
unsuccessfully on such problems.
We observe today the enormous growth of combinatorics. The
bibliography of graph theory is doubled at each compilation. We see numerous col-
loquia, symposia, periodicals, and even quicker communications media. I
believe Paul Erdos is the strongest motor of this development.
Paul Turan
xviii FOREWORD

A Combinatorial Problem m Geometry
Reminiscences
It is not altogether easy to give a faithful account of events which took place
forty years ago, and I am quite aware of the pitfalls of such an undertaking.
I shall attempt to describe the genesis of this paper, and the part each of us
played in it, as I saw it then and as it lived on in my memory.
For me there is a bit more to it than merely reviving the nostalgic past.
Paul Erdos, when referring to the proof of Ramsey's theorem and the bounds
for Ramsey numbers given in the paper, often attributed it to me personally
(e.g., in Reference 1), and he obviously attached some importance to this
unusual step of pinpointing authorship in a joint paper. At the same time the
authorship of the "second proof" was never clearly identified.
I used to have a feeling of mild discomfort about this until an amusing
incident some years ago reassured me that perhaps I should not worry about
it too much. A distinguished British mathematician gave a lunch-hour talk to
students at Imperial College on Dirichlet's box principle, and as I happened
to be with Imperial, I went along. One of his illustrations of the principle was
a beautiful proof by Besicovitch of Paul's theorem,! and he attributed the
theorem itself to "Erdos and someone whose name I cannot remember." After
the talk I revealed to him the identity of Paul's co-author (incidentally also a
former co-author of the speaker) but assured him that no historical injustice
had been committed as my part in the theorem was less than e.
The origins of the paper go back to the early thirties. We had a very close
circle of young mathematicians, foremost among them Erdos, Turan and
Gallai; friendships were forged which became the most lasting that I have
ever known and which outlived the upheavals of the thirties, a vicious world
war and our scattering to the four corners of the world. I myself was an
"outsider," studying chemical engineering at the Technical University, but
often joined the mathematicians at weekend excursions in the charming hill
country around Budapest and (in summer) at open air meetings on the benches
of the city park.
Paul, then still a young student but already with a few victories in his bag,
was always full of problems and his sayings were already a legend. He used to
address us in the same fashion as we would sign our names under an article
* This paper. [18], is in Chapter 1.—ed.
t The "Second Proof" in [18].—ed.
REMINISCENCES

and this habit became universal among us; even to-day I often call old
members of the circle by a distortion of their initials.
":Szekeres Gy., open up your wise mind." This was Paul's customary
invitation—or was it an order?—to listen to a proof or a problem of his. Our
discussions centred around mathematics, personal gossip, and politics. It was
the beginning of a desperate era in Europe. Most of us in the circle belonged
to that singular ethnic group of European society which drew its cultural
heritage from Heinrich Heine and Gustav Mahler, Karl Marx and Cantor,
Einstein and Freud, later to become the principal target of Hitler's fury.
Budapest had an exceptionally large Jewish population, well over 200,000,
almost a quarter of the total. They were an easily identifiable group, speaking
an inimitable jargon of their own and driven by a strong urge to congregate
under the pressures of society. Many of us had leftist tendencies, following the
simple reasoning that our problems can only be solved on a global,
international scale and socialism was the only political philosophy that offered
such a solution. Being a leftist had its dangers and Paul was quick to spread
the news when one of our number got into trouble: "A.L. is studying the
theorem of Jordan." It meant that following a political police action A. L. has
just verified that the interior of a prison cell is not in the same component as
the exterior. I have a dim recollection that this is how I first heard about
the Jordan curve theorem.
Apart from political oppression, the Budapest Jews experienced cultural
persecution long before anyone had heard the name of Hitler. The notorious
"nuraerus clausus" was operating at the Hungarian Universities from 1920
onwards, allowing only 5% of the total student intake to be Jewish. As a
consequence, many of the brightest and most purposeful students left the
country to study elsewhere, mostly in Germany, Czechoslovakia, Switzerland,
and France. They formed the nucleus of that remarkable influx of Hungarian
mathematicians and physicists into the United States which later played such
an important role in the fateful happenings towards the conclusion of the
second world war.
For those of us who succeeded in getting into one of the home universities,
life was troublesome and the outlook bleak. Jewish students were often beaten
up and humiliated by organized student gangs and it was inconceivable that
any of us, be he as gifted as Paul, would find employment in academic life.
I myself was in a slightly better position as I studied chemical engineering and
therefore resigned to go into industrial employment, but for the others even a
high school teaching position seemed to be out of reach.
Paul moved to Manchester soon after his Ph.D. at Professor Mordell's
xx REMINISCENCES

invitation, and began his wanderings which eventually took him to almost
every mathematical corner of the world. But in the winter of 1932, 33 he was
still a student; I had just received my chemical degree and, with no job in
sight, I was able to attend the mathematical meetings with greater regularity
than during my student years. It was at one of these meetings that a talented
girl member of our circle, Esther Klein (later to become Esther Szekeres),
fresh from a one-semester stay in Gottingen, came up with a curious problem:
given five points in the plane, prove that there are four which form a convex
quadrilateral. In later years this problem frequently appeared in student's
competitions, also in the American Mathematical Monthly (53(1946)462,
problem E740). Paul took up the problem eagerly and a generalization soon
emerged : is it true that out of 2"~2 + 1 points in the plane one can always
select n points so that they form a convex n-sided polygon? I have no clear
recollection how the generalization actually came about; in the paper we
attributed it to Esther, but she assures me that Paul had much more to do
with it. We soon realized that a simple-minded argument would not do and
there was a feeling of excitement that a new type of geometrical problem
emerged from our circle which we were only too eager to solve. For me the fact
that it came from Epszi (Paul's nickname for Esther, short for e) added a
strong incentive to be the first with a solution and after a few weeks I was able
to confront Paul with a triumphant "E.P., open up your wise mind." What
I really found was Ramsey's theorem, from which it easily followed that there
exists a number Ar < x such that out of X points in the plane it is possible to
select n points which form a convex «-gon. Of course at that time none of us
knew about Ramsey. It was a genuinely combinatorial argument and it gave
for Ar an absurdly large value, nowhere near the suspected 2"~2. Soon
afterwards Paul produced his well known "second proof" which was independent
of Ramsey and gave a much more realistic value for .V; this is how a joint
paper came into being.
I do not remember now why it took us so long (a year and a half) to
submit the paper to the Compositio. These were troubled times and we had a
great many worries. I took up employment in a small industrial town, some
120 km from Budapest, and in the following year Paul moved to Manchester:
it was from there that he submitted the paper.
I am sure that this paper had a strong influence on both of us. Paul with
his deep insight recognized the possibilities of a vast unexplored territory and
opened up a new world of combinatorial set theory and combinatorial
geometry. For me it was the final proof (if I needed any) that my destiny lay with
mathematics, but I had to wait for another fifteen years before I got my first
GY. SZEKERES

mathematical appointment to Adelaide. I never returned to Ramsey again.
Paul's method contained implicitly that X > 2"~2, and this result
appeared some thirty-five years later (Reference 2) in a joint paper, after Paul's
first visit to Australia. The problem is still not completely settled and no one
yet has improved on Paul's value of
M2;:24)-
Of course we firmly believe that N = 2"~2 + 1 is the correct value.
References
1. P. Erdos. Some set-theoretical properties of graphs. Rev. Univ. Nac. Tucuman, Ser A., 3
(1942)363-367.
2. P. Erdos and Gy. Szekeres, On some extremum problems in elementary geometry, Ann.
Univ. Sci. Budapest 3, -/(1960, 61)53-62. [Appears in Chapter 16.—ed.]
Gy. Szekeres
REMINISCENCES

Introductory Commentary
The papers in this volume have been arranged into chapters. Each chapter is
roughly chronological. The divisions are, of necessity, somewhat arbitrary.
The transcendence of boundaries is characteristic of great mathematicians—
many of the papers of Paul Erdos fit into the crevices between graph theory,
geometry, number theory, foundations, and other seemingly unrelated fields.
A short commentary written by the editor prefaces each chapter. These
are by no means complete. They consist rather of personal comments and
therefore are prejudiced by the editor's interests. References are given, when
known, to more recent work on the problems considered in the papers.
A common theme of many of the papers is the probabilistic or nonconstructive
method of proof. Probabilistic Methods in Combinatorics by Paul Erdos and Joej
Spencer (Academic Press/Hungarian Academy of Sciences, publishers)
devotes itself to this topic. It is referred to in the commentaries as ES.
An alphabetical list of the papers contained in this volume as well as a
complete chronological list of the papers of Paul Erdos may be found in the
back. Bracketed references refer to the chronological list.
Paul Erdos' contributions to mathematics cannot be measured through
his papers alone. Over the years he has traveled extensively among the
mathematical centers of the globe. Like the bumblebee, flying from flower to flower
transmitting pollen, Paul Erdos has created an enormous cross-pollinization
effect in mathematics. An Erdos visit to a mathematical center is marked by
intense work. Mathematicians gather round and discuss the current problems
in their various fields. The resulting interplay of ideas is exhausting and highly
productive.
The mathematical community congratulates Paul Erdos on his sixtieth
birthday. Long may he continue to prove, conjecture, and advance
mathematical knowledge,
Joel Spencer
INTRODUCTORY COMMENTARY

Parti
Papers of Special Interest

Chapter 1
An Early Gem
This is one of Erdos' first papers and one of the earliest papers in which the
mathematical consequences of Ramsey's theorem were explored. The study
of Ramsey's theorem has remained a consistent interest of Erdos and a section
of this book is devoted to it. In addition, Erdos and Szekeres prove the
following result (in slightly different form): given any permutation of the integers
1,. . . , n2 + 1 there exists a monotonic subsequence of length n + 1. The best
proof of this now classical result is given in 8 lines in "A simple proof of a
theorem of Erdos and Szekeres," A. Seidenberg, J. London Math. Soc, 34
(1959)352.
Paper in Chapter 1
[18] (with Gy. Szekeres) A combinatorial problem in geometry
3 AN EARLY GEM

A Combinatorial Problem in Geometry
by
P. Erdos and G. Szekeres
Manchester
Introduction.
Our present problem has been suggested by Miss Esther Klein
in connection with the following proposition.
From 5 points of the plane of which no three lie on the same
straight line it is always possible to select 4 points determining
a convex quadrilateral.
We present E. Klein's proof here because later on we are
going to make use of it. If the least convex polygon which
encloses the points is a quadrilateral or a pentagon the theorem
is trivial. Let therefore the enclosing polygon be a triangle ABC.
Then the two remaining points D and E are inside ABC. Two
of the given points (say A and C) must lie on the same side of
the connecting straight line DE. Then it is clear that AEDC
is a convex quadrilateral.
Miss Klein suggested the following more general problem. Can
we find for a. given n a number N(n) such that from any set
containing at least N points it is possible to select n points forming
a convex polygonl
There are two particular questions: (1) does the number N
corresponding to n exist? (2) If so, how is the least N(n)
determined as a function of nl (We denote the least N by N0(n).)
We give two proofs that the first question is to be answered
in the affirmative. Both of them will give definite values for
N(n) and the first one can be generalised to any number of
dimensions. Thus we obtain a certain preliminary answer to
the second question. But the answer is not final for we generally
get in this way a number N which is too large. Mr. E. Makai
proved that iV0(5) = 9, and from our second demonstration, we
obtain N(5) = 21 (from the first a number of the order 210000).
TJius it is to be seen, that our estimate lies pretty far from
5 AN EARLY GEM

464
P. Erdos and G. Szekeres.
[2]
the true limit N0(n). It is notable that 2V(3) = 3 = 2 + 1,
N0 (4) = 5 = 2Z + 1, 2V0(5) = 9 = 23 + 1.
We might conjecture therefore that N0(n) = 2re_2 + 1, but the
limits given by our proofs are much larger.
It is desirable to extend the usual definition of convex polygon
to include the cases where three or more consecutive points lie
on a straight line.
First proof.
The basis of the first proof is a combinatorial theorem of
Ramsey 1). In the introduction it was proved that from 5 points
it is always possible to select 4 forming a convex quadrangle.
Now it can be easily proved by induction that n points
determine a convex polygon if and only if any 4 points of them form
a convex quadrilateral.
Denote the given points by the numbers 1, 2, 3, ..., N, then
any k-gon of the set of points is represented by a set of k of these
numbers, or as we shall say, by a ^-combination. Let us now
suppose each re-gon to be concave, then from what we observed
above we can divide the 4-combinations into two classes (i. e.
into ,,convex" and ,,concave" quadrilaterals) such that every
5-combination shall contain at least one ,,convex" combination
and each w-combination at least one concave one. (Wc regard
one combination as contained in another, if each element of the
first is also an element of the second.)
From Ramsey's theorem, it follows that this is impossible for
a sufficiently "large AT.
Ramsey's theorem can be stated as follows:
Let k, I, i be given positive integers, k 5g i; Z JS: i. Suppose that
there exist two classes, a. and ft, of i-eombinations of m elements
such that each k-combination shall contain at least one combination
from class a unci each l-combination. shall contain at least one
combination from class ft. Then for sufficiently great m < mt(k, I)
this is not possible. Ramsey enunciated his theorem in a slightly
different form.
In other words: if the members of a had been determined as
above at our discretion and m Sg mt(k, I), then there must be
at least one /-combination with every combination of order i
belonging to class a.
1) F. 11. Ramsey, Collected papers. On a problem of formal logic, 82—111.
Recently Skolkm also proved Ramsey's theorem [Fundamenta Math. 20 (1933),
254—261].
6 PAPERS OF SPECIAL INTEREST

[3]
A Combinatorial Problem in Geometry.
465
We give here a new proof of Ramsey's theorem, which differs
entirely from the previous ones and gives for m^k, I) slightly
smaller limits.
a) If i = 1, the theorem holds for every k and Z. For if we
select out of m some determined elements (combinations of order 1)
as the class a, so that every fc-gon (this shorter denomination
will be given to the combination of order k) must contain at
least one of the a elements, there are at most (k — 1) elements
which do not belong to the class a. Then there must be at least
(ra — fc + 1) elements of a. If (m—fc + l) 2^ Z, then there must be
an Z-gon of the a elements and thus
m^k + 1^2
which is evidently false for sufficiently great in.
Suppose then that i > 1.
b) The theorem is trivial, if k or Z equals i. If, for example,
k = i, then it is sufficient to choose m = I.
For k = 1 means that all i-gons aie a combinations and thus
in virtue of m = Z there is one polygon (i. e. the Z-gon formed
of all the elements), whose i-gons are all a-combinations.
The argument for Z = i runs similarly.
c) Suppose finally that k > i; and suppose that the theorem
holds for (i — 1) and every k arid /, further for i, k, I — 1 and
i, K — 1,1. We shall prove that it will hold for i, k, I also and in
virtue of (a) and (b) we may say that the theorem is proved
for all i, k, I.
Suppose then that we are able to carry out the division of
the i-polygons mentioned above. Further let k' be so great that
if in every Z-gon of k' elements there is at least one /3 combination,
then there is one (k-l)-gon all of whose i-gons are /3
combinations. This choice of k' is always possible in virtue of the induction-
hypothesis, we have only to choose k' = mt(k —1,1).
Similarly we choose Y so great that if each k-gon of V elements
contains at least one a combination, then there is one (I —1)-
gon all of whose z'-gons are a combinations.
We then take m larger than k' and Z'; and let
(%, a2, . . ., ak,) =^4
be an arbitrary fc'-gon of the first (n — 1) elements. By hypothesis
each Z-gon contains at least one j3 combination, hence owing to
the choice of k', A contains one (k—l)-gon (am , am , ..., am _ )
whose i-gons all belong to the class /3. Since in (am , . . ., amit_ ■ n)
1 AN EARLY GEM

466
P. Erdos and G. Szekeres.
[4]
there is at least one a combination, it is clear that this must be
one of the i-gons
(aPi,aPt, . . .,aP{_x,n) =s B.
In just the same way we may prove by replacing the roles
of k and I by k' and I' and of a by B, that if
(b1,ba,...,bl,) = A'
is an arbitrary Z'-gon of the first (n—1) elements, then among
the i-gons
(br ,br, ...,br ,n) = B'
there must be a /5 combination.
Thus we can divide the (i — l)-gons of the first (n — 1) elements
into classes a' and B' so that each /fc'-gon A shall contain at least
one a' combination B and each Z'-gon A' at least one B'
combination B'. But, by the induction-hypotheses this is impossible
form 22 mi_1(/fc'Z') + 1.
By following the induction, it is easy to obtain for mf(fc, I)
the following functional equation;
m{(k, I) = mx_x [mf(fc -1,1), mt (k, I - 1)] + 1. (1)
By this recurrence-formula and the initial values
mi(k,l) = k + l-l j
mt(i, I) — I, mi{k,i) = k\ '
obtained from (a) and (b) we can calculate every m^k, I).
We obtain e. g. easily
m2(fc + l,/ + 1) = (^1). (3)
The function mentioned in the introduction has the form
N(k) = m(5,k). (4)
Finally, for the special case i = 2, we give a graphotheoretic
formulation of Ramsey's theorem and present a very simple
proof of it.
Theorem: In an arbitrary graph let the maximum number of
independent points 2) be k; if the number of points is N 22 m(k, I)
then there exists in our graph a complete graph3) of order I.
2) Two points are said to be independent if they are not connected; k points
are independent if every pair is independent.
3) A complete graph is one in which every pair of points is connected.
8 PAPERS OF SPECIAL INTEREST

[5]
A Combinatorial Problem in Geometry.
467
Proof. For 1 = 1, the theorem is trivial for any k, since the
maximum number of independent points is k and if the number
of points is (fc + 1), there must be an edge (complete graph of
order 1).
Now suppose the theorem proved for (I— 1) with any k. Then
jV k
at least edges start from one of the independent points.
Hence if
*^^m(k,l~l),
i.e., N^k-m(k,l — l)+k, (5)
then, out of the end points of these edges we may select, in
virtue of our induction hypothesis, a complete graph whose
order is at least (I—1). As the points of this graph are connected
with the same point, they form together a complete graph of
order I.
Second proof.
The foundation of the second proof of our main theorem is
formed partly by geometrical and partly by combinatorial
considerations. We start from some similar problems and we
shall see, that the numerical limits are more accurate then in
the previous proof; they are in some respects exact.
Let us consider the first quarter of the plane, whose points
are determined by coordinates (x, y). We choose n points with
monotonously increasing abscissae 4).
Theorem: It is always possible to choose at least \/n points
with increasing abscissae and either monotonously increasing or
monotonously decreasing ordinates. If two ordinates are equal,
the case may equally be regarded as increasing or decreasing.
Let us denote by /(n, n) the minimum number of the points
out of which we can select n monotonously increasing or
decreasing ordinates.
We assert that
/(n + l,n + l)=/(n,n) + 2n-l. (6)
Let us select n monotonously increasing or decreasing points out
of the f(n, n). Let us replace the last point by one of the (2n — 1)
new points. Then we shall have once more /(n, n) points, out
4) The same problem was considered independently by Richard Rado.
9 AN EARLY GEM

468
P. Erdos and G. Szekeres.
[6]
of which we can select as before n monotonous points. Now we
replace the last point by one of the new ones and so on. Thus
we obtain 2n points each an endpoint of a monotonous set. Suppose
that among them (w + 1) are end points of monotonously
increasing sets. Then if yl 2| «/fcfor I > ft we add Pt to the monotonously
increasing set of Pk and thus, with it, we shall have an increasing
set of (n + 1) points. If yk^yl for every ft < I, then the (« + l)
decreasing end-points themselves give the monotonous set of
(w + 1) members. If between the In points there are at least
(w + 1) end-points of monotonous decreasing sets, the proof will
run in just the same way.
But it may happen that, out of the 2n points, just n are the
end-points of increasing sets, and n the end-points of decreasing
sets. Then by the same reasoning, the end-points of the decreasing
sets necessarily increase. But after the last end-point P there is
no point, for its ordinate would be greater or smaller than that
of P. If it is greater, then together with the n end-points it forms
a monotonously increasing (n + 1) set and if it is smaller, with the
n points belonging to P, it forms a decreasing set of (n + 1)
members. But by the same reasoning the last of the n increasing
end-points Q ought to be also an extreme one and that is evidently
impossible. Thus we may deduce by induction
/(n + 1, ra + l) = ra2 + l. (7)
Similarly let /(i, k) denote the minimum number of points
out of which it is impossible to select either i monotonously
increasing or k monotonously decreasing points. Wc have then
/(i, ft) =(i-l)(A—1) + 1. (8)
The proof is similar to the previous one.
It is not difficult to see, that this limit is exact i. e. we can
give (i — 1) (A; — 1) points such that it is impossible to select
out of them the desired number of monotonously increasing or
decreasing ordinates.
Wc solve now a similar problem;
P1, P2, . . . are given points on a straight line. Let f^i, k)
denote the minimum number of points such that proceeding
from left to right we shall be able to select either i points so
that the distances of two neighbouring points monotonously
increase or ft points so that the same distances monotonously
decrease. Wc assert that
/i(i, ft) =/,(*-!, k) +/j(i, ft- 1) - 1. (9)
10 PAPERS OF SPECIAL INTEREST

[7]
A Combinatorial Problem in Geometry.
469
Let the point C bisect the distance A B (A and B being the first
and the last points). If the total number of points is/j(i— 1, k) -\-
+/i(i, fc —1) — 1, then either the number of points in the first
half is at least /i(i — 1, k), or else there are in the second half
at least /j(i, fc—1) points. If in the first half there are /j(i — 1, fc)
points then either there are among them k points whose distances,
from left to right, monotonously decrease and then the equation
for ft(i, k) is fulfilled, or there must be {i — 1) points with
increasing distances. By adding the point B, we have i points with
monotonely increasing distances. If in the second interval there
are fx{i,k — \) points, the proof runs in the same way. (The
case, in which two distances are the same, may be classed into
either the increasing or the decreasing sets.)
It is possible to prove that this limit is exact. If the limits
/i^i —1, fc) and ft(i, fc —1) are exact (i. e. if it is possible to
give [/j(i —1, fc) — 1] points so that there are no (z — 1)
increasing nor fc decreasing distances) then the limit f^i, fc) is
exact too. For if we choose e. g. [/i(i —1, fc) — 1] points in the
0...1 interval, and [f^i, fc —1) — 1] points in the 2...3
interval, then we have [fi(i, k) — l] points out of which it is
equally impossible to select i points with monotonously increasing
and fc points with decreasing distances.
We now tackle the problem of the convex re-gon. If there are
n given points, there is always a straight line which is neither
parallel nor perpendicular to any join of two points. Let this
straight line be e. Now we regard the configuration AtA2A3A4 . . .
as convex, if the gradients of the lines AtA2, A2A3, . . . decrease
monotonously, and as concave if they increase monotonously. Let
f2(i, fc) denote the minimum number of the points such that from
them we may pick out either i sided convex or fc-sided concave
configurations. We assert that
f2(i, fc) =/,,(1-1, fc) +Mi, k-i) - 1. (10)
We consider the first f2(i — 1, fc) points. If out of them there
can be taken a concave configuration of fc points then the equation
for f2(i, k) is fulfilled. If not, then there is a convex configuration
of (i — 1) points. The last point of this convex configuration we
replace by another point. Then wc have once more either fc
concave points and then the assertion holds, or (i — 1) convex
ones. We go on replacing the last point, until we have made
use of all points. Thus we obtain /2(i, fc —1) points, each of
which is an end-point of a convex configuration of (i — 1)
11 AN EARLY GEM

470
P. Erdos and G. Szekeres.
[8]
elements. Among them, there are either i convex points and then
our assertion is proved, or (k — 1) concave ones. Let the first
of them be A-^, the second A2. A-^ is the end-point of a convex
configuration of (i— 1) points. Let the neighbour of Ax in this
configuration be B. If the gradient of BAX is greater than that of
AXA2, then A2 together with the (i— 1) points form a convex
configuration; if the gradient is smaller, then B together with
AXA2 . . . form a concave fc-configuration. This proves our
assertion.
The deduction of the recurrence formula may start from the
statement: /2(3, n) = f2(n, 3) = n (by definition). Thus we
easily obtain
/2(^) = (^2) + ^ (11)
As before we may easily prove that the limit given by (11)
is exact, i. c. it is possible to give ( I points such, that they
contain neither convex nor concave k points.
Since by connection of the first and last points, every set of k
convex or concave points determines a convex A;-gon it is evident
that ( I -(- 1 points always contain a convex k-gon.
And as in every convex (2k — 1) polygon there is always
either a convex or a concave configuration of k points, it is evident
that it is possible to give I . I points, so that out of them
no convex (2k ~ 1) polygon can be selected. Thus the limit is
also estimated from below.
Professor D. Konig's lemma 5) of infinity also gives a proof
of the theorem that if k is a definite number and n sufficiently
great, the n points always contain a convex fc-gon. But we thus
obtain a pure existence-proof, which allows no estimation of
the number n. The proof depends on the statement that if M
is an infinite set of points we may select out of it another convex
infinite set of points.
(Received December 7th, 1934.)
E) D. Konig, Ober eine SchluBweise aus dem Endlichen ins Unendliehe
[Acta Szeged 3 (1927), 121—130].
12 PAPERS OF SPECIAL INTEREST

Chapter 2
Problems
These papers contain only a sample of the conjectures of Paul Erdos. A strong
factor in the weighing of his mathematical influence would be measuring the
work of other mathematicians on his conjectures. It is remarkable how
consistently difficult, but not impossible, the problems are.
Erdos induces work on these problems by offering cash prizes for the
solutions to some of them. He works on them himself first and offers money if
he has no solution and so those with attached prizes are indeed difficult. The
proposed prizes range from 10 to 300 dollars—but very few people have
actually won over 50 dollars. The inducement is not financial (C. C. Chang, after
becoming the only person to win 250 dollars, noted that the payment came
out to about 50 cents per hour) but rather prestigious. A lucky few
mathematicians have in their possession a bill or check that was their reward for a
solution (causing Erdos check book balancing problems).
Papers in Chapter 2
[346] Some unsolved problems (partial paper)
[419] Some applications of probability to graph theory and combinatorial
problems
[548] Some unsolved problems in graph theory and combinatorial analysis
[575] Problems and results in combinatorial analysis
13 PROBLEMS

UNSOLVED PROBLEMS
HEPA3PEUIEHHbIE FIPOEJIEMbl
We start with this number a new
section of this journal: that of unsolved
problems. In this section unsolved
problems will be proposed, at the same time
some information about previous results
in the direction of the problem in question
will be given. Problems for this section
as well as comments on published problems
should be sent to 0. Alexits, editor of
the section, to the address of the
redaction of the journal (Budapest, V. Tteal-
tanoda u. 13- 15.).
HaMimafl c HacTOHinero Hoiwepa noMe-
maeTCH HOBbiH pa3flejr b Hamesi >KypHajie:
Hepa3pemeHHbie npoGjieivibi. B stom pa3flejie
Mbi nyGJiHKyeM Hepa3pemeHHbie npoGjie.wbi
h oflHOBpeiweHHO yKa3biBae.M Ha Ccmee paH-
Hbie pe3yjTbTaTbi, CBH3aHHbie c flaHHoii npo-
CjreMOH. ripoGJieMbi, npeflHa3HaMeHHbie jinn
SToro pa3flejra, a TaK>Ke 3aMeMaHHH, CBH3aH-
Hbie c coo6maeMbiMH npoCjiewaMH npociiM
HanpaBHTb no aapecy pe^anumi >KypHajia
(Budapest, V. llealtanoda utea 13—15.)
fl.ifl peaaKTopa pa3flejia G. Alexits.
SOME UNSOLVED PROBLEMS
by
Paul ERDOS
II. Problems in combinatorial analysis and set theory
1) Let rij, a2, . . . , ak he n elements. Alt A2 An are k sets formed
from the a's so that no A can contain any other. Sperner proved that
(II. 1.1)
max k —
(II. 1.1) has several applications in number theory, e. g. Behrend's result
(1.25.1) is proved by using (II.1.1).
The question has been considered that in how many ways can one
.select sets A{ so that no A should contain any other. Denote this number
by A(n). From (II.1.1) we have
M~
A(n)
2"
T
1 /i
where T„
perhaps
c can he chosen to he
It seems that A(n) < exp (cTn),
(1 + e) log 2 for every e > 0 if n > n
How many sets Ax, A2, . . . , At can one give so that the union of two
of them never equals a third? (all three sets are supposed to be distinct i. e.
At C Aj, At\jAj = Aj is not permitted). I conjectured for a long time that
I = r;(2"). If I could prove this the following result in number theory woulrl
follow. Let ax < a„ < . . . be an infinite sequence of positive density, then
a.
a
a,,
satisfying
there are infinitely many triplets of distinct integers
\at, a ■] = ak (see problem 1 26).
It is possible that I < (1 + o(l)) Tn.
Several other problems can be asked e. g. How many sets can one give
so that the union of any two of them never contains a third? How many seis
15 PROBLEMS

240
eudGs
At can one give so that the symmetric difference of any two sets should
contain at least r elements ?
E. Sperner: »Ein Sat/, ubev Untermengen einer endlichen Menee.« Math. Zeit-
scltrift 27 (1928) 544-548.
2) As far as I know R. Peltesohn and Sutherland (unpublished)
were the first to construct an infinite sequence formed from the symbols
(J, 1, 2 where no two consecutive blocks were identical. It is easy to.see that
in a sequence of length four formed from the symbols 0 and 1 two consecutive
blocks will 1)0 identical, I understand that Euwe proved that in an infinite
sequence formed from 0 and 1 there will be arbitrarily large identical
consecutive blocks, but tha+ there do not have to be three consecutive identical
blocks.
Let us now call two consecutive blocks ,.identical" if each symbol
occurs the same number of times in both of them (i.e. we disregard order).
I conjectured that in a sequence of length l2k— 1 formed from k symbols
there must be two "identical" blocks. This is true for k ^ 3, but for k = 4
de Bbxjijn and I disproved it and perhaps an infinite sequence of four symbols
can be formed without consecutive "identical" blocks.
3) Littlewood and Offobd proved the following result: LetZ,,
1 i i $ n be « complex numbers. Then there exists an absolute constant
c so that the number of sums
n
(II. 3.1) ^etZ„ et=±l
/=1
which fall into the interior of an arbitrary eirelo of radius 1 is less than
2" log n
c — . I proved that if Zt ^ 1, 1 ^. i ±L n (i. e. the Z, 's are real) then
the number of sums (II.3.1) which fall into the interior of any interval of
length two is at most / _,\ and this estimation is best possible. The proof
uses the theorem oi Sperneb (see problem 11.1.). I do not know if this inequality
remains true if the Z, are complex numbers (my proof gives for complex z
c2"/yn), or more generally vectors of Hilbert space of norm Ig 1. In this case
I can only prove that the number of summands (II.3.1) falling into an
arbitrary unit sphere is o(2").
J. E. Littlewood and C. Offord, Mat. Sboriiik. 12 (1943) 277 — 285.
P. Erdos: "On a Lemma of Littlewood and Offord." Bull. Amer. Math. Hoc.
31 (1945) 898-902.
4) Ramsay proved that there exists a function /(;', k, I) so that if we
split the ^'-tuples of a set of f(i, lc, I) elements into two classes then either
there are k elements all whose i-tuplets are in the first class or I elements
al whose /-tuples are in the second class. Szekeres and I proved that
(2k — 21
k- 1
The best estimation for f{i, k, k), i > 2 is due to Rado and myself.
(II. 4.1) 2*2</(2,M)^:
^ = (^
16 PAPERS OF SPECIAL INTEREST

SOME UNSOLVED PROBLEMS
241
It would be interesting to determine f(i, 1, I) explicitely, this seems
very difficult even for i" = 2-1 have not even be able to prove that lim/(2, k, 1)V*
exists. I can prove that , = co
(II. 4.2) /(2, 3, 1-) > cl2/(log I)2
but could not decide whether /(2, 3, 1) > c2k2 is true.
I do not wish to mention here the many problems connected with the
generalisations of Ramsay's theorem to cardinal and ordinal numbers and
just state one of the simplest unsolved problems in this subject.
Let q> be t. well ordered set of ordinal number u>a, a. < Q. Split the
pairs a ¢99, b £<p into two classes so that there is no triplet all whose pairs are
in the first clfcss. Does there then exist a set <p' C q> of type u>a all whose
pairs are in the second class I
For a = 2 this- was proved by Specker, for 2 < « < to it was
disproved by him, for a ^ w the problem is open. The most interesting unsolved
case is a = a>.
E. P. Ramsay: '-On a problem of formal logic." Collected papers, 82—111. Sec
also T. H. Skolem: »Ein kombinatorischer Satz mit Anwendung auf ein logisches Ent-
seheidungsproblem.« Fund. Math. 20 (1933) 254 — 261.
P. Erdos and G. Szekeres: "A Combinatorial Problem in geometry." Compositio
Math. 2 (1935) 463-470.
P. Erdos: "Remarks on a theorem of Ramsay." Bull. Res. Council. Israel (1957)
21 — 24. See also "Graph theory and probability."Cod. Journal of Math. I and II, 11 (1959)
34-38,13(1961) 346—352.
E. Specker: »Teilmengen von Mengen mit Relationen.« Comm. Math. Helv.
31 (1956-57) 302-314.
P. Erdos and R. Rado: „A partition calculus in set theory."' Bull. Amer. Math.
«S'oc. 62 (1956) 427-489. (See also the forthcoming triple paper of Erdos—Hajnal—Rado.)
5) Let alt a2, . . . , an be n elements A1} A2, ..., Ak. k > 1 sets whose
elements are the a's. Assume that each pair (at, a,j) is contained in one and
only one A. Then k>n. This is a result of de Bhuijn and myself (also proved
by Szekebes and Hanani). We can not determine the smallest I so that
there should exist sets Alt A2, ..., At, / > 1 so that every triplet (ah ap ar)
is contained in one and only one A.
N. G. de Brui.jn and P. Erdos: "On a combinatorial problem." Ind, Math.
(1948) 421-423.
C. Steiner conjectured that if n = 61 -\- 1 or 61 + 3 there exists a
system of triplets of n elements so that every pair is contained in one. and
only one triplet (if n is not of the above form it is easy to see that such a
system can not exist). Steiner's conjecture was first proved by Reiss and
later independently by Moobe.
Let now 2 ^ r < s be any two integers. For which n is there a system
of combinations taken s at a time formed from n elements so that all r tuples
should be contained in one and only one s tiuple. The case r — 2, s = 3 is
Steiner's. The only other case which has been settled is r = 3, s = 4
H. Hanani recently proved that such a system exists if and only if n = 2
or 4(mod 6). (Very recently IJanani settled the cases r = 2, s = 4 and
r = 2, s = 5).
It has been known for a long time that if n = p21 -)- pl -)- 1 (p prime).
r — 2, s = pl -)- 1, then there exist n (pl + 1) — tuplets so that every pair
is contained in one and only one (pl -)- l)-tuplet. If n = I2 + 1 + 1, 1 =f= p"
16 A Matematikai Kutato Intezet Kiizlemenyei VI. \—2.
17 PROBLEMS

24 2 ERDOS
it has been conjectured that .such a .system of (k + l)-tuplets doe.s not exist.
Special eases of this conjecture have been proved by Bkuck and Uyser, the
first unsettled case is k = 10 (see 1.10).
Connected with this problem is the following conjecture of Sylvester:
For every n ~ 0 (mod 4) there exists an orthogonal matrix of order n all
whose elements are + 1 (it is easy to see that if n ^k 0 (mod 4) such a matrix
doe.s not exist.) If n = 2k Sylvester showed that such a matrix exists,
if p == — 1 (mod 4). Paley proved that such a matrix exists for p + 1, the
general ease is still unsolved.
Denote by Mn the maximum value of an n by n determinant whose
elements are + 1. From IIabamabd's theorem it follows that Mn < iin2
and if a Sylvester matrix exists Mn = nn 2. It follows easily from the prime
number theorem for arithmetic progressions that for every n > n0(e)
(H.6.1) Mn > (1 —e)" re" 2.
Coltcci and 13 abb a proved that if n =t 0 (mod 4) then
o H
--1 n"2
e I
(11.6.2.) il/„< (2n- l)'i(n-l)("-0/2= (1+0(1))
M. Reiss: »Uber eine Steinersche kombinatorischc Aufgabo.« J. reiue mid an-
jsu-andte. Math. 56 (1859) 32G-344.
E. H. Moore-. "Concerning triple systems." Math, Annalen 43 (1893) 271 — 285.
See also ,,Practical memoranda." Amur. J. Math, 18 (1896) 264 — 303.
II. HanAtsti: "On quadruple systems." Can. J, Math. 12 (19G0) 145—157.
J. II. Sylvester: "Thoughts on inverse orthogonal matrices." Phil. Mag. (4)
24 (18()7) 461-475.
R. E. A. C. Paley: "On orthogonal matrices." Journal of Math, and Phys (1933)
311-320.
UjIjUCCi: «Sui valori massimi clei rleterminanti ad elementi ± 1.» Gior. di Matem.
di Battaglini 54. See also G. Barb A ibid. 71.
See also G. Szekeres and P. Turax: ,,An extremal problem in the theory of
determinants." (In Hungarian, German summary) Sitzungsber. III. Klasse Ui\g. Akad,
54 (1937) 796-806.
7) Problem of Van der Waerden: Let \alk\ be an n by n doubly
stochastic matrix i. e. aik >0 and 2aik=- -— aik = 1 f°r every i and k .
i'^i ' fc-i '
«! 1
Then the value of the permanent is ^ — , equahty only for at,k = — . The
n" n
permanent (a terminology of Sylvester) is the sum of the expansion terms
of the determinant. The fact that the permanent of a doubly stochastic
matrix can not be 0 is a theorem of Fbobenius—Konig. Van der Waerden's
problem seems to be difficult.
I made the following two weaker conjectures: The value of at least
one term of the permanent is >. — , and the still weaker one: There is at
least one non-zero expansion term of the, permanent wdiere the sum the
factors is ^ 1. This was proved by Pi. Ree and S. Marcus in fact they prove
that the sum is ^ -- ^ af}k
18 PAPERS OF SPECIAL INTEREST

S0J1K UNSOLVED l'UOKLLMS 243
R, Ree and S, Marcus: ,,Diagonals of doublv stochastic matrices." Quarterly
Journal of Math. 10 (1959) 296-301.
8) A special ease of a theorem of TitbAn states that if in a graph of n
vertices the number of edges is greater than
+ 1
then the
graph always contains a triangle. He points out that the following analogous
problem is unsolved: Let there be given n elements what is the smallest
number /(h) so that to every system q? of f{n) triplets formed from the n
elements there are always four elements all four triplets of which occur in (p.
P. TurAn: "On the theory of graphs." Coll. Math. 3 (1954) 19 — 30.
D. Konig: Theorie der endlichen und unendlichen Orapheri.
9) Hajnal and I proved the following theorem: To every real x make
correspond a bounded set of real numbers f(x) whose outer measure is less
than 1. Then for every finite k there exists an independent set of k elements
{i. e. a set xx, x2. . . . , xk so that for every 1 ^. i, j ii k, i =f= /, .r, ¢/(-^-)).
We can not prove that there always exists an infinite independent set (not
even if we also assume that the sets f(x) are compact.)
If we assume that the sets f(x) are closed and of measure < 1, we can
not even prove that there are two independent points. (Recently Gladysz
proved in a very ingenious way the existence of two independent points.
The existence of an independent triplet is open).
P. Erdos and A. Hajnal: ,,Some remarks on set theory, VIII." Michigan
Math Journal 7 (1960) 187—191, for further problems inthis directionsee P. ERDosand
A. Hajnal: "On the structure of set mappings. Acta Math. Hung. 9 (1958) 111 — 131.
and P. Krd6s: "Some remarks on set theory." 3 (1953) 51—57.
III. Problems in elementary geometry
1) Let j^, x2 xn be n points in the plane. Denote by Mn(x1..i:,. . . . , x„)
the number of distinct distances between any two of the points. Put
/(«) = min M^x^ x2, . . . , xn).
where xlt ,r2, ..., xn ranges over all sets of n distinct points of the plane.
It seems to be difficult to get a good estimate for f{n), the best results (due
to Moseb and myself are)
(III. 1.) qn3 3 < /(n) < c2n/y\og n.
I would guess that the upper bound is the right one and perhaps even
the following result holds: There is one point x( so that amongst the distances
(Xj, xt) there are at least c3 H/]/log n distinct ones.
If the net xlt x2. . . . , xn is convex it seems that f(n) = — ; despite
its seeming simplicity I have not been able to prove this. A somewhat stronger
conjecture is: In every convex polygon there is a vertex which has no three
vertices equidistant from it.
How often can the same distance occur between n points of the
plane? Denote this maximum by g{n). 1 proved
„L+c./loglog„ <: g(„) < n3z,
I believe that the lower bound is close to being the correct one.
16*
19 PROBLEMS

244
ERDrtS
Coxeteb asked me how many points does one, have, to have in
/(-dimensional space so that one should be sure to have more than two distinct distances
between them. I stated that for c5 sufficiently large nc- points suffice, but.
my proof was wrong and if corrected it only gave exp (w1-6).
One can show that from 7 points in the plane one can always find three
of them which do not determine an isosceles triangle, it is easy to see that
this is false for 6 points. How many points does one have to have in n-dimen-
sional space to be sure that one can find three of them which do not determine
an isosceles triangle? This is not even known for n = 3.
P. Erdos: "On sets of distances of n points.'" Amer, Math, Monthly, 54 (1946)
248-250.
L. Moser: "On the different distances determined by n points." Ibid. 59 (1952)
85-91.
P. EkdOs: "On some, problems in geometry." (In Hungarian) Mat, Lapok (1954)
8()-92. Many further problems are stated in this paper.
2) Blumenthal's problem. Let there be given n points in the plane,
denote by A{xlt x2, ..., ,r.,) the largest angle (<- n) determined by the, n
points, and define
an = inf A{x-i, x2, ..., xn)
where the minimum is taken over all sets of n points. Szekebes proved
that ot2»+i > n 1 + - and that for every * > 0,2" points ean be
given with -4(^, x2,
n(2n + 1)2
, X.2n) > 71
1-1
this implies ot2.. < n 1
1
n
Szekebes and I recently proved that a2« = n
and in fact for
every 2" points A(x^ x2,
r9» > rr
we also showed a2»—i
71 1
l\
n
Let there be, given
r>n
1 points in n dimensional space. 1 conjectured!
that there are always three of them which determine an angle > --. This
is trivial for n = 2, for n = 3 it was proved by (unpublished) N. H. Kaiper
and A. H. Boebdijk. For n > 3 the problem is open. (Recently this
conjecture was proved by L. Danzeb and G. Gbunbaum in a simple and
ingenious way.)
G. Szekeres: "On an extremum problem in the plane." Amer. Journal of Math.
53 (1941) 208 — 210. Our paper with Szekeres will appear in the Annates of the Univ.
of Budapest 3 (1961).
3) Bobsuk's problem. Is it true that every set of diameter one in n
dimensional space is the union of n -j- 1 sets of diameter < \1 This is trivial
for n = 1, easy for n = 2. For re = 3 it was first proved by Eggleston and
later simultaneously and independently Gbunbaum and Heppes found a
considerably simpler proof. The problem is open for n > 3.
Bobsuk and Ulam proved that the n dimensional sphere is not the
union of n sets of smaller diameter.
20 PAPERS OF SPECIAL INTEREST

SOME TXSOLVED PROBLEMS 245
K, Borsuk: oDrei Satze fiber die, »-dimensionale euklidische Sphare.« Fundamenta
Math. 20 (1933) 177-190.
II. G. Eggleston; "Covering a three dimensional set with sets of smaller
diameter." Journal of Loud, Math. Soc. 30 (1955) 11 — 24.
B. Grunbaum: ,,A simple proof of Borsuk's conjecture in three dimensions."
Proc. of Cambridge Phil. Soc. 53 (1957) 77G-778.
A. Heppes — P. Revesz: >>Zum Bovsnksehen Zerteilungsproblem." Acta Math.
Acad. Set. Hung. 7 (1956) 159-162.
A. Heppes: ,,Terbeli ponthalmazok felosztasa kisebb arme.roju reszhalmazok
osszegere." MTA III. Oxzt. Kozl. 7 (1957) 413-416.
II. Hadwigef: »Uber die Zerstiickung eines Eikorpers.« Math. Zeitschr. 51
(1949) 161-165.
H. Lenz: "Zur Zerlegung von Punktmengen in solche kleineren Durchmessers.c
Arch. Math. 6 (1955) 413-416.
4) Sylvester conjectured and Gallat first proved that if we have
n points, not all on a line then there is at least one line which goes through
exactly two of the points. Denote by Gn the minimum number of such lines,
de Bbuijn and 1 conjectured that Gn -> oo as n ~> oo. This was proved
by Motzkin (his paper contains many more problems and results in this
direction). Moseb and Kelly proved that Gn 2: and this is best pos-
7 J
sible for n = 7. For n > n0 perhaps Gn = n — 1. For large n perhaps there
always is a triangle all whose lines goes through only two of our points (except
if n—1 of them are on a line).
Sylvester asked: Let there be given ;; points no four on a line. What
is the maximum number of lines which goes through three of them? He proved
1
that this maximum is greater than
3
1
en on the other hand the maximum
is <
3
n
2
Let there be given n points not all on a line I observed that it easily
follows from Gallat's result that these points determine at least n lines
(see also 11.5). G. Dibac conjectured that there always exists a point which
is connected with the other points by more than an lines.
Let there be given n points not all on a circle. What is the minimum
number of circles these points determine? This problem is unsolved (see also
II.5).
T. H. Motzkin; "The lines and planes connecting the points of a finite set."
Trans, Amer. Math. Soc. 70 (1951) 451 — 464. This paper contains many more problems
and results and also the history of this problem and many references to the literature.
5) Miss Kleik asked: Does there exist for every n an f(n) so that if
f(n) points in the plane are given no three on a line then there always exist
n of them which are the vertices of a convex polygon. She proved /(4) = 5
and Makai and Ttjban proved that /(5) = 9. Szekebes conjectured /(«) =
= 2n"2 + 1. He and I proved
(•2n 4)
(III. 5.1.) 2""2^/(«) <
n
•7
21 PROBLEMS

246
erdOs
P. Ekdos and G. Szekeres : "A combinatorial problem in geometry", Compositio
Math, 2 (1935) 463—470. The proof of the lower bound in (III. 5. 1) will appear in
the Annates of the Univ. of Budapest.
6) Heilbro^jn's problem. Let there be. given n points in the unit
circle. Denote by A(xlt x2, ..., xn) the smallest area of all the triangles
determined by the x{. Estimate max A(-xlt ..., xn) where the maximum is taken
over all the xt in the unit circle. An < cJn is trivial. Roth proved
A„ = o\— , more precisely
" \n\ r J
An < c2/7i(loglog«)'=,
and I observed that An > cJn2. It seems to be a difficult and interesting-
problem to improve these inequalities for An.
K. F. JIoth: "On a problem of Heilbronn." London Hath. Soc. Journal 26 (1951
198-209.
7) Recently it was asked if the piano can be split into four sets (p,, 1 < (^4
so that no r/, should contain two points whose distance is 1. Several
mathematicians observed that this certainly can not bo done with three sets.
(L can not trace, the origin of this problem.)
8) Axning and I proved that if in ar>infinite sot of points in the plane
all tho distances between the points are integers then the points ah are, on
a line. On tho other hand it is known that one can give an infinite sot of points,
not all on a line so that all the distances should bo rational. Ulam asked'.
Does there exist a set <p dense in the plane so that all the distances between
points of (f are rational? I think the answer is no, but the question seems
very difficult. Schoenberg asked if to every polygon and every e there
exists a polygon whose vertices are at distance < f from the corresponding
vertices of the original polygon and all whose sides and diagonals have rational
length. Clearly if Ulam's problem has an affirmative answer, then the same
holds for Schoexbebg's problem. Besicovitch dealt with some special
cases of this problem.
P. Eedos and A. Anting: "Integral distances." Bull. Amer. Math. >Soc. (1945)
598-600 and 996.
A. S. Besicovitch: "-Rational polygons." Matheinatiha 6 (1959). 98.
Further literature on these similar problems and results: L. Fejes Toth: Lagerun -
gen in der Ebeue. auf der Kugel und in\ Rawu. Berlin, 1953 and Ff. Hadwioer and II.
Debruxner: »Kombinatorisehe Geometric in der Ebono.« L'Enseignement Math- 1
(1955) 56 — 67. The paper also appeared in French a more detailed version of this paper
recently appeared in book form ^tonographies de E'Enseignement Math^matiquc No -■
See also a forthcoming book of Hadwiger on these subjects.
22 PAPERS OF SPECIAL INTEREST

SOME APPLICATIONS OF PROBABILITY TO GRAPH THEORY AND
COMBINATORIAL PROBLEMS
P. ERDOS
In the first chapter of my lecture I will discuss applications of probabilistic methods
to Ramsay's theorem, next I will speak on problems of chromatic graphs and
finally I will briefly mention some other problems. &n) will denote a graph of n
vertices; a graph is called complete if all its vertices are adjacent. &n) will denote the
complementary graph of ©(n) (i.e. two vertices in @(n) are adjacent if and only if
they are not adjacent in &n)). </c> will denote a complete graph of k vertices.
1. Denote by /(k, I) the smallest integer such that for n = f(k, I) either @(n)
contains a </c> or &n) an </>. It is known that [1]
By probabilistic methods [2], [3] I showed that
(2) f(k, k) > 2kl2
and that
(3) f{k, 3) > c/c2/(log fe)2 .
It is very likely that my method will show that for every I > 3 and e > 0, if k >
> k0(l, e) then
f(k,l)>k'-1-c,
but I have not worked out the formidable details. It would be very desirable to decide
whether/(/c, 3) > ck2 holds and to determine the limit of f(k, k)i/k. I cannot even
prove the existence of this limit. It would also be of interest to prove (2) and (3) by
a constructive method. I only succeeded to show by such methods that /(k, 3) >
> k1+c for a certain c > 0, [4].
2. Zykov and Tutte [5] were the first to construct, for every k, a graph which
contains no triangle (i.e. no <3>) and whose chromatic number is k. J. B. Kelly
and L. M. Kelly [6] showed that such graphs exist which contain no circuits of
length ^ 5 and they asked if there exist, for every k and I, /c-chromatic gTaphs which
133
23 PROBLEMS

contain no polygons of length f§ I. By probabilistic methods I proved that such
graphs exist [7] (it would be very desirable to give an explicit construction). In
fact I showed that there exists an eM > 0 so that for every n there is a ©(B) which
contains no circuit of length ^ I and for which @(B) does not contain an <[«'"■']>
(the chromatic number of our ©(B) is then clearly > n1 ~*"■').
More generally let S„ be a set of n elements and let At c= Sn be subsets of S„.
We say that the system {Aj} is /c-chromatic if the set S„ can be split up into k disjoint
subsets Sj, 1 S j S k so that no At is contained in an Sj and that such a splitting is
impossible into fewer than k subsets. Hajnal and I now showed that for every k, r
and I there exists an n0 = n0(/c, r, I) so that for every n > n0(/c, r, I) there exists
a /c-chromatic system {/1;}, As <= S„ where each A, has r elements and for u ^ I the
union of any u A's contains at least 1 + u(r — 1) elements (for r = 2 this condition
means that our graph contains no circuit of length f§ I). We use probabilistic methods;
our proof has not yet been published.
A well known theorem of Brooks [8] states that every graph which does not contain
a </c + 1> and every vertex of which has valency ^ /c is at most /c-chromatic. Grun-
baum constructed a 4-chromatic graph which does not contain a triangle and a
quadrilateral and every vertex of which has valency 4 (Griinbaum's construction has
not been published yet). Griinbaum now asked: Does there exist, for every k and I,
a /c-chromatic graph every vertex of which has valency k and which does not contain
a circuit of length ^ P. I am very far from being able to solve this difficult question
and can only show by probabilistic methods that there exists an absolute constant c
so that there exists a graph of chromatic number > c/c/log k every vertex of which has
valency k and the graph contains no circuit of length g I. I very much doubt whether
Griinbaum's problem can be settled by probabilistic methods.
Denote by f(m, k, n) the maximum of the chromatic numbers of all graphs ©{n)
every subgraph of m vertices of which has chromatic number ^ k. By probabilistic
methods I proved [9] that
(4) /(m,3, n)>c(-X'3 /log— .
\mj j 2m
(4) in particular implies that to every k there is an e > 0 so that if n > nQ(e, k)
then there exists a /c-chromatic @(n) every subgraph of which having [en] vertices
is at most 3-chromatic. The situation is radically different if we consider f(m, 2, n).
Gallai proved (his proof will appear in Publ. Math. Inst. Hung. Acad. Sci.) that
/([n1/2], 2, n) = 4,mother words there exists a four-chromatic @(n) every odd circuit
of which has length > n1/2. Gallai and I conjectured that the largest value of m for
which/(m, 2, n) = k is of the order of magnitude nllk+2, but we have not even been
able to prove that for every e > 0 and n > n0(e), /([en], 2, n) = 3.
3. A theorem of Redei [10] states that the directed complete graph of n vertices
always has an open Hamiltonian path. Szele denotes by T„ the maximum number of
134
24 PAPERS OF SPECIAL INTEREST

open Hamiltonian paths for all possible orientations of <n>. Szele [11] proves
by elementary probabilistic methods (calculation of the first moment) that
As far as I know this is the first use of probabilistic methods for a combinatorial
problem. Szele's paper unfortunately has been almost completely unknown. The
same method was rediscovered and used in several other papers [12]. It would be
very desirable to construct a directed <n> which has at least n!/2"_1 open Hamilton
iines. Szele proved that lim (T^,,,)1'" exists, he conjectures that this limit is \.
n~ oo
Schutte asked the following question: Determine the smallest integer f(k) for
which there exists a directed </(fc)> so that for every k vertices of our </(fc)> there
is another vertex of it from which edges go out to each of the k vertices. Trivially
/(1) = 3, Schutte observed/(2) = 7, I proved [13]
(6) 2k + 1 - 1 ^f(k) < ck22k
where the upper bound is obtained by probabilistic arguments; perhaps f(k) =
= 2k+1 - 1 always holds.
In connection with some work on set theory Hajnal and I raised the following
question [14]: What is the smallest integer m(p) for which there exists a family of
finite sets /1,, ..., /lm(p) each having p elements and such that every set S which has
a non-empty intersection with each A,, 1 5=/5= m(p) contains at least one of the A?.
We observed m(p) 5= ( ). By probabilistic methods I showed [15] m(p) >
\ P J
> 2" 1, and for p > pQ(e), m(p) > 2" (log 2 — e). I cannot even prove that
Ip
(7) lim m(p)1
P~ 00
exists. It is easy to see that m(2) = 3, m(3) = 7, m(4) is unknown.
Added in proof: Schiitte's problem was also raised by "Ryser and (6) was proved
by Moser independently. Moser and I will publish a paper on this subject in the
Bull. Canad. Math. Soc.
Recently I learned that a result substantially equivalent to (6) was proved by
Gleason, but he published nothing about this subject.
The fact that the limit in (7) exists was proved by Abbot and Moser (will appear
in Bull. Canad. Math. Soc). Later I proved that the limit is 2. In fact
(8) 2" U + 0 f-\\ < m(p) < cp22" .
The lower bound in (8) is due to W. Schmidt, the upper bound to me. Our papers
will appear in Acta Math. Hung. Acad.
135
25 PROBLEMS

References
[1] P. Erdos and G. Szekeres: A combinatorial problem in geometry. Comp. Math. 2 (1935)
463—470; for a slightly sharper result see C. Frasnay: Sur des fonctions d'entiers se rappor-
tant au theoreme de Ramsay. C.R. Acad. Sci. Francaise 256 (1963) 2507-2510.
[2] P. Erdos: Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53 (1947) 292—299.
[3] P. Erdos: Graph theory and probability II. Canad. J. Math. 13 (1961) 346—352.
[4] P. Erdos: Remarks on a theorem of Ramsay. Bull. Res. Council Israel 7 (1957) 21—24.
[5] A. A. Zykov: Mat. sbornik 24/66 (1949) 163—188.
[6] Blanche Descartes: A three colour problem. Eureka (April 1947, Solution March 1948);
see also Solution to Advanced Problem 4526, Amer. Math. Monthly 61 (1954) 352. J. B. Kelly
and L. M. Kelly: Path and circuits in critical graphs. Amer. J. Math. 76 (1954) 786—792.
[7] P. Erdos: Graph theory and probability I. Canad. J. Math. 11 (1959) 34—38.
[8] R. L. Brooks: On colouring the nodes of a network. Proc. Cambridge Phi!. Soc. 37 (1941)
194—197.
[9] P. Erdos: On circuits and subgraphs of chromatic graphs. Matematika 9 (1962) 170—175.
[10] L. Redei: Ein kombinatorischer Satz. Acta Szeged 7 (1934) 39—43.
[11] T. Szele: Combinatorial investigations about the directed complete graph (written in
Hungarian). Mat. es Fiz. Lapok 50 (1943) 223—254, see p. 242—243.
[12] P. Erdos: On the number of complete subgraphs contained in certain graphs. Publ. Math.
Hung. Acad. Sci. 7 (1962) 459—464; sec also a paper of Erdos and Moon which will appear
in Canad. Bull. Math. — L. Moser (unpublished) estimated by this method the maximum
number of cycles in a directed complete graph. The papers quoted in [2] and [15] also use
this method.
[13] P. Erdos: On a problem in graph theory. Math. Gazette 47 (1963).
[14] P. Erdos and A. Hajnal: On a property of families of sets. Acta Math. Acad. Sci. Hung. 12
(1961) 87—123; see in particular problem 12 on p. 119.
[15] P. Erdos: On a combinatorial problem. Nordisk Math. Tidskrift 11 (1963) 5—10.
26 PAPERS OF SPECIAL INTEREST

Some Unsolved Problems in Graph Theory and
Combinatorial Analysis
P. Erdos
Hungarian Academy of Sciences, Budapest, Hungary
Jn the present note I discuss some unsolved problems in graph theory and
combinatorial analysis which I have thought about in the recent past. I hope
that at least a good proportion of them are new.
First I introduce some notation. G(n;k) will denote a graph of n vertices
and k edges. Cr denotes a circuit having r vertices, K, denotes the complete
graph of I vertices. A graph is said to have girth k if it contains a Ck but no C,
for I < k. A clique is a maximal complete subgraph (i.e. it is a complete
subgraph which is not properly contained in any larger complete subgraph).
Kr(Pi, ---,/>,■) denotes the complete /"-chromatic graph where there are p:
vertices of the i'th colour and any two vertices of different colour are joined
by an edge. By an /--graph G(r) we shall mean a graph whose basic elements
are its vertices and /--tuples; for /-=2 we obtain the ordinary graphs.
G(r)(/7;£) will denote an /--graph of n vertices and k /--tuples. A set of/--tuples
is called independent if no two of them have a vertex in common, c, c1, c2, ...,
will denote position absolute constants.
1. In the colloquium on graph theory at Tihany, Bollobas and I stated the
following problem: is it true that every graph of n edges contains a subgraph
of at least en31* edges which has no rectangle? Folkman (in a letter) gave the
following counter-example. Let the vertices of our G be xu ..., xm; y1, ..., ym2,
every x is joined to every y. This graph has m3 edges and it is easy to see that
every subgraph having m2 + I ' J + 1 edges contains a rectangle (this
statement is false for m2 + I - I edges)
Perhaps our conjecture is true with en213 instead of en31*, but I cannot even
prove it with cn* + e (en* is trivial).f
2. Is it true that every graph of 5« vertices which contains no triangle can
t Note added in proof: Szemeredi proved cn2,i.
97
27 PROBLEMS

98 P. ERDOS
be made two-chromatic by the omission of at most n2 edges? It is easy to see
that if this is true it is best possible.
3. I proved that if k < en then every G(n; [i«2] + k) contains k edge-
disjoint triangles. The proof uses the following theorem of Gallai and myself:
every G(n;[^(n — 1)2] + 2) which has chromatic number 3 contains a
triangle.
I first thought that the theorem might hold for very much larger values of k,
but Sauer showed by a simple example that this is not so. Let the vertices of
G be xu ..., x„;y1, ...,y„;z1, ...,z4. Every x is joined to every y and z, every y
is joined to every z, and any two z's are also joined. This is a G(2n + 4;
(n + 1)2 + An + 2) or k = An + 2, and it is not difficult to prove that G
contains only An + 1 edge disjoint triangles.
It would be interesting to determine the largest value of k for which our
result holds; our proof only gives small values of k < cn(c < i). Perhaps the
following result holds: to every ct there is an /(Cj) so that every
G(n; [^n2] + k), k < c1 n contains at least k — /(cj edge disjoint triangles.
In view of my theorem with Gallai the following question could be asked:
what is the smallest integer ur so that every G(n;ur) which has chromatic
number ^ r, contains a triangle? u2 = [4/12] + 1 (this is the well known
theorem of Turan) and u3 = [j(n — 1)2] + 2. w4 is unknown [5].|
4. We proved that every Gl n; I J + 2 I is Hamiltonian and that this
is best possible. I showed that for n > na(k) every
i-C^HVH
contains a C„_fc. My proof is not quite trivial. The result is easily seen to be
the best possible. It would be interesting to determine or estimate n0(k) [6].
5. Renyi and I considered the following problem. Determine or estimate
the smallest f(n) for which all but 0(((2)11 of the graphs G(n;f(n)) on n
labelled vertices are Hamiltonian. This question seems to be difficult. Recently
Moon and Moser and I. Palasti proved f(n) < en212, but it seems certain that
/(«) <n1+£ [24].
6. Denote by g(3, n) the smallest integer for which there is a graph of
g(3, n) vertices which contains no triangle and has chromatic number n.
It is known that
c, n2 log n/log log n < g(3, ri) < c2 n2(log n)2. (1)
It would be desirable to improve (1) and to give an asymptotic formula for
t Note added in proof: Simonovits determined ur.
28 PAPERS OF SPECIAL INTEREST

UNSOLVED PROBLEMS IN GRAPH THEORY AND COMBINATORIAL ANALYSIS 99
g(3, n), also it might be of some interest to prove
lim <?(3," + 1)/0(3, n)= 1.
n = oo
Denote by/(/, n) the smallest integer so that every graph of/(/, ri) vertices
either contains a K, or a set of n independent points. It is known that
c3 n2 log n/log log n < /(3, n) < c4 n2(log n)2. (2)
It would be desirable to improve (2) and to obtain an asymptotic formula
for/(/, n). I cannot even prove
lim/(/,«+ 1)//(/, n)= 1.
n — oo
Recently Yackel proved
f{n,n)<c2^jjn (3)
I proved by probabilistic methods
An, n) > 2"'\ (4)
It would be very interesting to prove that
lim f(n, n)11"
n = oo
exists and to obtain a non trivial lower bound for/(«, n) by non-probabilistic
methods [4], [13], [27], [30].
Yackel's proof of (3) will appear in the Journal of Combinatorial Theory.
7. Clearly every graph can be directed so that it should contain no directed
circuit. The following problem is due to Ore. Let G be a graph. We want to
direct its edges so that it should contain no directed circuit, and further, that
it should contain no circuit which becomes directed if one reverses the
direction of one of its edges. What is the necessary and sufficient condition
that G can be directed in such a way?
Clearly G can contain no triangle. At first one may guess that every G
which contains no triangle can be directed in such a way, but Ore showed
that this is not true.
Gallai showed that the graph of Grotsch (Fig. 1) gives a counterexample.
I could find no graph whose girth is greater than four and which cannot be
directed in such a way.
The graph of Grotsch is 4-chromatic and has no triangle. Perhaps it is the
only graph of not more than 11 vertices with this property.|
t Note added in proof: I learned that this is known to several mathematicians.
29 PROBLEMS

100
P. ERDOS
8. Hajnal and I conjectured that every graph of infinite chromatic number
contains a subgraph of infinite chromatic number which contains no triangle
(or more generally has girth > k). If our graph is the infinite complete graph
this is a well known theorem of Tutte.) We also formulated the finite version
of these conjectures: is it true that there is an f(k, r) so that every G having
chromatic number ^ f(k, r) contains a subgraph of girth > k and chromatic
number r?
Finally we asked: let m be an infinite cardinal and G be an m-chromatic
graph. It is true that G has a subgraph of chromatic number m which contains
no triangle? If G is the complete graph of power in this is a result of Rado
and myself.
Perhaps these problems could be formulated for connectivity instead of
chromatic number, e.g. is it true that if G remains connected after the
omission of any finite set of its vertices and all the edges incident to them,
then G has a subgraph with the same property and which has girth 5= kl [3],
[22].
9. Moon [32] proved that if n = 3k + 2 5= 8 and we colour the edges of
a Kn with two colours, then there are always k vertex disjoint triangles whose
edges have the same colour (different triangles can have different colours).
Let /(/) be the smallest integer with the property that, if we colour the
edges of a K„ with two colours, then there are always [n//] —/(/) vertex
disjoint K,'s all of whose edges have the same colour. Ramsey's theorem
implies/(/) < Ab but it seems certain that/(/)1/! -> 1 and perhaps/(/) < c/,
or/(/) is bounded.
How many vertex disjoint K,'s are there, all edges of which have the same
colour? (Here different K/s must have the same colour.) It is easy to see that
for / = 2 the answers is \n + 0(1) [31].
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UNSOLVED PROBLEMS IN GRAPH THEORY AND COMBINATORIAL ANALYSIS 101
10. Denote by logr n the r-fold iterated logarithm and let L(«) be the
smallest integer k for which 1 < logtn < e. I state two simple problems in
graph theory, which seem to lead to this function L(n). Moon and Moser
proved that if/(«) is the largest integer for which there is a graph of n vertices
having f(n) cliques of different sizes, then
logn logn
n-- log logn </(«)< n - -. (1)
log 2 log 2
logn
I improved the lower bound to n — . — L(n), but could not improve
log L
the upper bound.
Bondy considered the following problem. Denote by h(n) the smallest
integer for which there exists a Gin; n + //(«)) which contains a Ck for every
3 < k < n. Bondy proved (not yet published)
logn log n
-A- < h(n) <T^ + L(n). (2)
log 2 log 2
It seemed to us that in (1) the lower bound, and in (2) the upper bound, is
close to the truth but we could not even prove [12]
i, \ log" lo8n ft \
log 2 log 2
Bondy's paper is not yet published.f
11. Goodman, Posa and I [18] proved that every G(n;k) is the union of at
most [4'J2] edge-disjoint complete graphs, where in fact the complete graphs
can be chosen as edges and triangles. It is easy to see that [in2] is best possible.
We thought that for k > \n2 our theorem could be sharpened. Lovasz proved
in this direction the following result: put e= (?) — k and let t be the
largest integer for which t2 — t < e. Then G(n;k) is the union of e + t
complete subgraphs. For e = t2 and e = t2 — t the result is sharp.
Lovasz does not assume that the complete graphs are edge disjoint and
observes that the result no longer holds if edge disjointness is insisted upon.
In this case no satisfactory non-trivial sharpening of our theorem is known.
Gallai and I considered the following further problems. Is it true that every
connected graph of n vertices is the union of [^(n + 1)] edge disjoint paths?
Lovasz proved this if all the vertices of G have odd valency.
Denote by h(n) the smallest integer so that every graph of n vertices is the
t Note added in proof: Spencer proved /"(«) > n — — c.
log 2
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102
P. ERDOS
union of h(n) edge-disjoint edges and circuits. We showed h(n) < cnlogn,
but probably h(n) < c^n. K2(3, n — 3) shows that h(n) > (1 + c2) n. Perhaps
every graph of n vertices is the union of n — 1 edges and circuits if we do not
require them to be edge disjoint [29].
12. Is it true that every G(n; [n1 + E]) contains a subgraph which is non-
planar and has at most cE vertices? It is not difficult to see that ce -> oo
as e -> 0.
13. It is well known that G(n; k) can be planar only if k < 3n — 6. A planar
graph G(n; 7>n — 6) is called saturated. A theorem of Turan implies that every
G(«;[i«2] + 1) contains a triangle, i.e. a saturated planar graph of three
vertices. It is easy to construct a G(n; [|n2] + [-J-(n — 1)]) which contains no
saturated planar graph of more than three vertices; perhaps this example is
the best possible, and every G(n;[%n2] + [\(n + 1)]) contains a saturated
planar graph of more than three vertices. Simonovits has just proved this
conjecture [15], [34].
14. T. Gallai and I proved that if
l>(k-\)n-{^ + ^ (1)
then G(n; I) contains k independent edges.
I proved that if n > 400A:2 and
^(JLZ*!^)+ <»_„._(* _,,. + (*-') (2)
then G(n;l) contains k vertex independent triangles.
Posa and I proved that if n > 24/c and
I > (2k - 1) n - 2k2 + k (3)
then G(n; I) contains k independent circuits.
It is easy to see that (1), (2) and (3) become false if the inequality is replaced
by equality.
It would be desirable to improve (2) and (3) so that like (1) they would be
valid for all n, but I have not succeeded in doing this, and did not even
formulate a reasonable conjecture.
Denote by f(n;r,k) the smallest integer so that every G(r)(n;f(n;r,k))
contains k independent /--tuples. The value of/(n; 2, k) is given by (1).
Denote by g(n;r,k — 1) the number of those /--tuples formed from the
elements xl,x21 ..., xn each of which contains at least one of the elements
xu ..., xk_x. Clearly/(n; /-, k) > g(n;r,k - 1).
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UNSOLVED PROBLEMS IN GRAPH THEORY AND COMBINATORIAL ANALYSIS 103
I proved that for n > crk
f(n;r,k)= 1 + g(n;r,k- 1).
Perhaps
f(n;r,k)= 1 + max((rfcr~ l},g(n;r,k - 1)). (4)
If r = 2, (4) is true and becomes (1), but I could not even settle r = 3 [11],
[17], [21].
J. Moon, found a simpler proof of (1) and (2).
15. Recently several papers were published on extremal problems in graph
theory; here I want to mention only a few of them. Let C be a graph. f(n; G')
is the smallest integer so that every G(n\f(n\ G')) contains C as a subgraph.
Kovari, the Turans and independently I, proved that
f(n;K2(r,r))< crn2~^. (1)
It would be very desirable to prove that
f(n;K2(r,r))> cr'n2~1,r. (2)
(2) is known for r = 2 and r = 3 but no good lower bound is known for
r ^ 4. For r = 2, Brown, Mrs Turan, Renyi and I in fact proved
f(n;K2(2,2)) = (1 + o(l))n3'2/2. (3)
Perhaps if G is a graph of n vertices which contains no triangle and
rectangle, then it has at most (l + o(l)) nil2j2^2 edges.
Very likely
cn)„t + i/, </(n;c2r) < c/2)«1 + 1/r. (4)
The upper bound is not hard to prove but the lower bound is not known
for r > 2.
Let Gk be the following graph: its vertices are x;yu ...,yk; z1; ...,Z(*\
x is joined to every y, and every z is joined to two j's; distinct z's are joined
to different pairs. Perhaps
f(,r,Gk)<ckn3'2. (5)
I proved (5) for k = 3, but had no success with k > 3. f(n\ Gk) > en3'2
is trivial, since for k ^ 3, Gk contains a rectangle (i.e. a K2(2, 2)).
Simonovits and I tried to investigate /(«; Gk — x) (i.e. we omit from Gk
the vertex x and all edges incident to it). It seems likely that/(«; Gk — x) =
o(«3/2) and in fact we conjecture
f(n;Gk-x) <n^l2)-Ek (6)
33 PROBLEMS

104 P. ERDOS
but we could prove none of these results. We showed that to every e > 0
there is a KE so that
f(n;GKe-x)>n(3'2)-< (7)
The proof of (7) is not published.
Perhaps the most interesting unsolved problems in this field are the original
problems of Turan which are perhaps not sufficiently well known. Therefore
I restate them here: denote by g(n; k, I) the smallest integer so that if |S| = n
and Ax, ..., As,s = g(n; k, /) are subsets of S, \A(\ = k, 1 =¾ i =¾ s, then there
is a B c S, \B\ = I all of whose subsets of k elements occur amongst the A's.
Turan determined g(n; 2,/) for every /, but for k > 2 the problem is unsolved.
It is easy to see that
lim g(n; k, /)
/0
exists for every k and /, but for k > 2 the value of the limit is not known.
In particular Turan conjectured
«7(3«; 3, 4) = 3/!^) + l' 9^2"' 3' 5) = Mt) + l
but the proof of (8) seems elusive [2], [10], [14], [25], [33], [34].
(8)
16. Rado and I considered the following question: let f(n;k) be the
smallest integer so that if At, 1 =¾ ;' =¾ /(«; k) \A,\ = n are sets, then one can
always find k. /Ts which have pairwise the same intersection. We proved
(k-lf+1 </(«;fc)
^^-^^1 -^)-3!^----^0^-) (1)
We conjectured that
f(n;k)<ck" (2)
Abbott improved (1), but (2) is far from being settled. Recently Sauer
determined/(2; k) (not yet published).
(2), if true would have several applications in number theory.
Rado and I also investigated these questions if n and k are infinite cardinal
numbers, but all the problems can then be solved completely.
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UNSOLVED PROBLEMS IN GRAPH THEORY AND COMBINATORIAL ANALYSIS 105
Finally, the following question could be considered: let g(n;k) be the
smallest integer so that if |S\ = n, At cz S,l =¾ ;' < g(n; k) then one can always
find k A's which have pairwise the same intersection. Determine or estimate
g(n;k); compute lim g(n;k)1/k [1], [23].
n — oo
17. Let {A,}, [j A, = S, \A,\ ^ 2 be a set system. This set system is said
1 k
to be /c-chromatic if one can divide S iftto k disjoint sets S,, [J St = S
i= 1
so that no Ax is contained in any S; and such a division into less than k sets
is impossible. (A set system is thus two-chromatic if it has property B
according to Miller.) I conjectured and Lovasz proved that if \At\ ^ 2 and for every
St < S there are fewer than S1 A's contained in S1 then the chromatic number
of the system is 2. The Steiner triplets of n = 7 show that this result is best
possible (even if \At\ > 2).
Probably the following result holds: there is a constant c„ c, -> oo as
t -> oo so that if {A,} is a set-system, \A,\ > t, and for every S1 c S there
are fewer than c,\St\ A's contained in St, then the system has chromatic
number 2.
Lovasz proved that if \A,\ = r > 2 and the set system is /c-chromatic and
does not contain all the /--tuples of a set of (k —[1) (r — 1) + 1 elements, then
there are/c A's any two of which intersect in the same element. This is a
generalization of a well-known theorem of Brooks.
Is it true that if \A,\ ^ t and the system is three-chromatic then there is an
element which is contained in at least (1 + c)' /4,'s? [20], [29].
18. Let |S| = n, At c S, 1 =¾ ;' =¾ /„. Assume that no At is the union of
other ^'s. Kleitman and I observed that (unpublished)
cx 2" / n3'2 <l„<c2 2"jn3/2
Probably there is a c so that
max/„ = (1 + o(l))c2"/«3/2.
19. Let |S| = n, At c S, 1 < ;' < u„, |/4,-| = 3. Assume that any subset
S; c S, \Sj\ = 6 contains at most two A's. How large can u„ be? No doubt
u„ = o(«2) and I expect that u„ < n2~c for some c > 0.
Let |S| = n, Ai c S, \At\ = 3, \A, n Aj\ < 1. It is easy to see that there
always exists an St a S, \St\ > c^n so that no At is contained in Su but
the above result becomes false if cly/n is replaced by c2 «2/3. A set St is
called independent if no A is contained in it. Denote by f(n) the minimum of
the largest independent set where the minimum is extended over all possible
35 PROBLEMS

106
P. ERDOS
choices of the sets Ab \At\ = 3, \At n Aj\ < 1. As stated c1^/n <f(n) < c2 n2/3.
Improve the estimation for/(«).
Hajnal and I considered the following question: let S have the elements
xl,...,x„. To each couple (x;,Xj) make correspond an element xh
xr = p(xh Xj), r # i,r #/. A set St c S is said to be independent if for any
xleSi,xJeS1,p(xl,xJ)$S1. Denote by g(ri) the minimum of the largest
independent set where the minimum is taken over all functions p(xh Xj).
We proved
c1 ni/3 < g(n) < c2v«log«.
Improve the estimations of g(ri) [20].
20. Kleitman proved the following conjecture of mine. Let |S| = n, At c S,
1 < i ^ fc. Assume that the union of two A's never equals a third. Then
max/c =(1 + 00))(^-,). (1)
Moser now asked: let Au ..., A, be k arbitrary sets. Denote by f(k) the
largest integer so that for every choice of the k sets there always are/(/c) of
them /4,-,, ..., Aif(k) so that the union of two of them never equals a third.
Riddel observed/(/c) > c^k and recently Komlos and I showed/(/c) < c2Jk.
Moser's beautiful question can clearly be modified in various ways. It can
also be given a number-theoretic interpretation, e.g. let at < ... < ak be k
integers, let g(k) be the largest integer so that one can always find g(k)
integers ah < ... so that the sums (or products) of any two are distinct (or
never equals a third etc.) [9].
21. Let |S| =n, k < njl, A, c S, 1 < i < r, \A,\ = k ^ nj2, \At n /LJ S?
1, 1 ^ i ^ j ^ r. Ko, Rado and I proved that then max r = I I. In fact
the same result holds if instead of \A,\ = k we only assume \At\ < k, At ¢: Aj.
Let us now assume \Atn A-\ ^ s > 1. What can be said about max r?
(it — .y\
I, but Min showed
that this is not always true and there does not seem to be an easy way to
determine max r. Denote max r = f(n;k, s). We conjectured [16]
a**H(2)-(?)2)A
It is easy to see that
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UNSOLVED PROBLEMS IN GRAPH THEORY AND COMBINATORIAL ANALYSIS 107
22. Szekeres and I proved that if xu ...,xm2 + 1 is any sequence of distinct
numbers one can always find n + 1 of them which form a monotonically
increasing or decreasing sequence; it is easy to see that this theorem is false
for n2 numbers.
I now asked: let/(5) be the largest integer so that every sequence of distinct
numbers xt, ..., Xy(n) can be decomposed into the union of n monotonic
sequences. Hanani proved that
„ , n(n + 3)
/(«) = —y— •
As far as I know the following question is not yet settled. Let xu ..., x„
be a sequence of distinct numbers, determine max (£ x;r) where the maximum
is to be taken over all monotonic sequences [26], [28].
23. Let \At\ =»,!<;</,/< c{T
IU.
= N. Is it true that if
n > «0lcil there are c22N(c2 = ¢2(1^)) subsets B of [j A; which have a non-
t= 1
empty intersection with every A, but which contains none of the /1,'s? If
/ = 2" I cannot even prove the existence of a single such set B [8].
24. Let the vertices of Kn be xx, ..., xn. Denote by (i,j) the edge joining
xt and Xj. Let /(7, j) = + 1,1 < f < j < «. Put
//(«) = min(max|X/(U)l) (0
where in (1) the maximum is taken over all complete subgraphs of
K„(l </•<«), the summation is extended over all the edges of Kr and the
minimum is taken over all the 2(2) functions f(i,j). I proved [7]
— < H{n) < en3'2. (2)
It would be desirable to improve (2).f
References
1. H. L. Abbott (1966). Some remarks on a combinatorial theorem of Erdos and
Rado. Canad. Math. Bull. 9, 155-160.
2. W. G. Brown (1966). On graphs that do not contain a Thomsen graph. Canad.
Math. Bull. 9, 281-285.
t Note added in proof: Spencer and I proved H(n) > en312.
37 PROBLEMS

108
P. ERDOS
3. P. Erdos (1959). Graph theory and probability I. Canad. J. Math. 11, 34-38.
4. P. Erdos (1959), (1961). Graph theory and probability I and II. Canad. J. Math.
11, 34-38 and 13, 346-352.
5. P. Erdos (1962). On a theorem of Rademacher-Turan, Illinois J. Math. 6,
122-127.
6. P. Erdos (1962). Remarks on a paper of Posa, Pulb. Math. Inst. Hung. Acad. Sci.
7, 391-412.
7. P. Erdos (1963). On combinatorial problems concerning a theorem of Ramsey
and Van der Waerden (in Hungarian). Mat. Lapok 14, 29-37.
8. P. Erdos (1964). On a combinatorial problem II. Acta Math. Acad. Sci. Hung.
15, 445^147.
9. P. Erdbs (1965). Extremal problems in number theory, Proc. Sym. Pure Math.
Vol. VIII; Theory of numbers, Amer. Math. Soc.
10. P. Erdbs (1965). On some extremal problems in graph theory. Israel J. Math.
3, 113-116.
11. P. Erdos (1965). A problem on independent /--tuples. Annates Univ. Sci. Budapest
8, 93-95.
12. P. Erdos (1966). On cliques in graphs. Israel J. Math. 4, 233-234.
13. P. Erdos (1967). Some remarks on chromatic graphs Coll. Math. 16, 253-256
14. P. Erdos (1968). Extremal problems in graph theory. Proc. Symp. Smolenice
1963, 29-36. Some recent results on extremal problems in graph theory, On some
new inequalities concerning extremal properties of graphs, Theory of Graphs,
Proc. Coll. held at Tihany, Hungary 1966, Academic Press and Akad. Kiado
83-98.
15. P. Erdos (1969). Uber die in Graphen enthaltenen saturierten planaren Graphen,
Math, Nachrichten 40, 13-17.
16. P. Erdos, Chao Ko and R. Rado (1961). Intersection theorem for systems of
finite sets. Quarterly J. of Math. 12, 313-320.
17. P. Erdbs and T. Gallai (1959). On the maximal paths and circuits of graphs.
Acta Math. Acad. Sci. Hung. 10, 337-357.
18. P. Erdbs, A. W. Goodman and L. Posa (1966). The representation of graphs
by set intersections. Canad. J. Math. 18, 106-111.
19. P. Erdbs and A. Hajnal (1958). On the structure of set mappings. Acta Math.
Acad. Sci. Hung. 111-131.
20. P. Erdbs and A. Hajnal (1958), (1966). On the structure of set mappings. Acta
Math. Acad. Sci. Hung. 9, 111-131 and On chromatic number of graphs and
set systems, ibid. 17, 61-99.
21. P. Erdbs and L. Posa (1962). On the maximal number of disjoint circuits of a
graph. Publ. Math. Debrecen 9, 3-12.
22. P. Erdbs and R. Rado (1960). A construction of graphs without triangles having
pre-assigned order and chromatic numbers. J. London Math. Soc. 35, 334-448.
23. P. Erdbs and R. Rado (1960), (1969). Intersection theorems for systems of sets I
and II, J. London Math. Soc. 35, 85-90 and 44, 13-17.
24. P. Erdbs and A. Rdnyi (1960). On the evolution of random graphs, Publ. Math.
Inst. Hung. Acad. Sci. 5, 17-61.
25. P. Erdbs, A. Rdnyi and V. T. Sos (1966). On a problem of graph theory. Studia
Sci. Math. Hung. Acad. 1, 215-235.
26. P. Erdbs and G. Szekeres (1935). A combinatorial problem in geometry. Comp,
Math. 2, 463^170; for a very simple proof see Seidenberg.
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UNSOLVED PROBLEMS IN GRAPH THEORY AND COMBINATORIAL ANALYSIS 109
27. J. E. Graver and J. Yackel (1968). Some graph theoretic results associated with
Ramsey's theorem, J. Combinatorial theory 4, 125-175.
28. H. Hanani (1957/58). On the number of monotonic subsequences. Bull Res.
Council Israel, Sec. F, 11-13.
29. L. Lovasz (1966). On covering of graphs. "Theory of Graphs", Proc. Coll. held
at Tihany, Hungary, Academic Press and Akad. Kiado, 231-236.
30. L. Lovasz (1968). On chromatic number of finite set systems. Acta Math. Acad.
Set Hung. 19, 59-67.
31. L. Lovasz (1968). On chromatic number of finite set-systems, Acta Math. Acad.
Sci. Hung. 19, 59-67.
32. J. Moon (1966). Disjoint triangles in chromatic graphs. Math. Mag. 39,259-261.
33. M. Simonovits. A method on solving extremal problems in graph theory,
Stability problems. "Theory of Graphs", Proc. Coll. held at Tihany Hungary
1966, Acad. Press and Akad. Kiado 1968, 279-320.
34. P. Erd5s (1969). On extremal problems of graDhs and generalized graphs. Israel
J. Math. 2, 183-190.
35. P. Turan (1941), (1954). Eine Extremalaufgabe aus der Graphentheorie (in
Hungarian) Mat. Fiz. Lapok 48, 436—452, see also: On the theory of graphs.
Coll. Math. 3, 19-30.
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PROBLEMS AND RESULTS IN COMBINATORIAL
ANALYSIS
P. ERDOS
This review of some solved and unsolved problems in combinatorial analysis
will be highly subjective. I will only discuss problems which I either worked on or
at least thought about. The disadvantages of such an approach are obvious, but the
disadvantages are perhaps counterbalanced by the fact that I certainly know more
about these problems than about others (which perhaps are more important). I will
mainly discuss finite combinatorial problems. I cannot claim completeness in any
way but will try to refer to the literature in some cases; even so many things will be
omitted. |S| will denote the cardinal number of S; c, cu c2, ■ ■ ■ will denote absolute
constants not necessarily the same at each occurrence.
I. I will start with some problems dealing with subsets of a set. Let \S \ =n. A well-
known theorem of Sperner [57] states that if /1(<=S, 1 <i<m, is such that no At
contains any other, then max ^ = (^]). The theorem of Sperner has many
applications in number theory; as far as I know these were first noticed by Behrend [2]
and myself [8].
I asked 30 years ago several further extremal problems about subsets which also
have number theoretic consequences. Let /ff<=S, l<i<mu assume that there are
no three distinct A's so that A{ U Aj = Ar. I conjectured that
maxm! = (1 +o(l))T " J,
but could not even prove max w1 = o(2"). This latter result was proved by Sarkozi
and Szemeredi but was never published because it was superseded by the result of
Kleitman [44], who first of all proved that max m < 23/2([„"2]) and recently (in this
volume) that
which is in fact stronger than my conjecture. It would be of interest to determine
max m1; maybe this question has no simple solution, but perhaps an asymptotic
Copyright © 1971, American Mathematical Society
77
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78
P. erdOs
formula for
max mi
\[«/2])
is not quite hopeless.
The second problem I asked was: Let At<= S, \<i<m2- Assume that there are not
four distinct A's say Ah Ar, As, At satisfying
At U Ar = A„ Ai n Ar = At.
KJeitman proved maxm2<c127»i"1 and I showed that max m2 > c22njnln.
Presumably
maxffij = (c + o(l))2njnm
but as far as I know this has not yet been proved. These results are not yet
published (they will appear in Proc. Amer. Math. Soc).
Here I would like to mention a question which goes back to Dedekind: How
many families of subsets of S are there where no set of a family contains any
other? Denote the number of such families by/(«). There may not be a simple
explicit formula for /(«), but Kleitman, sharpening previous results of several
authors, proved (not yet published)
Iogy(«) = (1 +o(l))([w"2]) log 2.
It would be interesting to give an asymptotic formula for f(n) but this is probably
rather difficult.
Kleitman proved several other conjectures of mine involving subsets, some of
which have not yet been published.
Rota observed that Dilworth's theorem [6] implies Sperner's theorem and many
other results in combinatorial analysis.
Ko, Rado and I [26] proved that if n>2k, A^S, \A,\=k, AinAj=£0,
1 <i<j<m, then
maxw = (fcll)-
Assume now \A{ n A,\ >r. Put max m=f(n, k, r). We proved that for n>nQ(k, r),
Min [26] observed (in the same paper) that (1) is not true in general; the
determination of fin, k, r) in general seems to be a difficult problem. We conjectured
but we could not decide whether (2) is true.
41 PROBLEMS

PROBLEMS AND RESULTS IN COMBINATORIAL ANALYSIS 79
We also observed that if A^S, Ai n At=£ 0, 1 <i<j<m, then max m = 2n~x. It
does not seem to be easy to determine the number of families A^S, Ai n A,^ 0,
1 <i<j<2n~x. We could not even get an asymptotic formula for the number of
these families.
Let \S\ =n, A^S, 1 <i<k. What is the smallest value of k so that there should
always be three A's any two of which have the same union? This question like some
other problems in this chapter has connections with number theory. (More
generally we can ask what is the smallest value of k = kr so that there always are r A's
any two of which have the same union.)
II. Some geometric problems. Let zt>l, \<i<n. Consider all the sums
Z"=i £:z:> £i= ± 1- I [9] proved as an easy application of Sperner's theorem that the
number of sums which fall into the interior of an interval of length 2 is at most
([#i/2])> w'th equality if z{ = 1, 1 < i < n. I conjectured that if the z{ are complex
numbers satisfying \zt\ > 1, then every circle of radius 1 contains at most ([„%]) sums
2"=i £jZj (this would sharpen a result of Littlewood and Offord); more generally I
conjectured that the above result may remain true if the zt are vectors in Hilbert
space or even in a Banach space.
Katona [40] and Kleitman [43] independently and almost simultaneously
proved my conjecture in the plane by giving an interesting generalization of
Sperner's theorem, and Kleitman [43] also proved that my conjecture holds in k-
dimensional space if n > n0(k), but the general conjecture has not yet been settled.
Sarkozi, Szemeredi and I have the following conjecture: Let |z,| < 1, 1 < i<n, then
there are at least clnjn summands 2"=i E\zu e\= ± 1, which are of absolute value
< \/2 (it is easy to see that \/2 cannot be diminished; let an odd number of z's be
1 and an odd number z). The order of magnitude c2"/n is easily seen to be best
possible if true. Analogous conjectures can easily be made for higher dimensions.
Sarkozi and Szemeredi proved that if — 1 <z{< 1 then there are at least ([„"2])
sums 2™=i £izi, £i= ±1» which are less than 1 in absolute value. It is easy to see that
([#i/2]) 's best possible.
Sarkozi and Szemeredi further observed that our conjecture in the plane is true
if the following purely combinatorial result holds: Let |S|=n, A^S, \<i<k,
B^S, \<j<l (the A's and B's are all distinct). Assume \Ah n Ai2\>2, l<i\<
h<k; \Bh n Bj2\>2, l<j\<j2<l; \At n Bt\>l, \<i<k; \<j<l. Then
k + l < 2n~l-c2nln.
Miss E. Klein raised in 1932 the following problem: Let f(n) be the smallest
integer with the property that from f(n) points in the plane one can always select
vertices of a convex «-gon. Miss Klein proved that/(4) = 5, Makai and Turan
proved/(5) = 9. Szekeres and I [31], [32] proved
2-- <™* (2;-4)
(our proof of the lower bound contained an inaccuracy which was corrected by
Kalbfleisch). It seems likely that/(«) = 2"~2 + 1 but this is not known for «>6.
42 PAPERS OF SPECIAL INTEREST

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P. ERDOS
Let there be given 2" points in the plane. Szekeres and I [32] proved that these
points always determine an angle greater than 77(1 — 1/«), an earlier result of
Szekeres [58] states that to every e there are 2" points so that every angle is less than
77(1 — ljn) + e. Thus for 2" points the problem of minimizing the maximum of the
greatest angle is completely solved. It is not impossible that if n > nQ then already
2n-1+l points always determine an angle >7r(l —1/n), but we only proved that
2"—1 points always determine an angle >7r(l —1/n). In higher dimensions sharp
results are known only for special values of n, thus Danzer and Griinbaum [5]
proved that 2"+ 1 points in «-dimensional space always determine an angle >tt/2.
Sylvester conjectured and Gallai first proved that if we have n points in the plane
not all on a line then there is at least one line which goes through exactly two of the
points. Denote by/(«) the minimum number of such lines. N. G. de Bruijn and I
conjectured that/(«) -> 00 as n -> 00. This was proved by Motzkin [47]. Kelly and
Moser [41] proved that f(n) > 3n/7 and this is best possible for n = l. Motzkin
conjectured that f(ri) > [«/2] and showed that for infinitely many n this is best
possible.
Let there be given n points not all on a line, I observed that it easily follows from
Gallai's result that these points determine at least n lines. G. Dirac conjectures that
one of the n points is such that it is connected with the other points by more than
en distinct lines.
Assume now that the n points are such that not more than n — k of them are on
a line. I conjectured that these points determine at least ckn lines. If k is fixed and
n > n0(k) then Kelly and Moser [41] determined the minimum number of lines which
these points determine.
Let there be given n points in the plane, not all on a circle. I conjectured that
these points determine always at least ("2') circles. B. Segre disproved this
conjecture for « = 8, but Elliott [7] proved it for n>na.
One can pose the following general problem: Let au...,an be n elements,
Ax,.. ., At, t> 1, be sets whose elements are the as. Assume \At\ >r, 1 <i<t, and
that each r-tuple is contained in precisely one of the A's. Put min t=f(n; r).
Hanani, Szekeres, de Bruijn and I [3] proved that/(n; 2) = « and Hanani proved
(see Erdos, [16])
clfi3'2 </(«;3) < c2«3/2.
Thus for r = 2 the combinatorial and geometric problem has the same solution
(in the geometric problem the A's are the lines joining the points) but for r = 3 this
is no longer the case (for r = 3 the r's are circles). The cases r>3 have not been
investigated.
Further geometric problems and results of a combinatorial nature can be found
in [10], [12], [42]. Many very interesting problems on combinatorial geometry are
found in the lithographed notes of Croft. Further I would like to refer to two books,
Hadwiger and Debrunner [35] and [42].
III. A well-known theorem of Ramsey [50] states that if \S\ >N0 and we split
the ('-tuples of £ into two classes then there is an infinite set all of whose /-tuples
43 PROBLEMS

PROBLEMS AND RESULTS IN COMBINATORIAL ANALYSIS 81
are in the same class. Many extensions and generalizations of Ramsey's theorem
have been published in the last few years (see my remarks under Ramsey [50]).
Here we will only be connected with the finite version of Ramsey's theorem.
Denote by f(i; k, I) the smallest integer so that if |S| =/(/; k, /) and we split the
/-tuples of S into two classes then there either is a subset of k elements all whose
/-tuples are in the first class or a subset of / elements all whose /-tuples are in the
second class. Ramsey was the first who obtained upper bounds for /(/; k, I).
Szekeres and I proved [31]
(3) f(2;k,l)< (^1)
and I [11] proved that
(4) f(2; k, k) > 2*'2.
It would be very nice to prove that
lim f(2;k,k)llk
fees CO
exists and to determine its value.
(4) was proved not by an explicit construction but by a simple probabilistic
reasoning. It would be very desirable to obtain a good lower bound for (4) by an
explicit construction.
I [15] proved by a more complicated probabilistic reasoning
(5) /(2; 3,/) > c/2/(log/)2
and Graver and Yackel [33] recently showed that
(6) f(2;k,l) < c/*"1 log log//log/.
My method which I used to prove (5) very likely will also give (k fixed, /-> oo)
(7) f(2;k,l) > c1/fc-VGog/)e»
but I have not worked out the formidable details.
Very little is known about the exact values off(2; k, I). Trivially/(2; 2,/) = / and
/(2; k, 2) = k. Further we have (see Graver and Yackel, [33])
/(2; 3, 3) = 6, /(2; 3, 4) =9, f(2; 3, 5) = 14, /(2; 3, 6) = 18,
/(2; 3, 7) =23, f(2; 4, 4) = 18.
As far as I know nothing is known about the exact values of/(/; k, I) for />3.
Hajnal, Rado and I [25] proved that/(/; k, /) is less than an (/— l)-times iterated
exponential and greater than an (/ —2)-times iterated exponential.
We can generalize the Ramsey numbers by division into more than two classes—
even less is known about these than about division into two classes (Greenwood
and Gleason [34]).
The following question which is related to Ramsey's theorem is perhaps of some
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82
P. erdOs
interest: Let Kn be the complete graph of n vertices. Denote its vertices by X±,...,
Xn. To the edge (Xu Xj) we make correspond eitj where etJ= ± 1. Put
F(ri) = min max ^ etj
where the maximum is taken over the edges of all the complete subgraphs of Kn
and the minimum over all the 2C*.2 (C is the binomial coefficient) choices of the
etJ. I can only prove [18]
«/4 < F(n) < en3'2.
It would be desirable to obtain better estimates for F(ri).
Added in proof. J. Spencer and I proved F(n) > c^3'2.
IV. Miscellaneous combinatorial problems. Miller [46] in the course of some
investigations in set theory introduced the following concept: A family of sets
{Aa} is said to have property B if there is a set S which has a nonempty intersection
with every Aa and does not contain any of the Aa's.
Hajnal and I [23] continued Miller's investigations and also asked the following
question about finite sets: What is the smallest integer m(n) for which there is a
family of sets {Ak}, \Ak\ = «, 1 <k<m(n), which does not have property B? Trivially
w(2) = 3. It is not difficult to see that w(3) = 7. The value of w(4) is unknown.
Schmidt [55] and I [19] proved
2"(1 +4/^)-1 < m(n) < n22n + 1.
It would be of interest to give an asymptotic formula for m(n) and to compute
w(4). Perhaps no simple formula for m(n) exists.
Gallai asked: Does there exist a family of sets {Ak}, 1 </c<Wi(«), not having
property B and satisfying |^fc|<« and \Aki n Ak2\ < 1, 1 <k1<k2<m1(n)? A
priori it is not obvious that m^ri) is finite, but Hajnal and I proved that Wi(«) < c"
for every n [24]. We cannot prove that lim,,.^ m1{n)lln exists.
Rado and I investigated the following question: A family of sets is called a
A-system if every two members of the family have the same intersection. Denote by
F(k, I) the smallest integer for which if {A{}, 1 <i<F(k, I), is a family of sets each
having k elements, then it always contains a subfamily Air, \<r<l, which is a
A-system. We proved
(/-1)* < F(k, I) < ^.(/-1^1-^- • • • -^^-
We believe that
(8) F(k, I) < cklk
holds. (8) would have many number-theoretic applications but is also of great
intrinsic interest.
We investigated the problem of A-systems also if k + l>X0. But this
set-theoretical problem is much simpler than the combinatorial one and we have determined
F(k, I) in this case [29].
45 PROBLEMS

PROBLEMS AND RESULTS IN COMBINATORIAL ANALYSIS 83
I conjectured that if au . .., a2h is a sequence of length 2k where as is one of the
integers 0, 1, .. ., k — 1 then there are always two consecutive blocks containing
each of the integers the same number of times. This is obvious for /c = 2 and de
Bruijn proved it for /c = 3. But for /c = 4 de Bruijn and I disproved it. Later Croft
constructed a sequence of length 50 for k = 4 without such consecutive blocks and
he suggested that for /c = 4 there probably is an infinite sequence without two such
consecutive blocks. With the help of the Atlas computer Churchhouse constructed
such a sequence of length about 1700 which gives a strong support to the conjecture
of Croft.
Steiner conjectured that if n = 6k+ 1 or 6k+3 then there exists a system of
triplets of n elements so that every pair is contained in one and only one triplet of our
system. It is obvious that if n is not of the above form then such a system does not
exist.
Steiner's conjecture was first proved by Reiss [51]. More generally the following
question can be asked: For which values of n is there a system of combinations
taken s at a time formed from n elements so that every /--tuple is contained in one
and only one of our ^-tuples. Hanani [37], [38] settled the cases r = 3, ^ = 4, r = 2,
^ = 4 and r = 2, ^ = 5. The general problem seems very difficult.
Finally I would like to call attention to an old conjecture of van der Waerden
which seems surprisingly difficult: The permanent of an n by n doubly stochastic
matrix is >«!/«". Equality only if all elements of the matrix are \jn.
V. Some problems in combinatorial number theory. Van der Waerden [59]
proved the following theorem: If we split the integers into two classes at least
one of them contains arbitrarily long arithmetic progressions. Here we are more
concerned with the following finite form of van der Waerden's theorem: Let/(«)
be the smallest integers so that if we split the integers not exceeding f(ri) into two
classes at least one of them contains an arithmetic progression of n terms. Van der
Waerden's proof gives a very poor upper bound for f(n). Sharpening a previous
result of Rado and myself, Schmidt proved f(n)>2n~cinlogn>li2 [54]. I understand
that recently Belrekamp proved f(n) > 2" (Canad. Math. Bull. 11 (1968), 409-414).
It would be very desirable to obtain better lower and especially upper bounds for
/(«). Undoubtedly lim7l = co/(«)1,,l exists—I expect the limit to be infinite. One
could try to estimate/(«, m) where/(«, m) is the smallest integer so that if we split
the integers not exceeding/(n, m) into two classes either the first class contains an
arithmetic progression of n terms or the second an arithmetic progression of m
terms. Also one can consider splittings into more than two classes. R. Schneider
and R. Ecks have certain results in these directions.
Define fs(n) to be the smallest integer so that if g(m)= ± 1, 1 <m<fe(n), is any
number-theoretic function then there is an arithmetic progression of n terms
0<a<a + d<---<a + (n-l)d< f£(ri)
for which
2 g(a + kd)
I k = 0
> en.
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84
P. ERDOS
I proved/£(«)> (1 +17)", 17 = 17(e) [18]. The proof is probabilistic and similar to my
proof with Rado. I would guess that/£(n) < (1 +7/0" and perhaps 17! -> 1 as e -> 0
but I cannot disprove
lim fs(n)Vn = 00
for every e > 0.
Roth [52], proved that if g(m) = ± 1, \<m<n, there always is an arithmetic
progression 1 <a< • • • <a + kd<n for which (for every e>0 if n>n0(e))
(9)
2 Si^ + ld)
and he conjectured that in (9) «1/4-e can be replaced by nll2~e. I proved [22] that
there is a constant C and a g(m)= ± 1 so that for every progression
(10)
2s(a+ld)
1=0
< Cn
1/2
and I conjecture that (10) holds for every C> 0 if n > nQ(C).
Added in proof. This conjecture was just proved by J. Spencer.
An old conjecture of mine states that if g(m)= ±1, 1 <m<oo, then to every c
there is a d and an m so that
(11)
2 g(kd)
> c.
The proof of (11) seems to present great difficulties.
Let «!<••• <ak be k distinct real numbers. Denote by f(n; au .. ., ak) the
number of solutions of
k
n = 2 £<a<, e, = 0 or 1.
Moser and I [21] proved that
(12) f(n; alt..., ak) < c2Vfe3/2(log kf'2
and we conjectured that in (12) (log k)3'2 can be omitted. This conjecture was
proved by Sarkozi and Szemeredi [53].
Moser and I further conjectured that if k = 2l+ 1 then
(13) /(»;(fli,...,fla,+i)) </(«;-/, -/+1,..-, ^1,0,1,...,/-1,/).
As far as I know (13) has not yet been proved. Van Lint [45] found an
asymptotic formula for/(«; —/,...,/).
We further conjectured that the number of solutions of
k k
n = 2 £'fli> 2 £' = ?
is for every t less than c2kjk2.
47 PROBLEMS

PROBLEMS AND RESULTS IN COMBINATORIAL ANALYSIS 85
Donald Newman conjectured that for every n and m there is a function hn_m
having the following properties: hn_m is defined for I <i<n and
hn,Jj) ^ K.JJ), 1 < i <j < n, m < hn-m(i) < m + n,
OWm(O) = 1, I < i < n.
Baines and Daykin [1] proved that for n = m such a function exists, but the general
case is not yet settled. One would think that Hall's theorem can be applied here,
but this seems to present great difficulties.
Schur [56] proved the following result: Denote by //(«) the least integer so that
if we split the integers from 1 to H(n) into two classes, the equation x + y = z is
solvable in at least one class. Schur proved H(n) < [en! ]. It would be very interesting
to decide whether lim,^^ H(n)iln is finite or not.
Saunders in his dissertation written under Ore proved the following result: To
every k there is an F(k) so that if we split the integers from 1 to F(k) into two
classes there are always k integers at< ■ ■ ■ <ak so that all the 2*— 1 sums
k
(14) 2 eifl<. £, = 0 or 1 (not all £j = 0),
(-1
are in the same class. Graham and Rothschild then asked the following question:
Split the set of all the integers into two classes. Does there then exist an infinite
sequence a^< ■ ■ ■ so that all the sums
GO
(15) 2 £<a» e< = 0 or 1,
( = i
are in the same class, where in (15) not all the e( are 0 but only a finite number of
them are different from 0?
I do not know the answer to this question. It easily follows from Ramsey's
theorem that there is an infinite sequence where all the sums
CO CO
2 E<a» Ej = 0 or 1, 2 E> = ?'
i=l i=l
are in the same class. But I cannot decide the following question: Does there exist
an infinite sequence ax< ■ ■ ■ where all the aif 1 </<oo, and all the sums a( + a,,
1 <;'</< oo, belong to the same class. This would of course follow from Graham's
conjecture. The following weakening of Graham's conjecture also does not seem
to be completely trivial: There is an infinite sequence at< ■ ■ ■ so that all the sums
CO CO
2 e(tfi, e( = 0 or 1, 2 ei = ?> l<t<*x>,
( = i (=i
belong to the same class but the class may depend on t.
Finally I would like to ask the following question which I could not decide
even if we assume the continuum hypothesis. Split the real numbers into two
48 PAPERS OF SPECIAL INTEREST

86
p. erdOs
classes. Does there then exist a set of power K1; {aa}, 1 <a<w1, so that all the sums
belong to the same class?
For further problems of combinatorial number theory I refer to [22] and my
first paper on extremal problems in number theory [17]. See also [20].
VI. Finally I would like to call attention to some curious results and problems
of Czipszer, Hajnal and myself [4] which are partly graph-theoretic and partly
analytic—in fact they deal with Tauberian theorems.
Let G be an infinite graph whose vertices are the integers and g(n) the number of
edges of G both vertices of which do not exceed n. A monotone path of length k is
a sequence of integers iL < ■ ■ ■ <ik + i where it and it + L, j = 1,..., k, are joined by
an edge. We conjecture that if for every e > 0 and n > n0(e)
then G contains infinitely many monotone paths of length k.
We proved this conjecture for /c = 2 and /c = 3, but could not settle the general
case.
For k = 2 we proved the following stronger theorem: Assume that for n>nQ(e)
,, n2 (1 \ n2
8(n)>Y+[T2 + E)(teinr'
then G contains infinitely many monotone paths of length k. This result is best
possible since it fails if we only assume
. . __ n2 n2 I n \
gW ~ ¥ + 32(log«)2+°\(log«)7
We further proved that there is a G for which
lim inf g(n)/n2 > \
n=co
but G does not contain an infinite monotone path. On the other hand we showed
that there is an a > 0 so that if
lim inf g(n)/n2 > \ — <*
n= od
then G contains an infinite monotone path. We do not know the largest value of a
which will insure this conclusion.
Several further problems of a combinatorial nature can be found in my two
papers [13], [14] on unsolved problems.
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41. L. M. Kelly and W. O. J. Moser, On the number of ordinary lines determined by n points,
Canad. J. Math. 1 (1958), 210-219. MR 20 #3494.
42. V. L, Klee (editor), Convexity, Proc, Sympos. Pure Math., vol. VII, Amer. Math. Soc,
Providence, R.I., 1963.
43. D. Kleitman, On a lemma of Littlewood and Ojford on the distrbution of certain sums,
Math. Z. 90 (1965), 251-259. MR 32 #2336.
44. , On a combinatorial problem of Erdos, Proc. Amer. Math. Soc. 17 (1966),
139-141. MR 32 #2337.
45. J. H. van Lint, Representation of 0 as 2"=-„ ckk, Proc. Amer. Math. Soc. 18 (1967),
182-184. MR 34 #5789.
46. E. W. Miller, On a property of families of sets, C.R. Soc. Sci. Varsovie 30 (1937), 31-38.
47. Th. Motzkin, The lines and planes connecting the points of a finite set, Trans. Amer.
Math, Soc. 70 (1951), 451-464. This paper contains many further interesting problems some of
which have been settled in the meantime. See, e.g., Hansen [39]. MR 12, 849.
48. C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc,
Cambridge Philos. Soc. 61 (1965), 33-39. MR 30 #3850.
49. R, Rado, Studien zur Kombinatorik, Math. Z, 36 (1933), 424-480.
50. F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1929),
264-286, also Collected Papers, 82-111, See also Erdos and Rado [28]. For generalizations of
Ramsey's Theorem see Erdos and Rado [27] and Nash-Williams [48]. I would further like to
mention the following unpublished result of Galvin: Let F be a family of finite subsets of the
integers so that every infinite set contains a set in F. Then every infinite subset S of the integers
contains an infinite subset Si, so that every 5"2cS"i, |52| = Xo has an initial segment in F.
51. M. Reiss, Vber eine Steinersche kombinatorische Aufgabe, J. Reine Angew. Math. 56
(1859), 326-344.
52. K. F. Roth, Remark concerning integer sequences, Acta Arith. 9 (1964), 257-260.
MR 29 #5806.
51 PROBLEMS

PROBLEMS AND RESULTS IN COMBINATORIAL ANALYSIS 89
53. A. Sarkozi and E. Szemeredi, Vber ein Problem von Erdos und Moser, Acta Arith. 11
(1966), 205-208. MR 32 #102.
54. W. Schmidt, Two combinatorial theorems on arithmetic progressions, Duke Math J.
29 (1962), 129-140. MR 25 #1125.
55. , Ein kombinatorisches Problem von P. Erdos, Acta Math. Acad. Sci. Hungar.
15 (1964), 373-374. MR 29 #4701.
56. I. Schur, Vber die Kongruenz xm + ym = zm (modp), Jber. Deutsch. Math.-Verein. 25
(1916), 114-117. See also Rado [49].
57. A. Sperner, Ein Satz uber Untermengen einer endlichen Menge, Math. Z. 27 (1928),
544-548.
58. G. Szekeres, On an extremum problem in the plane, Amer. J. Math. 63 (1941), 208-210.
MR 2, 263.
59. B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. (2)
15 (1927), 212-216. See also R. Rado [49].
University of Colorado
Hungarian Academy of Sciences
52 PAPERS OF SPECIAL INTEREST

Part II
Graph Theory

Chapter 3
Representation of Graphs
All that the papers in this section have in common is the term "representation."
Paper [344] considers the minimal set of vertices in a graph such that
every edge is incident to that vertex set. Bounds are given on the size of the
minimal set in terms of the number of vertices and edges in the graph. Further
results are given in [434] (in Chapter 5).
Paper [426] is of interest in connection with "voter paradoxes." It has
long been known by political scientists that in a three candidate race it may
be possible for a majority of voters to prefer A to B, a majority prefer B to C,
and a different majority to prefer C to A. In general, given n candidates, if
there are sufficiently many voters any pattern of binary preference may be
achieved. Erdos and Moser give fairly tight bounds on the number of voters
needed. Bounds are also found on the largest transitive subtournament
contained in a tournament on n players. Further results on this problem appear
in [424] and [450] in Chapter 12, Tournaments.
In [464] it is noted that given any family of sets the intersection graph
can be defined by letting the sets be vertices and considering them adjacent if
they intersect. Conversely, any graph may be represented as an intersection
graph. An exact bound is given on the number of points required in the union
of the sets of such a representation in terms of the number of points of the
graph. L. Lorasz (University of Budapest) has further (unpublished) results.
In [426], on page 127, it is asked if/(15) = 4. This has been decided
negatively by K. B. Reid and E. T. Parker, "Disproof of a conjecture of
Erdos and Moser on tournaments," J. Combinatorial Theory 9(1970)225-238.
Papers in Chapter 3
[344] (with T. Gallai) On the minimal number of vertices representing the
edges of a graph
[426] (with L. Moser) On the representation of directed graphs as unions of
orderings
[464] (with A. W. Goodman and L. Posa) The representation of a graph by
set intersections
55 REPRESENTATION OF GRAPHS

ON THE MINIMAL NUMBER OF VERTICES REPRESENTING THE EDGES
OF A GRAPH
by
Paul ERDOS and Tibor GALLA1
Introduction
In this paper we will only consider non-directed graphs which do not
contain loops and where two vertices are connected by at most one edge1
(see 11] and [7]). We permit isolated points and we do not exclude the empty
graph i. e. the graph without vertices and edges. 71(G) and v(G) denotes the
number of vertices respectively of edges of the graph G. G' _c G denotes
that G' is a subgraph of G. (If G'cG and G' =j= G, we write G'czG.)
We shall say that the vertices Plt ..., Pk(k ^ 1)2 re-present the edges
ei< ■ ■ ■ > ej(j ^ 1) of G if every edge e,(l g i g j) contains at least one the
points Ph(l g. h g. k). If the vertices Plt . . . , Ph represent all edges of G
we call R = {P1} . . . , Pk} a representing system of G and say that R represents
G. We denote by fi(G) the minimal number of vertices representing every1
edge of G (i. e. we can find fi(G) vertices in such a way that every edge of
Gcontaints at least one of these vertices, but there do not exist /j,(G)—1
vertices with this property). If G has no edge, then by definition fi(G) = 0.
The chief object of this paper will be to give various estimations from
above of fi(G).
In § 1 we shall obtain estimates for fi(G) in terms of 71(G), v{G) and other
characteristic data of G. One of our results (Theorem (1.7)) which will be an
easy consequence, of a result of Turan states that
/i(G) g, , if v(G) > 0 .
In § 2, 3 and 4 we shall estimate /i(G) in terms of n(G') where G' runs
through certain subgraphs of G. Our principal results are:
If n(G') g pfor all G' c G with n(G') g 2p + 2, then /i(G) g p.
(Theorem (3.5)).
1 Every edge "contains" exactly two vertices, which are "connected" by it.
2 Numbers which are denoted by letters are always assumed to be non negative
integers.
181
57 REPRESENTATION OF GRAPHS

182
EIID0S—GALT.Al
Let h ^ 2, p>p0(h). Assume that 71(G) ^ -lp — h -f 3 and that G has
■n<) isolated vertices, further assume that for every G' C G with tt(G') ^L p + h
ire have u(G') ^ p. Then fi(G) ^ 2p—h (Theorem (2.2)).
In general the above results are best possible.
In i? 5 we generalise our problems to ,,multidimensional graphs". Instead
of graphs we consider sets of /^-tuples (A' ^. 2) and study the minimal number
of elements which represent each of our given ^-tuples.
§ I-
(1.1) First of all we need some definitions and notations.
G will always denote a graph, and if in the following it is not explieitely
Indicated to which graph some symbols and notations belong, we a1 ways
assume that they refer to the graph denoted by G.
o.(M) will always denote the number of elements of the finite set M.
We shall denote by PQ the edge connecting the vertices P and Q. The
graph which consists of the vertices P and Q and the edge PQ will also be
called an edge. The graph which consists of the vertices Pv P2, P3 and the
edges PXP^ P^s- P3P1 M'i'l 0e called a triangle and will be denoted by PXP2PS.
If P is a vertex of G, then we call the number of edges of G which
are incident to P the valency of P (in G).
If any two vertices of G are connected by an edge G will be called
complete. The graph consisting of one point will be called complete too.
If G is complete and 71(G) = n we sbalJ call G a complete n-graph.
Assume that G has at least two vertices. The complementary graph G of
G is defined as follows: G has the same vertices as G and two vertices of G
are connected if and only if they are not connected in G.
For the definition of path and circuit see |7] (path = Wog, circuit =
= Krci.s),
A graph G — having at least two vertices — is said to be connected if
any two of its vertices are on a path of G. The graph having one vertex is
called connected.
The components of (the non empty) G are its maximal connected
subgraphs .
Denote by S the set of vertices of G. Let M c S. We denote by \M]
the subgraph of G whose vertices are the elements of M and whose edges
are all the edges of G which have both vertices in M.
If M c S and N c S then we call the edges one vertex of which is in
M and the other in N the MN-edges.
I J/, N) denotes the subgraph of G whose vertices are the elements of
M\jX and whose edges are the jlfAT-edges of G.
G is even if there is an M and .V for which M\jN=S, Mp\N= 0 and
[AT, X] = G,
Let P^S. G—P denotes the graph which we obtain by omitting from
G the vertex P and all the edges incident to P.
The vertices Plt . . . , F-(;>1) of G are called independent (in G) if no
two of them are connected by an edge (in G). One vertex is always called
independent, [1(G) denotes the maximal number of the independent vertices
of G. If G is empty, then by definition Jl(G) = 0.
58 GRAPH THEORY

MINIMAL .VUJIBER 01" REPRESE.NTIXG VERTICES 183
The edges ev ■ ■ ■ , e; ()>1) are called independent if they have no
common vertex. One edge is always called independent. The maximum number
of the independent edges of G is denoted by e{G). If G has no edges we have
by definition e(G) = 0.
Wc shall call G k-fold connected (k 2i 1) if in ease k = 1 G is connected
and for k>\ if 71(G) > k + 1 and G remains connected after the omission
of any k—1 of its vertices (and all the edges incident to them).
(1.2) It follows from our definitions that if Glt . . . , Gy (j ^ 1) are the
components of G then if cp = n, v, /t, /7 or e
cp(G) == 2\(Gt) ■
1 = 1
(1.3) It is easy to see that (see [6], p. 134.)
(1) p(G)+-ji(G)=n(G).]
If G is non empty then /1(G) Jg 1, equality here holds if and only if
G is complete. From this remark and (1) we obtain
(1.4) If G is non empty then uIG) rg n(G) — 1. Equality holds if and only
if G is complete.
If we make special assumptions about G we can improve the above
estimation. Thus the following trivial inequalities hold:
(1.5) If G is even /u(G) g, — n<G).
If we assume that G does not contain a triangle (or a complete fc-graph
(£>3)) then the problem of giving a sharp upper bound for fi(G) in terms of
rc(G) is difficult and will not be discussed in this paper. Because of (1.3) (1)
this is really Ramsay's problem ([8], [5]).
(1.6) n(G) 5i v(G) is trivial. Equality holds if and only if no two edges
of G have a common vertex.
We can obtain non trivial upper estimates of /j,(G) using both rc{G)
and v(G).
Theorem (1.7). Assume that G has- edges. Then
2v(G)+tz(G)
or in other words: /'(G) is less than or equal to the harmonic mean between
— 71(G) and r(G). Equality holds if and only if G is a complete graph, or if each
component of G is a complete graph each of which has the same number of vertices.
59 REPRESENTATION OF GRAPHS

184
ERDGS—GALLAI
Proof. Our theorem is an easy consequence of a result ofTuran. Turan
proved ([9], p. 26.) that if n(G) = n and G does not contain a complete
tj + 1)-graph but contain a complete y'-graph, then
(1) v(G) g ^- (w2 - r2) + ( r
2 / I 2
where n = jf -f r (0 S r<y)- If r = 0 equality occurs if and only if G (G is
tlie complement of G) has j components and each of them are complete
t -graphs3.
Applying this theorem we obtain
(2) v(G) >
n\ - \hll. („a - ,2) + f r\\ = iZL- r)Sn zJ+ r\
2j | 2; \2 } 2j
where .t(G) = n, (i(G) = j and n = jt + r (0 5i r<;'). Further if r = 0
equality occurs if and only if all components of G arc complete ^-graphs.
Let fi(G) = k. By (1.3) j = n—k, thus from (2)
(;„ V{G)> 1»-JW±H.
2 (n — k)
From 0 rg. r<n — k we have k<n—r 52. n. Thus
(4) (n — r){k + r)^ nk ,
equality only if r = 0. From (3) and (4) we obtain, assuming that v(G) =
= m>h
k^ 2 -.
n m
Equality can hold only if we have equality both in (4) and in (2). This
completes our proof since every graph G with tz(G) > 2 is the complementary graph
of a certain graph.
From (1.7) we easily obtain
Theorem (1.8)
(1) M^+'W.
Equality holds if and only if G is empty or if the components of G are edges and
triangles.
Proof. If G is empty the theorem is trivial, henceforth we shall assume
tt(G)>0. It follows from (1.2) that it will suffice to prove our theorem for
3 Turan gave also in the case r ;; ■ 0 the necessary and sufficient condition for
equality in (1).
60 GRAPH THEORY

MINIMAL NUMBER OF REPRESENTING VERTICES
185
connected graphs and that equality can hold for 67 only if it holds for every
component of 67.
Henceforth we shall assume that 67 is connected. Put n(G) — n,
v(G) = m.
For n — 1 (1) clearly holds with the sign < . Thus we can assume m>_ 1.
From (1.7) we have
(2) ,(G)^-2-™"-,
2 m + n
equality holds if and only if G is complete. For positive m and n the inequality
2mn/(2m + n) r£ (m + n)/3 is equivalent to
(3) 0 .S (m — t?) (2 m — n) .
Therefore if m S; n, (1) is implied by (2) and (3), further we can deduce that
equality holds if and only if m — n and 67 is a complete K-graph. But this is
possible only if n — 3.
If m<n, then since 67 is connected, m — n — 1 and 67 is is a tree (see
[7], p. 51.). Since every tree is even, we have by (1.5)
fi(G) ^ — n .
2
For n ^ 2 we have (m + w)/3 = {in — 1)/3 > V(2w), equality only for n = 2.
This proves (1) for m <n and shows that equality holds if and only if 67 consits
of a single edge. This completes the proof of our theorem.
(1.9) Next wc estimate ^(67) in terms of e(G).
Assume, v(G) > 1 and let P1 P[ , . . . , Ps P's(s = e(67) ^ l) be a maximal
system of independent edges of 67. Clearly the vertices Plt ..., Ps, P[, . . . , P's
represent the edges of 67. On the other hand we clearly need at least s vertices
for the representation of the edges of 67. Thus we obtain the following trivial
inequality
(1.10) £(67)^(67)^2 6(67).
(1.10) trivially holds for v(G) = 0 too.
The following theorem which we will often use is due to Konig ([7],
p. 233.).
(1.11) (Konig). For even graphs fi{G) = e(G).
For the upper bound in (1.10) we have the following
Theorem (1.12). ^(67) = 2 e(67) holds if and only if 67 is empty or each
component 67,- of G is complete and ti(67,) is odd.
Proof. The sufficiency of the above conditions is evident. To prove
the necessity observe that because of (1.2) it will be sufficient to show that
for a connected 67 satisfying ti(67) > 2, /i(G) = 2 e(67) holds only if 67 is complete
and 7i(67) = 2 e(67) + 1. This immediately follows from (1.4) and from the
following
61 REPRESENTATION OF GRAPHS

186
KUDOS -G.4LLAI
Theorem (1.13). Lei G be 1--fold connected (k > 1). Assume ti{G)>2e(G) + 1.
that k ^. e(G) and
/i{G) < 2e{G) - k .
The above bound for fi(G) is best possible.
Our proof of theorem (1.13) uses the theory of alternating paths. The
proof can he deduced easily from the properties of alternating paths stated
in § 4 of [4]. We do not give the details of the proof.
We remark that one can give a simple proof of (1.12) without using
(1.13).
The following example shows that the bound 2 e(G) — k in theorem (1.13)
is best possible: Let G0 be a complete /j-graph and G, a complete (2a t + 1)-
graph (k ^ 1, a, 2r 0, i = 1, . . . , I. I > k + 1). The graphs GQ and Gt have
no common vertex. The vertices of G are the vertices of G0 and those of the
G, (i =-- 1, . . . , I), the edges of G are the edges of G0, the edges of G, (i = 1,
...,/), and overv edge which connects a vertex of G0 with a vertex of G,
(1 __' ;' _'_ I). We have'
/ i
s(G) =-- A: + ^ a, ■ ^(G) = k + J£ 2 a, ,
;=i i-i
/
%{G) = & + ^" (2 a,- + 1) =- Z — & + 2 e(G) > 2 e(G) + 1 .
i' = i
G is A'-fold connected, /i(G) = 2 e(G) — 4. Observe that in our example n(G)
can lie made arbitrarily large for given s(G).
Remark. If G satisfies n;(G)>3 e(G)— 2 (e(G) 2; 1) and is connected
then we can prove
(1) t*(G) ^ 2e(G) — d
where r/ is the minimum of the valency of the vertices of G. If G is A'-fold
connected and rr(G)>l, then clearly d > k, thus (1) is a sharpening of (1.13).
The proof of (1) is similar to that of (113) and will be suppressed.
Finally we obtain bounds for /.t(G) in terms of s(G), v(G) and n(G).
Theorem (1 14)
(1) MO)^(G)+^e(G),
(2) rMZeW+^-^+W-JLW.
2 4
Remarks. These bounds arc best possible. For (1) we see this by
considering a graph whose components are edges and triangles, and it is not
difficult to see that this is the only case of equality.
For (2) the situation is more complicated. The only connected graphs
(with >'(G)>0) known to us for which there is equality in (2) are: 1.) an edge,
62 GRAPH THEORY

MINIMAL NUMBER OF REPRESENTING VERTICES
187
2.) a triangle, 3.) a complete 4-graph, 4.) two triangles connected by an edge.
It is possible that there are no other cases. Clearly if all the components of
G are the above ones then G satisfies (2) with the sign of equality.
Proof. We use induction for v(G). (1) and (2) are trivial if v(G) <^ 1.
Let m>\ and assume that (1) and (2) holds for every G* satisfying v(G*) <m.
In what follows assume that G is an arbitrary graph for which v(G) = m.
We are going to show that (1) and (2) holds for G too.
We clearly ean assume that G has no isolated points. If G is not
connected, let its components be Glt ..., G- (j >. 2). Clearly v(G,)<rn (/ = 1,
...,/). Thus by our induction hypothesis and (1.2) it follows that G satisfies
(1) and (2).
Henceforth we shall assume that G is connected.
Assume first that G has a vertex P of valency 1 and let PQ be the
edge incident to P- There clearly exists another edge incident to Q say QQ'
[Q' ^= P). Omit the edge QQ' from G, and denote the graph thus obtained
by G'. Let R be a representing system of G' with a(R) = /j{G'). Clearly R
contains P or Q, hc-nee we can assume Q^R. But then R is a representing system
of G too, thus fi{G) = fi{G'). A simple argument further shows that s(G') = s(G)
(i. e. if a set of independent edges of G contains QQ', we can replace QQ' by
QP and obtain a set of independent edges of G'). From this and from ;r(G") =
= ti(G), v(G') = v(G) — l and from the induction hypothesis we obtain (1)
and (2).
Henceforth we are going to assume that the valency of even- vertex
of G is ^ 2.
If 7i{G) — 2e(G) = 0, then (2) clearly implies (1). Sext we show that
(2) implies (1) also if 7t(G) — 2e(G) = />(). Let P,- P\ (/ = 1, . . . , s; s = e(G))
be a maximal system of independent edges of G. Further put A" = {Plt ...,
Ps, P[, . . . , P's}, N = S—N {S denotes the set of vertices of G). [Ar] = G".
By our assumptions
(3) 1 ^ e(G) < v(G') < v(G) .
The vertices of N are independent (in G) and all of them have valency > 2.
Thus we have
v(G)^v(G') + 2j
and hence
2 ~~ 4 ~~ 4
which shows that (2) implies (1).
Thus it will suffice to prove (2).
Assume for the time being that n(G) — 2 e(G) = />0 and let us use
our above notations. Clearly if Ii is a representing system of G' then B\jX
represent all edges of G, thus /u(G) < /li(G') + /. Further clearly e(G') = s(G)
and n(G) = n(G') + j. These equalities together with (3) and (4) imply (2)
by the induction hypothesis.
Henceforth we can assume n(G) = 2e(G).
Assume first that G contains a path with the edges /\F2, P.2P3, /J3f4.
where P2 and P3 have valency 2 in G. Let G" = (G — P2) — P3. If G contains
63 REPRESENTATION OF GRAPHS

188
ERDOS—GALLAI
the edge P^\ put 67" = 67', if not G" is obtained from G' by adding the edge
P-iP^ to it. It is easy to see that
(5) n(G") = n(G) - 2, v(G') < v(G) - 2 , e(G") = e(G) - 1, fi(G") = //(67) - 1 .
(0) and our induction hypothesis implies (2).
Henceforth assume that G does not contain a path of the above type.
Let P( P'i (i = 1, . . . , s; s = e(67)) be a maximal system of independent
edges of G. By our assumptions the valency of both P,■ and P'f(i =1,...,6-)
are greater than one and by our last assumption the}' can not both be two.
Thus without loss of generality we can assume that the valency of Pt is
2^3 ((' = 1, . . . , s). Assume that for some i(l fS i < s) the sum of the
valencies of P, and P'i is greater than 5. Put 67* = (G — P,) — P's. Thus
(6) ti{G*) = 7i{G) - 2, v(G*) ^ v{G) - 5, e(67*) -= s(G) - 1, //(67*)^ //(&') - 2 .
(6) and our induction hypothesis proves (2).
Thus finally we can assume that the valencies of the vertices P( are
all 3 and the valencies of the vertices P'( are all 2(i = 1, , . . , s). But then
P'i and P'j(i =f= j, 1 < i 5i s, 1 g, j ^ s) can not be connected by an edge,
since otherwise G would contain the path with the edges PtP\, P\P\, P'\P\
where P't and P) having valency 2 in 67, but this contradicts our assumptions.
Hence we see that the vertices Pt{i =1,..., s) represent all edges
of G, which clearly proves (2).
Thus the proof of Theorem (1.14) is complete.
§ 2.
(2.1) e(67) ^ p is equivalent to the statement that //(67') ^ p for every
67' c67 with v{G') g> p + l.Thus the trivial relation //(67) g. 2e{G) can be
restated in the following form:
Assume that for every 67' c 67 with v(G') g p -f 1 we have //(67') < p.
Then //(67) < 2p.
It is now a natural question to ask: what can be said about //(67) if for
every 67' (z G with v(G') 5i q (q>p + 1) //(67') 5i pi Here we prove
Theorem (2.2). Let h ^. 2. Then there exists a smallest integer p0(h) with
the following properties : If p>p0(h) and G is a graph with n(G) ^ 2p — h -f- 3
which has no isolated points, and for every 67' c G with v(G') 5i p + h we have
ft{G') < p, then
(1) //(67) ?L2p — h.
Before proving our theorem we make some remarks.
1.) 2p—h is best possible. To show this let (¾ be a complete (2p—h)-
graph. The graph 672 is defined as follows: Its vertices are the vertices of Glt
another vertex P, and the vertices of a set ill (which may be empty, but
which does not contain P and the vertices of G^. The edges of 672 are the
edges of Gx and every edge which connects P with a vertex of Gx or ill. It is
easy to see that //(672) = 2p—h. Now- we show that for every 67' _c 672 (which
does not contain an isolated vertex) satisfying v(G') ^ p + h we have
//(67') ^ p. To see. this observe that if 67' does not contain P we have n{G') 5i
64 GRAPH THEORY

MINIMAL NUMBER OF REPRESENTING VERTICES
189
<, 2p—h and therefore by theorem (1.8) [i(G') ^ p. If G' contains P then
the number of the not isolated vertices of G'—P is not greater than 2p—h,
v(G'—P) ^p+ h — 1. Thus from theorem (1.8) fi{G' —P) g p — l or n(G') g p
which completes the proof.
We remark that in our example 2e(G2) equals one of the values 2p—h,
2p—h -j- 1, 2p—h + 2. This is not an accident, since if 2e(G) ^ 2p—h, then
because fi(G) ^ 2e(G) (1) trivially holds, equality only if 2e(G) = 2p—h.
Further a simple modification of our proof of Theorem (2.2) shows that if
2e(G)>2p — h + 2 we can improve fi(G) 5i 2p—h to /n(G)<2p — l where /
tends to infinity with p but is of much lower order than p, we can give only
very rough estimates for I = l(p, h).
2.) In (2.3) we shall show that if p is not "sufficiently large" compared
to h then (1) does not always hold. More precisely we shall show that if c is
an arbitrary constant and h>h0(c) then p0(h)>ch.
3.) If h = 2 our proof could be simplified considerably, and we can
show p0{2) =z 2.
Proof, of (2.2). (I) According to a well known theorem of Ramsay)
(see [8] and [5]) to every k there exists a <p(k) so that every G with n(G) > cp(k)
either contains a complete fc-graph or G has k independent points (i. e. /i{G)>k).
Clearly <p(Jc) > k.
We are going to show that
(2) p0(h) < h + y(cp(2h + 4)).
Clearly
(3) h + <p(<p{2 h-{-4)) > 3 h + 4 .
Our proof will be indirect. WTe are going to show that the following
conditions lead to a contradiction:
(4) G has no isolated point.
(5) h > 2.
(6) p>h + cp(cp(2h + 4)).
(7) n{G) > 2p-h + 3.
(8) IfG'cG and v(G') ^p + h then /i(G') ^ p.
(9) fi(G)>2p-h.
Let G satisfy the above conditions and put
7r(G) = n, e(G) = s.
It is easy to deduce from our conditions and (3) that for every h > 2
p > 11, n > 21, /j,(G) ^ 19, s > 9.
From (8) it follows that s ^ p. Let
p =z s + a.
65 REPRESENTATION OF GRAPHS

190 KUDOS— OALA^r
Clearly a 2i 0. (9) implies because of fi(G) g 2s that
(10) 2a ^ h — \ .
In the most important cases we will obtain the contradiction by showing
that G contains a subgraph G' whose components are triangles and edges
and for which v(G') < p + h and fi(G') = p + 1 (these facts contradict (8)).
Assume that such a G' has x + y components, x triangles and y edges.
Clearly
v(G') = 3j + y <I p -j- h and p{G') = 2x + y = p + 1.
Thus
(11) x ^ A- 1 .
Conversely if (11) is satisfied then because of (3) and 2x + y = p f 1 we
obtain y>0. G' further clearly satisfies
x + y g^ s ■
Thus from y = p -(- 1 — 2x
x ^ a + 1 .
(From (5) and (10) a + 1 1J-1.)
In the following we will only use the G' for which x and y takes on the
following values:
(12)
(13)
(14)
(15)
In case
In case
(II) Let e,
■>a ^ /i—3
2a ^ h — -2
^r,P',(i = i.
x =
x =
a* =
.r =
= 2a + 2,
= 2a + 1,
= 2a,
= a + 1,
, s) be a
y = «-(3« + 3)
j/ = «-(3« + 1)
y = s—(3a —1).
«, = «-(« + 1)-
maximal system of
edges. Those edges will bo considered, fixed during the rest of the proof, bet
M - U\ Ps), M' = {PI ■ ■ ■ . P's}. N = M\JJ\I', G. = [.V].
JV = S—X (S is the set of vertices of G.)
If X is non empty (i. e. n>2s), then put
X = {&. ..., Q„_2j}.
From the fact that s = e(G) it trivially follows that
(16) the vertices of X are independent,
(17) the edges /JA.Q, and P'k Qj (Pk £ M, P'k^M',i=j= j, {Q„ Qi} C X) can not
both occur in G,
(18) if !\Qk and PjQ, are in G (i =J= j, k=f=l, {Ph P/} c M, {<?,, <?,} c X),
then i3,'/3'- is not in G.
66 GRAPH THKORY

MrSI.MAL .VUMBEIl OF REPRESENTING VERTICES
191
From (4) and (16) we obtain
(19) every vertex of A7 is incident to A7AT-edges.
From (17) and (18) it follows that
(20) if both Pt and P't{\ g i g s) are incident to an iVJV-odgo then PJ^
and these two ATAT-edges form a triangle, (this means that there can be
only one ATA'-edge incident to f, and P'i).
(Ill) We prove that
(21) ji{Gc) g 2A —3« —2.
If J\T is empty then Gs = G, n = 2s and because of (9)
(22) Ji(Ge) = n — /j{G) < 2s - (ip—h + 1) = h— 2a-l.
In this case from (22), (5) and (10) follows (21).
For the rest of (111) we assume that N is non empty. Put G0 = \X, A'].
G0 is an even graph which, because of (19), is non empty. Thus by the theorem
(1.11) of Konig
(23) p(G0) = e(G0).
Let (\, ..., e'So(s0 = e(G0)) be a maximal system of independent edges
of G0. By (17) we can assume that
<*; =-/',-<?,■ (»' = i *«.)■
Put Al[ = {P[ , . , . , P'Sa}. By (18) the vertices of JP1 are independent
and because (20) if P\ £M[ then the only vertex of A' with which P\ can be
connected by an edge is Q,. Denote by M'2 the vertiees of M[ which are
connected with the corresponding Qt and put a(M'2) = t.
Assume t ;> a -f 1, without loss of generality we have M'2= {P[, . . . , P't}.
bet 4- =: 1\ P'iQ^i = 1,...,/). Then the triangles zl,(/ = 1, . . . , a -(- 1)
and the edges <'a+i, ■ ■ ■ , <% form a subgraph G' of G whose existence because
of (15) contradicts (8).
Assume next t g a. The, vertices of i\I's = M'^ —M'2 arc independent
(assuming that M'3 is non empty) and the only edges incident to them belong
to GE. Therefore the vertices of A^ = M (J {M' — M'3) represent the edges of
G. Thus
fi(G) g a(AT1) = 2s - (s0~t) g 2p - (a + *„)■
Thus from (9)
(24) sQ < h—a—l.
Let Ra respectively Rt be a representing system of G0 respectively GE
having minimal number of elements. R0\jRe clearly represents G and thus
by (23) s0 + ii(GE)>/.i(G). Thus from (9) and (24) we' obtain fi(Ge)>2p — 2h ~
+ a + 2. Thus by (1.3) (1) we obtain (21).
From now on the, triangles zl, and the sets M[, j\['2 will not occur any
more. Thus we will vise these symbols and the symbols used for their vertices,
for other purposes.
67 REPRESENTATION OF GRAPHS

192
ERDfiS—GALLAI
(IV) Now we shall show that both [if] and [At'] contain suitably
related complete graphs having sufficiently many vertices. From (6) and
(10) we have
?i{[M]) =- s>cp(cp(2h + 4)).
By /2([M]) < ji{Ge) we have from (21)
/I([M]) <2h + 4 <I cp{2h + 4).
Thus by Ramsay's theorem there is an M1 c M so that [ifj] is complete and
^([iYj]) = 99(2¾ + 4).
Let
M1 = {Plt ..., Pu}, M[={Pl ..., P/,} (u = <p(2h + 4)).
By (21) //([if^]) < 2¾ + 4. Thus by n([Mi)) = <p(2/i + 4) we obtain from
Ramsay's theorem that there exists an M'2 c M\ so that [M%~\ is complete
and n{[M^]) = 2h + 4. Put
m; = {PJ,... ,P2h.4}.
(10) implies 3(a + 2)<2h + 4. Thus since [ifj] and [ifg] are complete,
the triangles
4 = 4-2^,-^3, , ^ = ^3,-2^3,-1^3, (i = 1, . ■ . , a + 2)
are all subgraphs of G.
By (10) 2a 51 h — 1. Now we distinguish three cases, 2a < #—3, 2a =
— A — 2 and 2a = A —1.
(V) Assume 2a 5i A — 3. Then the pairs of triangles (Ah A'f) (( = 1, ... , a+1)
and the edges e3a+4. . . . , es form a subgraph of G which by (12)
contradicts (8).
(VI) Assume next 2a = h— 2. By (7) n ^ 2s -[- 1, thus N is non empty.
By (19) there are AW-edges. Now the following statement holds:
(25) Any two vertices of N which are not incident to AW-edges are
connected by an edge.
For if two such vertices would not be connected, the other vertices
of AT would represent the edges of G. Thus n <i 2s —2 = 2p^h, which
contradicts (9). Thus (25) is proved.
Assume first that there is a j (1 <i j <; s) so that both P; and P'j are
incident to A7AT-edges. By (20) the vertices of these NN-edges which arc in A"
must coincide. Denote this common vertex by Qv Consider the triagles (Ait A';)
(i = 1, . . . , a 4- 2) defined in (IV). We can find a of these pairs in such a
way that none of them should have a common vertex with e-. These pairs
of triangles together with the triangle PjP'jQ1 and together with all the
edges e, (1 <L i <1 s, i =j= f) which have no common vertex with our a triangle-
pairs form a subgraph of G whose existence by (13) contradicts (8).
For the rest of part (VI) we can assume that no vertex of if' is connected
(by an edge) to a vertex of N. Thus we obtain by (25) that [M'] is a complete
graph. No we prove the following statement:
68 GRAPH THEORY

MINIMAL NUMBER OF REPRESENTING VERTICES
193
(26) Assume that G contains an edge PjQi (1 5a j 2s s, Q, £N), assume further
k =/= /(1 51 k <; s), then the edge P'jPk is not in G.
If (26) would bo false, then since [31'] is complete the triangle P'jPkP'k
is a subgraph of G. From the triangle-pairs (/1,-, A[) (i = 1, . . . , a -{- 2) we
can again find a of them so that none of them have, a common vertex with
e,j or ek. These triangles together with the triangle P'jPkP'k, the edge PjQh
and all the edges p,(1 <. i ^L s, i =/= j, i =f= k) which have no common vertex
with our a triangle-pairs form a subgraph of G whose existence by (13)
contradicts (8). __
We, now show that every vertex of 31 is incident to AW-odges. To see
this observe that if Pk(l <i k <. s) would be, a vertex which is not incident
to an AW-edge, then by (25) this would be connected to every vertex of 31'.
Among these vertices there clearly is a vertex P'j so that the corresponding
Pj is incident to an AW-edge, which contradicts (26).
From (18) and from the fact that [31'] is complete it follows that the
AW-edges incident to the vertices of M are all incident to the same vertex
(¾. Therefore by (19) N == {(¾}. From (26) we further deduce that the only
vertex of 31' to which P ■ can be connected is P'j(l 51/51 s).
Next we show that no two vertices of 31 are connected. To see this
assume that G contains the edge PjPk{j =/= k, {F-, Pk} cl), Choose a of
the triangle-pairs (/1,, A\) (i = 1, . . . , a -j- 2) so that none of them contain
a common vertex with the edges e;- and ek. These triangle-pairs together
with the triangle (¾ Pj Pk and together with all the edges e,(l 51 i 51 s, i =/= j,
i =/= k) which hove no common vertex with one of our a triangle-pairs form
a subgraph of G which by (13) contradicts (8).
From what has been said it follows that the set B = 31'[JN represents
G, further a(B) = s -j- 1 <1 2p^h and this contradicts (9).
(VII) Finally assume 2a = A—1. Then by (7) n ^ 2s + 2, or A' contains
at least two vertices. Every vertex of A7 is incident to AW-edges. For if AT
would have a vertex which is not incident to an AW-edge then the other
vertices of A7 would represent G, their number is 2s — 1 = 2p — h which
contradicts (9).
By (19) and (20) there is a / and k{\ <1 ) < «> 1 < k 51 s, / =/= k) for
which the triangles A' = Q-^PjP'j and A" = QiPkP'k are subgraphs of G.
We now select from the triangle-pairs (/1,, A't) (i = 1, . . . , a -\- 2) a —1
pairs so that none of them contain a common vertex with e;- or ek. These pairs
together with A', A" and with all the edges e,(l 51 ; 51 s, z =/= j, i =/= k) which
have no vertex in common with the seleeted pairs form a subgraph of G.
By (14) this contradicts (8).
This completes the proof of Theorem (2.2).
Now we show that if c (c> 1) is any constant and h>hQ(c) then p0(h)>ch.
More precisely we shall show
Theorem (2.3). Let c(c>l) be any constant, then there exists an h0(c)
so that for every h>h0(c) there exists an integer p>ch and a graph G satisfying
the following conditions :
1.) G contains no isolated vertex.
2.) 71(G) ^ 2p—h + 3.
69 REPRESENTATION OF GRAPHS

194
ERDOS—GALLAI
3.) For every G' c G vhich satisfies v(G') <; p + h u-e have /n{G') < p.
4.) fi(G)>2p—h.
Proof. (I) A theorem of Ebdos ([3], p. 34. (4)) implies that to every
c(r>l) there is an n0 (c) so that for every v>n0(c) there exists a graph G,
having no isolated vertices, for which
3 n
(1) ti(G) = «, fi(G) <—■ -
and for which
(2) every circuit contains more than 28c vertices.
We are going to show that
(3) A0(c) = max|28,W°^i
satisfies the requirements of our theorem.
Let h>h0(c), and choose p so that
(4)
Let further ;; satisfy
(5) 2p
Let G be a graph having no isolated vertices and satisfying (1) and (2)
with the above choices of c and n. We shall show that G satisfies the conditions
1.), 2.), 3.) and 4.) of Theorem (2.3).
Conditions 1.), 2.) and 4.) are clearly satisfied. Thus to complete our
proof wo only have to show that 3.) is satisfied.
(II) Lot G' c G, v(Gr) < p + h. We .shall prove that
(6) fi(G') < p.
To prove (6) wo define by recursion for every k >. 0 a subgraph Gk of
G' as follows: GQ = G'. If Gk has no vertex of valency > 2 we put Gk^1 = Gk-
If Gk has a vertex of valency > 2, let Pk such a vertex and put Gk^ = Gk^Pk.
Since G was finite there is a smallest I' say I so that Gl+1 = Gv Gl has no
vertex of valency greater than 2, and wo obtained G, from G' by the omission
of I vortices of valency ^ 3. Thus from (4). (5) and r(G') < p -f- h wo obtain
9
(7) .-r(G,) = n - Z < 2 p p - I,
14 c
(8) v(G,) =- v(G') -3Z <p+ P - 3/.
c
Since all vortices of Gl have valency <, 2, the components of G, can
only bo circuits, paths and isolated vortices. Assume that there arc j circuits
ch < p < — ch
6
3 6 3
— p < n < 2 p p .
4c 7 4c
70 GRAPH THF.ORY

MINIMAL NUMBER OF REPRESENTING VERTICES
195
among the components of Gv By (2) every circuit of G, contains more than
28c vertices, thus by (7)
or
(9)
/ • 28 c < n(G,) < 2p
V
1 <
Uc
The edges of a circuit or path of k vertices can always be represented
by [h/2] or [Jc/2] -+- 1 vertices respectively. Thus from (7) and (9)
^(0,) ^-^(0,) +j < p - -^- - ^- .
2 4c 2
The edges of G' which do not occur in Gl we represent by the I vertices
which do not occur in G,. Thus we obtain
/i(G') < p(G,) + I < p - ^- + -- .
4c 2
Thus if p/(4c) ^ 1/2 we obtain /x{G') < p. If p/(4c) < 1/2, then by ^(£,)^1-(0,)
and by (8) we have
fi(G') ^ rfG,) + 1 <p,
which proves 3.) and thus the proof of Theorem (2.3) is complete.
§ 3.
(3.1) In connection with the general problem raised in (2.1) the following
questions can be asked:
Does there exist to every p a smallest f(p) so that if G has the property
that for every G' c G with v(G') < f(p) we have /i(G') S p, then /n(G) < p\
This question can be answered affirmatively. From the Theorem (3.5)
we easily deduce
[2 v 4- 21
Theorem (3.2). Assume that for every G'czG with v(G') <
ire have fi(G') ^ p. Then fj(G) < p
1-2P+2
The estimate f(p) ^
Conjecture (3.3).
2
seems to be a poor one.
t(P) =
p+2
2
We can prove our conjecture for p ^ 4 (see the remark 1. made to
Theorem (3.10)). The example of the complete {p -+- 2)-graphs shows that
ip 1. -i\
1 since if G is a complete (p -+- 2)-graph for every proper sub-
lip) >
graph 6" of it we have ft{G') < p and v(G') ^
(P+2\
1, but fi(G) = p+ 1.
13*
71 REPRESENTATION OF GRAPHS

196
EltDSS—GALLAI
(3.4) Now we ask the following question:
Assume that for every G' c. G satisfying ti(G') <. q we have fi(G') <I p
what upper bound can be given for ,u(G)?
If q = 2p 4- 1, p{G) can be arbitrarily large. To see this consider the
following even graph G*: The vertices of G* are /q, . . . , Pm, Qlt . . . , Qn
and its edges are PjQj(i = 1, . . . , m; j=l, . . . ,n). Clearly fi(G*) = min (m, n),
but a simple argument shows that for every G' c G* with n(G') _g
< 2p 4- 1 we have /x(G') ^ p. Here we have for m = n ti(G*) = 2n,
/i(G*) = re. The more complicated examples given in [2] and [3] show that
a graph G with n(G) = «, fi(G)>n—o(n) exists so that for every G' C.G
with 7t(G') < 2p + 1 we have ,u(G') < p.
On the other hand we are going to prove that for q = 2p -\- 2 we have
/j(G) < p (which is clearly best possible).
Theorem (3.5). Assume that for every G' c G with ti(G') ^ 2p 4- 2 we
have /,(G') < p. Then ji(G) ^ p.
We will prove Theorem (3.5) in § 4. It is curious to observe the sharp
change between q = 2p 4- 1 and q = 2p -)- 2. This change can be seen also
in the order of magnitude of the number of edges.
If q = 2p 4- 2 (3.5) immediately gives
(1) v(G) < P(tz(G) - 1) .
(1) is best possible. To see, this let the vertices of G be Plt ..., Pp ,
Qn-p (the set of the Q's may be empty). The edges of G connect each
ft,
of the vertices Pv
and v(G) = p (n — 1).
P with all the other vertices of G. Clearly /i(G) = p
If q = 2p 4- 1 then G* shows that v(G) can be as large as
71(G)
I7i(G)l
for m =
2
this is best possible. Here we have
Theorem (3.6). Let n(G) :> 4(p
rr(G') ^ 2p 4- L fi{G') < p. Then
v(G)<
For sufficiently large values of 71(G)
1). Assume that for every G' c G w#/i
:r[G)i2
2
Disregarding the condition 71(G) ^ 4(p -f 1), for p = 1 this theorem
is identical withTuBAN's theorem ([9], p. 26.) for / = 2. The proof of Theorem
(3.6) uses this special case of Tuban's theorem. We supress the details.
Perhaps we can digress for a moment and call attention to the following
interesting class of problems. Let 71(G) = n and assume that for every G' czG
with tt(G') ^ q we have fi(G') < p. Denote by g(n, p, q) the maximum value
of i'(G). We wish to determine or estimate g(n, p, q). The cases q ^ p 4- 1
are trivial sines there trivially g(n, p, q) =
q^2pJr 2 implies by (3.5)
a(n< P- <7) == p(n —1)- The interesting range is p -\- 2 ^ q ^, 2p -\- 1- The case
q = 2p 4- 1 is settled by Theorem (3.6). 9 = p -\- 2 means that G docs not
contain a complete (p + 2)-graph and is thus settled by Turan's theorem.
72 GRAPH THEORY

.MINIMAL NUMBER OF REPRESENTING VERTICES
197
The determination of g(n, p, q) for general p and q seems to be a difficult
problem and we made very little progress with it. Ebdos can show that for
sufficiently large n
'In 4- IW
(1) g(n,p,2p)= L^^ _!.
The methods required for the proof of (1) and Theorem (3.6) are quite different
than those used in this paper.
It is easy to see that conjecture (3.3) and theorem (3.5) can be restated
in the following form:
(3.7) Conjecture. If //(G) > p then there is a G' c G for which fi(G') > p
and v(G') <iP+2\.
(3.8) If n(G)>p then there is a G' cG for which /j(G')>p and n(G') g,
< 2p -f 2.
A graph G is said to be edge-critical if it has edges and for every G' c G,
fi(G') < ii(G).
G is point-critical if it has edges and for every G' c G for which n(G')<
< 71(G) we have n{G') < fi(G).
Clearly every G which has edges has .subgraphs G' which are edge-,
respectively point-critical and for which /z(G') = fi(G).
(3.7) and (3.8) are substantially equivalent to the following statements:
Conjecture (3.9). For every edge-critical graph G we have v(G) < \
Theorem (3.10) For every point-critical G we have n(G) 5i 2 (i(G).
The proof of the equivalence is left to the reader. The proof of (3.10) will be
given in § 4.
Remarks. 1.) Conjecture (3.9) holds for fi(G) <L 4.
2.) In § 4 we shall show that in (3.10) equality can hold only if 2e(G) =
= 71(G).
3.) From (3.10) and from the fact that an edge-critical graph is also
point-critical we obtain that for an edge-critical graph G wc have v(G) <
(2/*(G)|
2 I '
<
§ 4.
In this § we are going to prove Theorem (3.10) (and thus also Theorem
(3.5)).
Our definitions trivially imply
(4.1) A point-critical graph can have no isolated vertices. If G is
nonempty and not point-critical, then it has a vertex P with fi(G-P) = /"(G).
73 REPRESENTATION OF GRAPHS

198
BRDOs—GALLAI
(4.2) Let S he the set of vertices of G, further let
(1) iSf,■ c S (i = 1, . . . , k ; k > 2); S^Sj = 0 (i =f= j, i, / =- 1, . . . , A") and
U S,■ = S .
Let B be a set of /x(G) vertices which represent every edge of G. Clearly
B f) S/ represents all edges of G, = [£,], thus
k
(2) 2 t*(Gi) ^ H(G) ■
i=i
If there exists a decomposition of 5 into now empty subsets £, satisfying
(1) for which
k
>V(G,)=^(G)
holds, then we say that G is decomposable and we call the set {G1; . . . , Gk)
a decomposition of G. The following two statements trivially follow from our
definitions:
(4.3) If G is decomposable we have n(G) > 1 and G has a decomposition
{Glt ..., Gk} where all the G,(l < i. < k) are indecomposable.
(4.4) If (the non-empty) G is not connected, it is decomposable.
(4.5) If 7i(G) > 1 and G is indecomposable, then G is point-critical.
Proof. If (4.5) would be false, there would exist by (4.1) a P £ Ssothat
for Gx = G — P we would have /*(G1) = /<(G). Clearly neither G1 nor G2 = [P]
are empty and /*(G2) = 0. Thus fi{Gx) + fi(G2) = /i(G), but then {G1, G2}
would be a decomposition of G.
(4.6) d-,et G be point-critical and {G1 Gk} a decomposition of G. Then
the Gl- (i = 1, . . . , k) are also point-critical.
Proof. Assume say that G1 is not point critical. Since G1 is non emptv
it has by (4.1) a vertex P so that fi^—P) = //(G^. But then by (4.2) (2)
k k
fi(G - P) ^ ti(G1 - P) + 2V(«,) = 2V(G,) =■ /i(G),
(=2 i = l
which is a contradiction since G was assumed to be point-critical.
Theorem (4.7). If n(G) > 1 and G is indecomposable, then
n(G) < 2(,(G)
where equality stands only if G consists of a single edge.
Proof. (I) Because of (4.4) G is connected and therefore it has no
isolated vertex. If G consists of a single edge :t(G) = 2p(G) trivially holds.
Henceforth we assume n(G) > 2. Let B be a set of n(G) ~ r vertices which
represent every edge of G. Put S — B = T. Clearly neither B nor T are empty
74 GRAPH THEORY

MIXIMAL NUMBER OP RiSPRESEXTIXci VERTICES
199
and the vortices of T are independent. Thus every vertex of T is incident
to TR-edges.
Consider the graph G' = [R, T]. Clearly G' is even and contains edges.
Put fi(G') = r'. Clearly 0 < r' g r.
We are going to show in (II) that the only representing system of G'
with r' elements is T. This easily implies :r(G) < 2fi(G), since R is a representing
system of G' and therefore, r > r', or n(G) = r -\- r' < 2r as stated.
(II) Let R' he any representing system of G' which has r' elements.
R' is non empty. Put
R' n R = R-l, R' f)T = Tv
Assume that Rx is empty. Then from R' (zT and from the fact that
every vertex of T is incident to T_B-edges it follows that R' = T.
Thus to complete our proof we only have to show that the assumption
cl{R{) = rx > 0 leads to a contradiction.
By theorem (1.11) of Konig G' contains r' independent edges, sav
(\ = PiP'i (P^R, P'i£T, i = l. . . . , r'). Each of these edges is incident to
exactly one vertex of R' ■ Denote by c\, . . . , eri the edges incident to the
vertices of R1 and put {P[, . . ._, P'ri}'= T2. We evidently have 1^0^ = 0 .
Let R — R1 = RV T—T2 = T2. G' clearly does not contain an /T^-edge.
Put
G1 = [R, U T2). G, = [H, U T.,\
a) Assume first that G2 is empty. Then
i\ = /■' = ;■ = o.(R) = a(T).
Since x(G) > 2 we have r > l.ThcnifG3 = [{Plt P[}] and G,t ^ [S — {P^P'A]
we, have //(G3) = 1 and /i(G4) > r — 1 (since 6'4 contains e2, . . . . er). Thus
{G3, G4} is a decomposition of G and this is a contradiction.
b) Assume, now G2 non empty. The vertices of R1 represent all edges
of 67j and since G1 contains the edges e1( . . . , eri we obtain
(1) /.(GO = /v
/t(G2) 2: /■ — r1 is impossil)lp since {G1; G2} would then be a decomposition of
G. But fi(G2) < r—r1 is also impossible, for in this case if R2 would be, a
representing system of G2 having /i(G2) elements, then R1 (J R2 would represent
all edges of G and thus
/«(G) g /.(Gi) + MG2) < r,
which is impossible. This completes the proof of (4.7) .
Finally we prove (3.10) and our remark 2.) belonging to it.
(4.8) If G is point-critical then n(G) g 2fi(G), equality can hold only if
■2e(G) = :i(G).
Proof. If G consists of an edge, (4.8) it trivial. We can therefore assume
that 7i(G) > 2. If G is indecomposable, then by (4.7) n(G) < 2/i(G). Assume
now that G is decomposable and let {G1: ..., Gk} be a decomposition of G
75 REPRESENTATION OF GRAPHS

200 BRDOS—GALLAI
where all the G, (1 < i < k) are indecomposable. By (4.6) G, (i = 1, . . . , k)
is point-critical and thus
k k
ji(G) = 2n(G,) -^ 2 .2^(^,) = 2 KG) ■
Equality occurs if and only if every G,. (1 <; i 5i k) consits of a single edge-
In this case the edges of G1, . . . , Gk are independent, which implies 2e(G) =
= .-r(G).
§ 5.
(5.1) In this § we generalise our problems to "graphs of several
dimensions" i. e. to k-tuples. Let S be a set (its elements we will call points) and
II a certain finite set of ^-tuples formed from the elements of S. (For k = 2
H was G and the points of S which occur in the 2-tuples of H, i. e. in the
edges of G were called the vertices of G. This G has no isolated vertices.)
Denote by n(H) the number of elements of S which occur in the ^-tuples
of H and by v(H) the number of ^-tuples of H. If B c S and if every £-tuple
of H containts at least one point of B we say that the points of B represent
H or that B is a representing system of H. Denote by fj,(H) the minimal number
of points which represent H.
Generalising the problems considered in (3.1) and (3.5) (i. e. in (3.7) and
(3.8)) we wish to determine the smallest values f{k,p) and g{k,p) whieh
satisfy the following conditions:
Every H for which ft(H) > p contains a subset H' and a subset H"
for which "fi(H') > p, fi{H") > p and v(H') < f{k,p), n(H") < g(k,p).
We now obtain upper estimates ior f{k,p) and g(k,p) further wo
determine f(k, 1) respectively g(k, 1) for every k > 2.
Theorem (5.2)
p
W,p)< 2 v.
Proof. For p = 0 our statement is trivial. Assume henceforth p 5; 1.
Let H be an arbitrary finite set of fc-tuples with /j.(H) > p and let t0 =
= {-P1; ..., Pk} be an arbitrary element of H. Put H0 — {t0}. Since (i{H) >
> p > 1 a single element can not represent H and therefore to every Pit (1 ^
5a h S k) there is a tti in H which does not contain Pit. Let tti — {Pt^, ■ ■ ■ ,
Piik) {i1 =1,..., k) and put H1 — [tx, ..., tk). If p ;> 2 we need at least
three points for the representation of H and therefore we can find to every
pair of points i\, i\,-8 (1 < ix < k, 1 5S. i2 < k) a £-tuple ttiii — [P,,,-^ ! j =
= 1, . . . , &} which does not contain Pti and i3,,,-,- Put //2 = {^,/, | ^.½ =
= 1,..., fc}. Continuing this process for every j (1^/^. p) we obtain the
fc-tuples ti ,-. (of H) and the points /** ijt. and the sets of ^-tuples
Hj ft, ...,'i), i]+1 =1,..., *). Put
#' - U //,
/ = 0
76 GRAPH THEORY

MINIMAL NUMBER OF REPRESENTING VERTICES
201
Since v(Hj) g k> we have v(H') < jj? kj. Now wo show n(H') > p. To
see this let B be a representing system of H'. B must contain an element
of t0 say Px. By our construction Px^t-^, thus B must contain an element of
ij say P12- If p > 2 then P1 and P12 are not contained in i12 and B must contain
an element P123 of ^12- This process can be continued (p -f 1) times and wo
(B) > V
obtain that B contains the elements Plt P12, • • • , P-i* ■ ■ ■ nJ.i or a
as stated.
Theorem (5.3) /(¾. 1) = ¾ + 1 (k > 2).
Proof. Bv (5.2) /(¾. 1) ^ £ + 1. The following example shows f{k, 1) =
= k + \. Lot's = {P0, .... Pk}. H consists of tho (k + 1) ^-tuples formed
from S. Here (x{H) > 1 but for every H' C H [j,(H') = 1.
(5.4) In general we know little about the value of /(¾. p). Conjecture
/» + 2
This and (5.3) might permit us to eonjoe-
(3.7) states that /(2, p)
ture f(k, p)
of all the
In any case f{k, p) >
(p + k)
k
. To see this let H consists
p + k\
k
k )
p + 1, but a simple argument shows that for every H' <zH ^{H^gp, which
(p+k
^tuples formed from p + k elements. Clearly ft(H)
proves /(¾. p) >
A trivial argument shows that g(k,p) <. kf(k,p). Thus we have
p+i
Theorem (5.5). g(k,p)g > k' [k > 2) .
/= i
We know only a little more about g(k, p) than about /(¾. p). (3.8) states that
g{2,p) — 2p -f- 2. Further we have
Theorem (5.6). g{k, 1)
(¾ 4- 2)2
(¾ > 2)
Proof. (I) First we show g(k, 1) < [(k + 2)2/4]. To see this let H be a
set of ^tuples for which /i{H) > 1, let further t' and t" be two ^tuples of
H for which a(t' C)t") = a is minimal.
If a = 0, then putting H' = {t', t"} we have ix{H') > 1 and n{H') =
= 2k < [(¾ 4- 2)2/4]. Thus we can assume a>0. Put i'n*" = {A. • • • . pa)-
To every P( (1 <; i g a) wo can find a i,- of H which does not contain Pr
Put H' = {t', t", t1: . . . , ta}. Clearly /i(H') > 1. Further for every i {\ g
gi < a)
(1)
a(t'nti)>a, a{t" r\tt) > a, a(t' f\t" f\tt) < a — 1.
Denote by ai the number of elements of tt which do not belong to t' f)t". We
have by (1)
«(*,-) — a(t'nti) — a(t"ntt) + a(t'nt" ntt) g k
77 REPRESENTATION OF GRAPHS

2 02 EEDOS—GALLAI
Thus
a
.t(#') ^a(t' U t") + ^ai<2k~-a+ a(k — a — 1) =
-= 2k + a{k — 2 — a) ^ 2 k -f |--~- " = — -
I 2 / 4
which proves our assertion.
(II) To show g(k, 1) > [{k + 2)2/4] put [fc/2] = I and Af = (7^,, ■ ■
where <? = Z + 1 if & is even and q = Z + 2 if & is odd. Let further
and
Mt = M - {P,}, M\ = {Pn ,...,!>„}. t, = M,- U -MJ 0" = 1.
(the P's with different indices denote different points).
Here we have a(t,) = k(i = 1, . . . , I 4~ 1) and
z(H) = q + qI
4
Clearly /i(77) > 1, but for #,. = H - {tt} (i = 1, . . . , Z + 1) we have
u(H,) = 1 since P, clearly represents 77,-. This completes our proof.
(Received November 25, 1960.)
REFERENCES
[1J Berge, C: Theorie des grapltes et ses applications. Paris, 1958.
[2] Erdos, P.: "Remarks on a theorem of Ramsay." Bull. Besearch Council of Israel
Section F 7 (1957) 21-24.
[3] Erdos, P.: "Graph theory and probability." Canadian Journal of Mathematics 11
(1959) 34-38.
[4] Erdos, P. —Gallai, T.: "On maximal paths and circuits of graphs." Acta Mathe-
matica Academiae Scientiarmn Hungaricae 10 (1959) 337—357.
|_5J Erdos, P, —Szekeres, G.: "A combinatorial problem in geometry." Compositio
Math. 2 (1935) 463-470.
[(!] Gali.ai, T.: "tJber extreme Pankt- und Kantenmengen." Annates Universitatio
Scientiarwn Budapestiensis de Bolando Eotvos Xominatae, Sectio Mathematica
2 (1959) 133-138.
| "J Koxig, D.: T/ieorie der endlichen und unendlichen Graphen. Leipzig, 1936.
(8] Ramsay, F. P.: Collected papers. 82—111.
[9] TurAn, P.: "On the theory of graphs". Colloquium Mathematicum 3 (1954) 19 — 30.
78 GRAPH THEORY

ON THE REPRESENTATION OF DIRECTED GRAPHS
AS UNIONS OF ORDERINGS
by
I'. EKDOS unrl MOSEK 1,.'
Introduction
Consider an mxn matrix in which each row consists of a permutation
of the integers 1,2, ..., n. Such matrices will be called ^-matrices (thev
really should have been called mxn _R-matrices, but where there is no danger
of confusion we omit the rax n). Corresponding to such a matrix R we define
aa oriented graph on the vertices 1,2, . . ., n, in which there is an edge oriented
from i to j (notation: i -> j) provided i precedes j in a majority of the rows
of B. If i precedes j as often as j precedes i the vertices i and j are not joined
by an edge. It has been known for some time [1] that every directed graph
in which every pair of vertices are joined by at most one oriented edge can be
realized as a graph associated with some i?-matrix in this manner. The
principal object of this paper is to obtain relatively sharp estimates for the smallest
number m(n) such that every oriented graph on n vertices corresponds to some
mXn matrix of the type described.
This as well as some related problems which wo will treat arise from
questions concerning methods of combining individual transitive preferences
on a set of alternatives by means of majority decisions. Thus we may think
of the rows of the matrix B as representing orderings by individual voters,
of a set of n candidates 1, 2, ...,■« in order of preference. Although caeh
voter thus expresses a sot of transitive preferences, the majority opinion
need not be transitive and indeed we will prove that every preference pattern
(ties permitted) may be achieved by no more than cx njlog n voters, (¾ a fixed
constant), i.e. m(n) <L cx nj\og n. On the other hand it was shown in a relatively
simple way by Stearns [2] that some preference patterns on n candidates
cannot be schieved by c2 njlog n voters (where c2 is another fixed positive
constant) so that m(n) > c2 n\ log n.
In § 1 we consider the following problem: What is the largest number
f(n) such that every oriented graph on n vertices in which every pair of
distinct vertices is jointed by a directed edge has at least one subgraph of /(■»)
vertices in which the orientation is transitive, i.e. in which i->j and j->k
implies i —>■ fc. Our result here is that f{n)< 2[log2w] +1. Stearns has
shown that/(«) ^ [log2 ri\ +1.
In § 2 we will develop some lemmas concerning oriented graphs which
can be represented by 2xn .R-matrices. In the voting terminology this means
1 University of Alberta, Edmonton, Alberta, Canada. This paper was written
while P. Erd6s was visiting the Universit}' of Alberta in Edmonton.
125
79 REPRESENTATION OF GRAPHS

126
EKDfiS —MOSKU
that we study the preference patterns of candidates that can be achieved by
a pair of voters — wo will call them a eouple. The point in considering such
pairs of voters is that by balancing their transitive preferences in a certain
way the pair of voters can achieve a preference between certain pairs of
candidates in the manner in which these pairs are to be preferred by the
majority, while with respect to all other pairs the profeicncos of the couple
cancel one another.
In § 3 we relate the graph theoretic lemmas of § 2 to the problem of
estimating m(n) and obtain the result
ci Wl°g n > wi(w) > (■■> /i/log n .
We conclude with a number of unsolved problems.
§1-
The problem discussed and partially solved here is independent of our
main problem the estimation of m(n). By a complete oriented graph or
complete paired comparison we mean a graph in which every pair of vertices is
joined by one oriented edge. As mentioned in the introduction, Steaens has
proved that every such graph on n vertices contains a subgraph on [log2 v] -f- 1
vertices on which the orientation is transitive. For the sake of completeness
we sketch the relevant argument: Consider a complete oriented graph on ;;
vertices. Let w(i) be the number of edges oriented away from vertex i. Relabel
the vertices so that w(\) 2l w(2) 2l . . . ^ w(n). Since every pair of vertices
n I 71 \ lb 1
contributes 1 to E w(i) we have "V/Wi") = so that w(l) 2t . To con-
S W \2) 2
struct a transitive chain of [log2«] -f- 1 vertices place vertex 1
at the beginning of the chain and use induction to find in the subgraph
n — 1
of vertices which arc joined to 1 by edges oriented avay from 1. a
yi \W
log2 + 1 vertices. These together with the ver-
1 2 J
tex 1 form the required set.
To obtain a lower bound for the largest transitive set in some
complete oriented graph on n vertices, assume that every such graph has a
transitive subset of k elements. Now such a transitive subset must be one
71 \
subsets of k of the vertices, and any one of these subsets in order
to be transitive, can be ordered in k\ ways. Having fixed the transitive
subset (including its order) we observe that such a transitive subset can appear
transitive subset of
of
in exactly 2^2' ^2' complete directed graphs, since the complete graph is
Ik
2
171 I
edges.
■ 2l 8
of which have alreadv
been fixed. Finally, since each of the 2v2' oriented graphs has a trar>itive
subgraph of k vertices we have
["|*!2(?H^2(*n>,
SO GRAPH THEORY

ON A JtEl'KKSKNTATlOJf OF DIRECTED OKAI'H.s 127
and using < nk/k ! we are lead to
\k)
log 2
which completes the proof of
Theorem 1.
Llog2H] + l^/(n)^2Llog2«] + l.
Wo remark that /(7) = 3. That /(7) ^ 3 follows from the left hand
side of the inequality above while /(7) gL 3 is obtained by considering the
directed graph on 1, 2, . . ., 7 in which i -> j iff the number i — j is a quadratic
residue (mod 7). We would like to call the attention of the reader to the fact
that we have been unable to disprove the conjecture that/(n) = [log2n] -j- 1.
In particular we cannot decide if /(15) = 4.
§ 2.
In what follows G will denote a directed graph in it vertices, not
necessarily complete, i.e. each pair of vertices is joined by at most one directed
edge. The graph H will be called bipartite and unidirected if the vertices of
H can be split into two disjoint subsets A and B (one of which can be empty)
such that every vertex of A is joined to every vertex of B in the direction
from A to B and no other edges exist in H, Suppose the vertices of A are
av . .., ak and those of B are bu b2, ■ ■ . ,b, {k -\-l = p). A and B will be called
the levels of our subgraph (A the top level, B the lower level).
Lemma 1. A bipartite and unidirected graph H can be represented by a
2 X p R-matrix.
Proof. Consider the matrix
' axa2 . . . akb^b2 . . . bl j
[akak^1 . . . ax b} b,^ . . . bj
The graph induced by this matrix has edges directed from each veitex
in A to each vertex in B. However there are no edges joining vertices of A
to vertices of A (or vertices of B to vertices of B) since for it j < k, a, precedes
0-j in one row and follows it in the other.
Next, if a graph H can be decomposed into disjoint bipartite and
undirected graphs it will be called bilevel.
Lemma 1 can be generalized to yield
Lemma 2. A bilevel graph H with n vertices can be represented by a 2 x »
R-matrix.
Proof. If the top level of H consists of the disjoint sets of vertices
Av A2, . . ., Au and the lower level of the corresponding sets Bv B2, . . ., Bu
and if At = {aitl,aii2, ...}, Bi= {biA,bi2t ...} then the required matrix
has first row consisting of the elements of .^'in some order followed by those
of B1 in some order. These are followed by the elements of A2 in some order
and the elements of B2 in some order, etc. In the second row we have first
the elements of An in the reverse order to that which they had in the first
row, followed by the elements of Bn again in reverse order. Then come t|w.
elements of An_1 followed by those of Bn_1 again in the order opposite to tl t
81 REPRESENTATION OF GRAPHS

128
ERDOS-MOSSEU
in which the\" appeared in the first row. We continue in this way up to the
elements of Ax in the reverse order to that in the first, row, followed finally
by the elements of B1 in reverse order. It is easily seen that this matrix induces
the required graph.
We proceed to prove
Lemma 3. If G is a directed graph with n vertices and e edges with
log n
nr IV
< e <
22r + 4 22r '
where
20 r
1
> 1
and
then G contains a bipartite unidirected graph with levels A and B having [ \n ]
log n 1
vertices respectively, and in which the valences of the vertices of
20r+lJ t a J J
A in the graph G do not exceed 16«/2r.
Proof. Consider first the vertices of G (if any) of valence at least 16w/2r.
If their number is x then we must clearly have x-16«/2r5i 2w2/22r+1 or
.r < nh2r~i. Thus the number of edges eontaining two such vertices does not
■xceed
52. n'-/22r+9. Hence if we omit all these edges there remain more
than ^2/22^6 edges at least one endpoint of which has valence < 16w/2r.
Denote the vertices of valence < 16w/2r (in G) by i\, v2, ■ . .,v( and let their
valences he yx, y.,, ....?/,. Clearly
71(1
and
- 2-C:4>) <t ^n
t
2Vt >
TV
)2r + 5
Without loss of generality we may assume that
y.y't >
n-
22r-,S
where y\ is the number of edges directed away from vt. Lot k(k ^ 1) be an
indeterminate for the time being. A &-tuple of vertices will be said to belong
to r,- (1 ^ i <. t) if every vertex of the fc-tuple is adjoined to i>, by an edge
[y't
directed away from vt. There are exaotly
k
^-tuples belonging to v,-. Denote
by S the system of ^-tuples belonging to one of the v,-(1 5i i 52. t) (if a &-tuple
belongs to exactly r v's it oocurs in S r-times). Clearly S has 5"
w<
elements.
Now V
will be a minimum if all the y't are equal and if t is as large as
n
possible. This is achieved by letting t = n and y\ =
I n
I n
> n
22r + 6
Thus
t
2
i* I
22r+6
k
>
TV
k\ I
M(22r^)k
82 GRAPH THEORY

O.N A llKI'liE.SKNTATlOX OF DIKKCTED GRAI'HS
129
Now the total number of /'-tuples that can be chosen from n points
Ti Tl
< — so that the same &-tuple must occur in S at least
~ k ! 2<2r 7>*
times. If k
Jogn_
20r + 1
will occur at least [Yn] times, or there will be at least [fn] vertices form a set
I lofr Tl
A each connected to each vertices of a set B which has — n
| 20 r + 1
Note that the set .1'was chosen from the vertices whose valences did not
exceed 16n/2r so the lemma is proved.
Wo next prove the crucial
a simple computation shows that the same /"-tuple
elements.
u he re
Lemma 4. Let n ,-- «„. If G is a directed graph with n vertices and e edges
— < ( ^-
22r :<. 22r '
and r < 10 loglog u
,, ,,, , r I " 1°S 1l 7
then G contains a bilevel graph of at least --— edges.
Proof. First we omit all edges connecting vertices with valences at
n-
least 16n/2r. As before the number of omitted edges is at most -'- - Hence
' a 22r 9
we are left with at least
22r- 1
9-'
1
28
>
edge-s and by Lemma .'i we have a bipartite imidireeted subgraph (Av Bx)
with levels /1, and Bl previously described. Since the vertices of /1, have
valence ^L 16»/2r and those of Bx have valence ^. n — 1 and .since
r < 10 < loglog n the number of edges incident to Ai \j Bt is at most
>i -r log n
I 2r
20 n3''2
<
2r
We remove these edges and there still remain
«'-' 20 /<3/2 m2
2r
>
edges, provided « > riu. Lemma 3 can therefore be used again and we obtain
a bipartite unidirected graph (.4.,,. B2) with levels A2 and B2 of the required
type. (In the bipartite graphs (A.. B,) it is not necessarily assumed that the
edges go from At to B,. their direction may depend on ?'). Now we repeat
the procedure and omit the edges incident to A2 U B2. If we repeat thic
9 A Matematikai Kuiato lntewt Koxlennnjei TX. A/l-
83 REPRESENTATION OF GRAPHS

130
KKD<%-MO.*EK
procedure
in
20 • 2r
times we are left with a graph which has at least
in
n-
,2r
20-2r
>
22r +
edges. We can therefore a-pply Lemma 3 onee more and thus obtain a bilevel
graph with the components (At, B,)
of at least
in
20 ^+6
1 ^
+ 1
i ^
in
'. 1" ...'
20 • 2' '5
20 r ~ 1
>
n log n
+ 1)2'-
edges and the proof of the lemma is complete.
Lemma 5. Let G be a connected directed graph of m vertices. Then G has a
m — 1
bilevel subgraph of edges.
I 4 J
Proof. We prove first that if T is a directed tree then it can be
decomposed into four bilevel graphs. For this purpose consider first the corresponding
undirected tree T*. Let x^ be an vertex of T*. Number I all edges of T* which
can be reached from xx in an odd number of steps. Number II all edges which
can be reached from xx in an even number of steps. The edges labelled I form
a union of disjoint stars (a star is a tree in which all but one vertex has valency
1) which can be split into two bilevel graphs and similarly for the edges
labelled II. The lemma now follows by considering for G a spanning tree T, i.e. a
tree whose edges are a subset of the edges of G and which contains all the
vertices of G. Such a tree clearly has r> — 1 edges.
Lemma 6. Let G be a directed graph of e edges. Then G contains a bilevel
fe
graph of at least
edges.
Proof. A graph G of e edges must have at least [ ]/2e ] vertices. Consider
the connected components G,- of G having u, vertices, i = 1,2, . . . ,1c.
bilevel graph of —'
4
Now by Lemma 5, each G, contains a
so that G contains a bilevel graph of
edges,
1
>
\e
edges.
We are now ready to prove our main result, namely that every
preference pattern on n candidates can be achieved by not more than ca «/log n
voters. For this purpose it will suffiee, by Lemma 2, to show that the directed
graph G corresponding to the preference pattern can be decomposed into
edge-disjoint bilevel graphs Gv G2, .... 6',, the set of whose vertices is
identical with the set of vertices of G, and t < c, «/(2 log n).
84 GRAPH THEORY

ON A REPRESENTATION OF DIEECTED GBAPHS
131
Wo are going to define the graphs
G, and G« 1 < i <
2ie.
log n
by induction. We will put G<" = G-G,UG2U ... U G, (i.e. we obtain
G(,) by omitting from G the edges of Gx, G2, ..., G,). Gj is one of the bilevel
subgraphs of G having the maximun number of edges and if G1 G,- are
already defined then G,+1 is one of the bilevel subgraphs of G(,) having tho
maximum number of edges. Denote by e, the number of edges of G(,). Let r
run through the integers r = 0, 1, ..., [10 loglog n]. Denote by ir the smallest
integer for which
n-
fir ^ —- — •
Wo shall prove that for r < [10 loglog n]
r 4- 1 n
(1
h + i
i, <215-
If e,t <-
then
2r+1 log re
0 and (1) is satisfied thus we can assume
e,- > . Let ir < j < ir,, then e, > and hence by Lemma 4 G^1'
contains a bilevel subgraph of at least
n log n
(r + l)2r+15
edges and hence by the maximality property of Gj
n log n
(2)
ej - eJ+1 ^
(r + 1) 2' + 15
(2) immediately implies (1).
Prom (1) we obtain that by the removal of at most
215 re "r + 1
2 (vn-*r)<-:—2 ~
log re r=0 2r+I
0Sr£[10Ioglogn]
216 n
log «
l)ilevel graphs G,-, 1 ^ i <
216 ft
we obtain a
logn
GO = G - U G,, 1 ^ i ^
where G(<) has fewer than
edges.
9*
216 re
log n
n-
220 loglog n
<
n-
(log «)i3
85 REPRESENTATION OF GRAPHS

132
KUDOS—MONK It
To complete the main result we need to show that a graph with this many
edges is the union of o edge — disjoint bilevel graphs and this is an
Uogwj
almost immediate consequence of Lemma 6.
As already stated the proof of m(n) >c2 w/log n is relatively simple but
we include it for completeness. Since each voter can vote in n ! ways the number
of distinct ways in which m voters can vote is (n\)m.
The number of preference patterns on n candidates is (since ties are
permitted) 3«'. If all these patterns can be achieved we must have (w!)m> 3^2'
from which the required result follows by a simple computation.
One might conjecture that that m(n) log n\n tends to a limit but this
conjecture is clearly well beyond the methods used in this paper. \\> cannot
even prove that lim m(n) log njn > — — .
«-*„ 2
Still another problem suggested by the present considerations is to obtain
good estimates for the largest number s = s{e) such that every ordinary graph
of e edges contains a bilcvel (undirected) graph of s edges. By more complicated
arguments than those used here we can prove s > c^e log e.
(Received November 25. 1963^
REFERENCES
[1] McGarve\\ D. C: "A theorem on the construction, of voting paradoxes". Econo-
metrica 21 (1963) 608 —(ilO.
|2] Stearxs. R.: "The voting problem". Ainer. Math. Monthly 66 (1959) 701 — TlV.i
86 GRAPH THEORY

THE REPRESENTATION OF A GRAPH BY SET
INTERSECTIONS
PAUL ERDOS, A. W. GOODMAX, AND LOUIS POSA
1. Introduction. Geometrically, a graph is a collection of points (or
vertices) together with a set of edges (or curves) each of which joins two distinct
vertices of the graph, and no two of which have points in common except
possibly end points. Two given vertices of the graph may be joined by no edge
or one edge, but may not be joined by more than one edge. From an abstract
point of view, a graph G is a collection of elements [xi, x-i, . . . j called points or
vertices, together with a. second collection ^of certain pairs (xa, xa) of distinct
points of G. It is helpful to retain the geometric language, and refer to any pair
in 9? as an edge (or a curve) of G that joins the points xa and xg.
A family of sets Si, 52, . . . gives a graph in a natural way, if to each set Sa
we associate a point xa and agree that
(1) xa and xB are joined by an edge of G if and only if at -^ /3 and Sa(~\ Sa t6- 0,
where 0 denotes the empty set. As far as we know it was E. Szpilrajn-
Marczewski (2) who first proved that the converse is also true; see also Culik
(1)
Theorem SM. Let G be an arbitrary graph. Then there is a set S and a family
of subsets Si, 52, . . . of S which can be put into one-to-one correspondence with the
vertices of G in such a way that (1) holds.
Notice that Theorem SM remains true if we replace Sa f~\ Sa 7½ 0 by
Sa (~\ SB = 0 in (1), because we can always replace G by its complement.
Our objective in this paper is to determine the minimum number of elements
in the set 5. In fact we shall prove the following theorem.
Theorem 1. If G is any graph with n vertices, then there is a set S with [»2/4]
elements and a family of n subsets of S such that (1) holds. Further [»2/4] is the
smallest such number.
2. Coverings by complete graphs. A graph G is said to be complete if
every pair of points of G is joined by an edge of G. A complete graph on two
points is just a line segment, and a complete graph on three points is just a
triangle. We define the sum G = Gi + Gi of two graphs as follows: (1) x is a
vertex of G if it is a vertex of G\ or of Gi, (2) xa and Xg are connected by an
edge in G if they are connected by an edge in G\ or in G2. We remark that if
they are connected in both G\ and Gi, then they are still connected by just a
single edge in the sum. If a graph G is the sum of graphs G\, Gi, ..., Gk, we
Received September 14,1964.
106
87 REPRESENTATION OF GRAPHS

REPRESENTATIONS OF GRAPHS
107
shall say that these graphs cover G. An isolated point of a graph is a point that
does not belong to any edge of the graph. The number of vertices of G is called
the order of G. With these agreements we have the following theorem.
Theorem 2. A ny graph G(7l) of order » > 2 with no isolated points can be covered
by at most [»2/4] complete graphs. Further, in the covering we need to use only
edges and triangles.
Proof. We use induction, going from index n to index n + 2. The theorem is
obviously true for n = 2 and n = 3. Further, we note that for any positive
integer n
(2) [(» + 2)74] = [»2/4] + n + 1.
Now let G(7l+2) be a graph of order n + 2 and let Xi and X2 be any two points
that are connected by an edge of G(7l+2). Let G("° be the subgraph consisting of
the vertex set V — {x3, xt, . . . , xn+2\ and those edges of G(n+2) that connect
pairs of points in V. By hypothesis, this graph can be covered by at most
[»2/4] triangles and edges. Consider xk £ V. If xk is joined to both X\ and x2 in
G(7l+2), then we introduce a triangle Xi x2 x4 and call this Gk. If xk is connected
to Xi or x-i, but not both, then we introduce for Gk an edge x\ xk or xi xk. If x^ is
not connected to either x\ or xz, then there is no need to introduce a line
segment or triangle. Hence for £ = 3,4 n + 2 we have at most » complete
graphs Ga. Finally we need G\, the edge connecting x\ and Xi. Since G(7l) is a sum
of at most [»2/4] edges and triangles, G(7l+2) is the sum of at most [»2/4] + (» + l)
edges and triangles. From (2), this completes the proof of the theorem.
Theorem 2 was also proved independently by L. Lovasz (oral communication).
It is easy to prove that the number [»2/4] that occurs in Theorem 2 cannot
be replaced by any smaller number. Let n = 2k or 2k + 1 and let A be a
collection of k points and B be a collection of the remaining points (either k or
k + 1 in number). We define T{n) to be the special graph of order n in which
xa and x$ are joined by an edge, if and only if one of the points is in A and the
other point is in B (3). Clearly Tm has no triangles and the number of edges is:
k2 = [(2£)2/4] = [«2/4], if n is even,
and
k{k + 1) = [{2k + l)2/4] = [«2/4], if n is odd.
Hence the graph T{n) will always require [»2/4] complete graphs for a cover.
We shall give a refinement of Theorem 2 in §4.
3. Proof of the main result. If the graph G of Theorem 1 has any isolated
points xa, we can select the empty set for Sa and for such points condition (1)
will be satisfied. Hence in proving Theorem 1 we may assume that G has no
isolated points. We next cover G with N complete graphs Gi, Gi, .-., GN,
88 GRAPH THEORY

108 PAUL ERDOS, A. W. GOODMAN, AND LOUIS POSA
where N < [»2/4]. By Theorem 2, this can be done and in fact in such a way
that each Gk must be either an edge or a triangle. With each graph Gk we
associate an element ek and with each point xa we associate a set Sa of elements
ek, where
(3) ek (z Sa if and only if xa £ Gk,
i.e. Sa is the collection of those elements for which the corresponding complete
graphs contain xa. If we set
then clearly 5 contains N elements. Further, Sa C\ S$ ?± 0 if and only if there is
a common element ek. This means that xa and x$ belong to the same complete
graph Gk, and this means that xa and Xp are joined by an edge in G. Conversely,
if Xa and x$ are joined by an edge in G, this edge will appear in some Gk in the
cover and hence ek will be in both Sa and 5^. Consequently, the sets constructed
by condition (3) will satisfy condition (1). This concludes the proof of the first
part of Theorem 1.
To see that [»2/4] is the smallest number for which Theorem 1 is true, we
turn again to the special graph T{n). Here each edge must give rise to at least
one element, for if xa and x$ are joined, then Sa (~~\Sp contains some element
ea0. But if this element were present in any other intersection, such as Sy f~\ 5j,
then the points xa xg xy or xa x$ xt would be vertices of a triangle in T{n). But
T{n) contains no triangles. Hence each edge in Tm gives rise to at least one
distinct element. Hence any representation of T{n) by the intersection of sets
satisfying the condition (1) must use at least [»2/4] elements.
In the reverse direction of Theorem 2 we can prove:
Theorem 3. Let G be a graph and suppose that for each point xa £ G there is a
set Sa such that condition (1) is satisfied. If the set S = VJ Sa contains N elements,
then G is the sum of N complete graphs.
Proof. For each fixed ek in S we form a complete graph Gk using those points
xa for which Sa contains ek. Clearly each GK is a complete subgraph of G and
G = d + G2 + . . . + GN.
4. A refinement. Let Ga and Gp be two of the complete subgraphs
constructed in the proof of Theorem 2. It is easy to see that Ga and G$ may have an
edge in common. With a little more labour we can avoid this.
Theorem 4. Any graph Gm of order » > 2 with no isolated point can be covered
by at most [»2/4] complete graphs Gi, Gi, ..., GN, and no two of the graphs Ga, Gs
will have an edge in common. Further, in the covering we need to use only edges and
triangles.
Proof. We say that a vertex x has valence k if k edges terminate at x.
The theorem is obviously true for n = 2. We assume that it is true for all
89 REPRESENTATION OF GRAPHS

REPRESENTATIONS OF GRAPHS
109
graphs of order less than n and note that for any positive integer n
(4) [«74] = [(»- l)»/4] + [n/2].
Hence in the induction we must show that in going from G{n~1} to Gin) we need
add at most [n/2] complete graphs that are pairvise edge disjoint.
Suppose G(n) has a vertex of valence < [n/2]. Call this vertex x\ and let G("_1)
be the subgraph on the points j.x2, x3, . . . , xn\. Then in going from G("_1) to
G(7l) we only need to use the edges joining x\ to the other points of G(n) for our
complete graphs, and there are at most [n/2] of these. In this case the proof is
complete.
In the contrary case, every vertex of G(7l) has valence > [n/2]. Let X\ be the
vertex with the smallest valence t, and set t = [n/2] + r, where by hypothesis
r > 0. Let Xi be joined to the vertices yi, y2, . . . , y t and let G('> be the subgraph
of G(7l) spanned by yx, y2, . . . , yt. Suppose that G(,) has r independent edges;
that is, no two edges have a common vertex. Call these edges
(yi. y2). (ys, y<0,..., (y2r_i, y2r)
and remove them from G(B_1). Cover the resulting graph with at most
[(» — 1)74] edges or triangles, that are pairwise edge disjoint. Then G(7l) is the
sum of these complete graphs together with the triangles
(xi, yi, y2), (xi, y3, yl), ..., (xi, y2T-i, yi/)
and the edges (xi yk), k = 2r + 1, 2r + 2, . . . , (. The number of graphs in the
sum is at most
[(« - 1)74] + r + t - 2r = [(« - 1)74] - r + [n/2] + r = [«74].
To complete the proof, we shall show that Gm must have r independent edges.
Assume that G('> has only r — 1 independent edges
(yi, yi), (ys, y0 (y2r_3, y2r-2).
By h>'pothesis, y2r_l has valence > [»/2] + r. It can be joined to at most
2r — 2 of the points y1( yi, . . . , yi,-i, and to at most n — t of the points not in
G(<). Hence the valence of y2r-i is at most
2r-2 + n-t = 2r-2 + n- [n/2] - r = n - [n/2] + r - 2 < [n/2] + r.
But this is the minimum valence. Hence y2r_i is joined to some other point in
G( ° and G(t) has at least r independent edges. This completes the proof.
The graph T{n) shows that the number [»2/4], mentioned in the theorem,
cannot be replaced by any smaller number.
5. Open questions. These results suggest a number of related problems.
For example, suppose that the graph G("° has [n2/4] + k edges, where k is a
fixed positive integer. Then it is clear that G(7l) can be covered by fewer than
[»74] complete graphs. What then is the new minimum as a function of k}
90 GRAPH THEORY

110 PAUL ERDOS, A. \V. GOODMAN, AND LOUIS POSA
Here it may be advantageous to use complete graphs of order greater than 3 if
k is large.
In another direction it seems as though every Gm can be covered by at most
n — \ circuits (here a single edge is counted as a circuit) but so far we have not
been able to prove this. If we add the side condition that the circuits be pair-
wise edge disjoint (no two circuits have an edge in common), then n — \
circuits will not suffice as T. Gallai proved in the following way (oral
communication). Let the vertices be denoted by xu Xi, x3, J\, yi, y3, . . . , y„-3 and let
G*(n) be the particular graph with the 3(» — 3) edges (xa, y$), a = 1, 2, 3,
/3 = 1, 2, 3, . . . , n — 3. Aside from the trivial circuits consisting of single
edges, all other circuits have either the form d : (xi, y$, xi, yy, X\), or the form
C2: (xi, y$, Xi, yy, x3, yt, xx), or suitable permutations of these. The requirement
that the cover be edge disjoint forces the inclusion of the single edge circuits
(x3, yB) and (x3, yy) in any cover that includes C\. Hence if all pairs y$, yy are
covered by circuits of type C\, the number of circuits required would be
3(» - 3)/2 if wis odd, and 3 + 3(» - 4)/2 if n is even.
If the edge-disjoint cover includes a circuit of type C2, then it must also
include the single-edge circuits (xi, yy), (x2, yj), and (x3, y$). Suppose that
n = 0 (mod 3) and the yt vertices are grouped in sets of three and that the
covering is made up of circuits of type Ci and single-edge vertices. Then the
number of circuits is 4(» — 3)/3. Since this is less than the number of circuits
used in the first case, it is clear that for n = 0 (mod 3), the smallest number of
edge-disjoint circuits needed to cover the special graph G*(n) is 4(» — 3/)3.
The cases n = 1, 2 (mod 3) lead to a similar result.
Let/(») denote the smallest integer such that every graph with n vertices can
be covered by fin) or fewer edge-disjoint circuits. The graph G*(n) proposed by
Gallai shows that lim inf/(»)/» > 4/3. It can be shown that
/(») < \n logw +0(n),
but it may be true that/(») < en for some suitable c.
6. Representation of a graph by distinct sets. In the proof of Theorem 1,
the sets obtained need not be distinct. Indeed there may be two different
vertices xa and x$ for which Sa = Sp, Both James H. Reed and G. Sabidussi have
pointed out that if the n sets corresponding to the vertices of G are required to
be all different, then the proof of Theorem 1 is not sufficient. However, if n > 4,
we obtain the same minimum as in Theorem 1.
Theorem 5. Let d{n) be the smallest number of elements in S with the property
that for each graph on n vertices, there is a family of n different subsets
Sa(a = 1, 2, . . . , n)
of S such that the relation (1) holds. Then d(2) = 2, rf(3) = 3, and
91 REPRESENTATION OF GRAPHS

REPRESENTATIONS OF GRAPHS
111
(5) d(n) = [»2/4]
ifn > 4.
Proof. The exceptional cases n = 2 and n = 3 are trivial. We proceed by
induction from n to n + 2, and we first prove that if <i(») < [»2/4] for » = 4
and n = 5, then the same inequality holds for all n > 4.
If the graph G(7l+2:) on » + 2 points has no edges, then we set S = jea} for
a = 1, 2, . . . , n + 2. This selection is satisfactory because n < [m2/4], for
n > 4. Suppose that in the induction from « to n + 2 the graph G(7l+2) has
an edge (xn+i> xn+2) and that neither of these two points are terminal points of
any other edge. Let G(n) be the graph on the n points \xx, x%, . . . , xn}. Then we
add two new elements e* and e2* to the set for G(n) and take for our new sets
Sn+i = jei*}, 5n+2 = jei*, e2*} while leaving the sets for G(7l) unchanged. In this
case the induction is complete.
In all other cases the graph will contain an edge (x„+i, xn+2) that is connected
with at least one other point of the graph. By equation (2), we have n + 1 new
elements at our disposal. Call them ex*, e2*, e* e*+i. We form the set Sn+i
by putting in ea* (a = 1, 2, . . . , n) if and only if xa is connected to xn+\ by an
edge. Similarly, 5„+2 is the set of all ea* for which ixa, xn+2) is an edge in G(7l+2).
It may happen that one of the two sets Sn+x and Sn+2 is empty. In this case we
form the sets 5^+i and 5^+2 by adding the element e*+i to both Sn+i and 5„+2.
Then 5*+i ^ 5*+2. If 5'„+iPi57l+2 is empty, we also adjoin the element e*+i to
both sets. In any other case we can set 5*+i = 5„+i and 5*+2 = 5„+2 VJ {e*+i!-
Let Si, 52, ..., Sn be the sets that satisfy Theorem 5 for G(7l). For
a = 1, 2, . . . , n, we form the new sets Sa* by adding ea* to 5« if xa is connected
to either xn+i or x„+2 by an edge. Then the sets Si*, 52*, ..., 5*+2 satisfy the
requirements of Theorem 5.
If n = 4, it is a simple matter to draw pictures of the 11 different graphs on
four points, and to construct the necessary sets with at most [42/4] = 4
elements. The same technique can be used if n = 5, but in this case the number
of different graphs is sufficiently large to make a short cut desirable. Since
[52/4] — [42/4] = 6 — 4 = 2, we have available two new elements for S in
passing from n — 4 to n = 5. If G(5) has one vertex with valence 2 or less, it is a
simple matter to proceed from G(5) to G(4) by deleting this vertex and its edges
and then to go back to G(5) using the two new elements. Hence one needs to
consider only those graphs on five points for which each vertex has valence
greater than or equal to 3. But there are only three such graphs and these are
easy to discuss.
This proves that din) < [»2/4]. But the same special graph Tm used in
Theorem 1 also proves that d(n) > [»2/4].
References
1. K. Culik, Applications of graph theory to mathematical logic and linguistics, Proc. Symp.
Graph Theory, Smolenice (1963), 13-20.
92 GRAPH THEORY

112 PAUL ERDOS, A. W. GOODMAN, AND LOUIS POSA
2. E. Szpilrajn-Marczewski, Sur deux proprietes des classes d'ensembles, Fund. Math., 33 (1945),
303-307.
3. P. Turan, On the theory of graphs, Colloq. Math., 3 (1954), 19-30.
University of Alberta, Edmonton,
The University of South Florida, and
Michael Fazekas High School, Budapest
93 REPRESENTATION OF GRAPHS

Chapter 4
Coloring of Graphs
Paper [296], "Graph Theory and Probability," which is reprinted in Chapter
8, could have also appeared in this section.
Paper [525] is an expository paper which could have been placed in
Chapter 2, the Problem chapter.
In [386] and [487] some unintuitive results are obtained on chromatic
number. (Some of these results are in ES.) In [386] it is shown that graphs
exist of "high" chromatic number though all of their subgraphs of "moderate"
size may be three colored. In [487] the maximum ratio of the chromatic
number to the clique number of graph is considered. Intuitively, we think of a
graph requiring k colors because some "small" subgraph (e.g., a complete
graph on k points) requires k colors. However, the chromatic number of
"random" graphs is bounded by "global" considerations. If the graph has n
points and no set of t independent points then the chromatic number is at
least n/t. In [487], P573 was settled negatively by J. E. Graver and J. Yackel,
Some Graph Theoretic Results Associated with Ramsey's Theorem, J.
Combinatorial Theory 4(1968)125-175.
Papers in Chapter 4
[386] On circuits and subgraphs of chromatic graphs
[487] Some remarks on chromatic graphs
[518] (with D. J. Kleitman) On coloring graphs to maximize the proportion
of multicolored A:-edges
[525] Problems and results in chromatic graph theory
[i] (with J. Spencer) Imbalances in A:-colorations
95 COLORING OF GRAPHS

170
ON CIRCUITS AND SUBGRAPHS OF CHROMATIC GRAPHS
P. Erdos
A graph is said to be ^-chromatic if its vertices can be split into k
classes so that two vertices of the same class are not connected (by an
edge) and such a splitting is not possible for &— 1 classes. Tutte was the
first to show that for every k there is a ^-chromatic graph which contains
no triangle [1].
The lower girth of a graph is defined as the smallest integer t so that
our graph has a circuit of ( edges. J. B. Kelly and L. M. Kelly [2] showed
that there exist graphs of arbitrarily high chromatic number and lower
girth 6. I proved [3] that for every t and k there is a graph of chromatic
number k and lower girth t. In fact I showed the following sharper
result: To every k there is an e so that for n>n0(e, k) there is a 0^
(Q(ro wj]i denote a graph of n vertices, Cr/™' will denote a graph with n
vertices and I edges) of chromatic number k and lower girth Js e logw.
We shall show that in some sense this result is best possible. First we
introduce some notations. f(m, k; n) denotes the maximum of the
chromatic number of all graphs Gin), every subgraph of m vertices of which
has chromatic number not exceeding k; gk{n) is the largest integer for
which there is a G{n) of chromatic number k and lower girth gk{n). Clearly
g3(n) is the largest odd integer not exceeding n (since every odd circuit
has chromatic number 3). For k > 3 the determination of gk(n) seems
very difficult. In [3] I proved! (c1; c2, ... will denote suitable positive
constants)
Now I shall prove
Theorem 1. For k Js 4 we have
.,.2 log n , ,
qk(ri) < = /76 „, +1.
Theorem 1 and (1) shows that for k >4 the order of magnitude of
gk(n) is logw (it would be easy to replace (1) by an explicit inequality).
It seems likely that for k > 3
lim .9^.(½)/log w
exists, but I have not been able to prove this.
Theorem 1 shows that the chromatic number can be "large" only if
the lower girth is ^elogw. Theorem 1 further implies that every Gin)
| In [3], (1) is proved in a slightly different form.
[Mathematika 9 (1962), 170-175]
97 COLORING OF GRAPHS

On circuits and subgraphs of chromatic graphs
171
which is 4-chromatic must contain a circuit of length sg: 1 + 2 log2?i. I
thought that every 4-chromatic GM must also contain an odd circuit of
length < c2 log n. In other words, I conjectured that for a sufficiently
large constant c2 we have /([c2 log n], 2 ; n) = 3 (a graph all of whose circuits
are even is 2-chromatic). T. Gallai (not knowing of my conjecture)
constructed a 4-chromatic G(n) the smallest odd circuit of which has length
[«.*]. Gallai's example is not yet published. Gallai and I then conjectured
that the largest value of m for which f(m, 2; n) = k is of the order of
magnitude mW, but we have not even been able to prove that for every
e > 0 and n > n0(e), f([en], 2 ; n) = 3.
The situation seems to change quite radically if we consider f(m, 3, n)
instead of/(m, 2, n). In fact I shall prove
Theorem 2. To every k there is an e > 0 so that if n > n0(e, k) there
exists a k-chromatic G{n) every subgraph of which having [en] vertices is at
most 3 chromatic.
Instead of Theorem 2 we shall prove the following stronger
Theorem 3. For m > 3 we have
b\1'3 /, n
/(«,3;»)>^-; (log^j . (V)
For /(m, k; n) at present we only can show a trivial upper bound :
f(m, k; n)sC
-=-+1
m
k. (2)
(2) is indeed trivial since we can split the vertices of G(n) into at most
|w/m] + l sets each having ^m elements, and by assumption the graphs
spanned by these vertices are at most /c-chromatic.
(2) is certainly very far from being best possible. It is easy to deduce
from a result of Szekeres and myself [4] that for m > k [f(>n, k, n) in
fact is meaningful only for m > k]
f{m, k; n) </(&-fl, k; n)<cin1-wk\ (3)
The deduction of (3) from [4] is easy and can be left to the reader (to
simplify his task we only remark that if every subgraph of lc-\-1 vertices of
G{n) is at most /achromatic then G{n) cannot contain a complete(&-(- l)-gon
4¾).
I further proved that [5]
/(3, 2; w)>c5wi/logw. (4)
It seems probable that
f(k+l,k; n)>n1-<1iv-e,
98 GRAPH THEORY

172
P. Erdos
for every e > Oif n > n0(e, k). I do not know to what extent the exponent
| in Theorem 3 can be improved for all values of m.
Proof of Theorem 1. A simple induction argument shows that every
fc-chromatic G{n) contains a subgraph G(m) every vertex of which has
valency ^ &— 1 (the valency, or order, of a vertex is the number of edges
incident to it). Assume now that Gin) is ^-chromatic and is of lower
girth t. Let G{m) be a subgraph of G{n) every vertex of which has valency
>&— 1 and let X0 be any vertex of G(m). Consider the set of vertices
of G{m) which can be reached from X0 by a path of [(t—1)/2] or fewer
edges. Clearly every such vertex can be reached by only one such path
(for otherwise G{m\ and therefore G{n\ would contain a circuit of fewer
than t edges). Since, further, every vertex of G(m) has valency >&— 1,
we obtain by a simple argument that there are more than (£-2)^-1^
vertices which can be reached from X0 by a path of [(t—1)/2] or fewer
edges. Hence
(&-2)<'-2>'2 < (&_2)«'-U!2l <m <w,
which proves Theorem If.
The proof of Theorem 3 will use simple probabilistic arguments and
will be similar to previous proofs used by Renyi and the author [5].
First we need two Lemmas which are of independent interest. Denote
by G£n) a graph having n vertices and I edges. If the vertices are labelled
then the number of different graphs (?/"> clearly equals ( ? ). A set of
vertices of (x/n> is said to be independent if no two of them are connected
by an edge.
Lemma 1. Let l=[m], r>c6: then for all except possibly -j^-f ? )
graphs G/n' the maximum, number of independent vertices is less than (njr) log r.
Let be the vertices of G/™'. The number of graphs G{n) for
which xiv ..., xia is an independent set is clearly
/(S)-(5)\
Since the vertices can be chosen in Q ways, the number of graphs 0/">
for which the maximum number of independent points is Js u is not
greater than
C)((S,7S,)<"(Q78,)<(?)"('-1)'(,!)) «->
<((P)(f)"^"
f This idea is used in [3] and also in Lemma 3 of P. Erdos and L. P6sa. " On the
maximal number of disjoint circuits of a graph ", Publ. Math. Debrecen, 9 (1962), 3—12.
99 COLORING OF GRAPHS

On circuits and subgraphs or chromatic graphs 173
By (5) the proof of our lemma will be complete, if we show that, for
u > (njr) logr, r > c6, we have
(")'
6-taV < ^. (6)
r
(6) can be shown by a simple computation and is left to the reader.
It would be easy to drop the condition r > c6, but then (njr) log
would have to be replaced by, say,
n log(r+2)
r+c7
It seems that the order of magnitude (njr) log r is not far from being
best possible at least for certain ranges of r.
Corollary. Let I— [rn], r>c6. Then for all except ^-( yj graphs
(x/n) the chromatic number of G[n) is greater than r/logr.
The corollary immediately follows from Lemma 1, since if G*-n) is
^-chromatic the maximum number of independent vertices must be
^njk (since the n vertices can be split into k independent sets).
Lemma 2. Let I = [rn] < 1/8 . Then for all but -jjj-l ?') graphs
(x/™' every subgraph spanned by u of its vertices, 4 $; u ^ 10~6wr~3, contains
fewer than \u edges.
In particular the lemma implies that these (x/n) contain no complete
quadrilateral. This result is contained in my paper with Renyi quoted
in [6].
Denote by N(u, t), 4 <w ^ 10~6M7—3, \u ^t ^min ((%), A the number
of graphs Gt{n) which contain a subgraph Gt("\ To prove our lemma
we have to show that
XXN(u,t)<-is(®), (7)
where the summation is extended over 4 $Jm $; 10~6wr~3,
§w<*<min (®, iy
First we estimate N(u, t). Let xh, ..., xia be any u vertices of (?,<">.
The number of graphs G/n' for which the subgraph spanned by xiv ..., xiu
contains t edges clearly equals
/(2))/(2)-(2:
Since the vertices xiv ..., xiu can be chosen in (") ways, we evidently have
"(-o-GK'DCt,'8)- «8>
100 GRAPH THEORY

174
P. Ebdos
From (8) we obtain by a simple computation
(t > §w, I = [rn], u < 10~6 Mr-3),
^•o(ir<""wW<((?rs(
/10m1'3 A' /10^(10-^-3)1^1 _
I ..J/3 i ^ I . J/3 10
Prom (9) we easily obtain by u > 4, t > f w, that (7) holds and hence our
I w4'3
lemma is proved, f r $; 1!3 . — was needed to make sure that 10_6wr~3 Js 4
should be true ; in other words, that the range for u should not be empty. 1
w4'3 I (B)\
Corollary. Let I = [rn] $; 13 , then for all but -jjf ( ?'I graphs
0/M> every subgraph spanned by u of its vertices u $; 10~6nr~3 is at most
^-chromatic.
As stated previously a simple induction argument shows that every
(?("> of chromatic number > 4 contains a subgraph GW every vertex of
which has valency > 3. Thus G^ has at least %v edges and the corollary
follows from Lemma 2.
The constant 10~6 could easily be replaced by a larger one and the
exponent —3 in 10~6wr~3 could also be slightly increased, but I do not
pursue these investigations since the corollary is sharp enough to deduce
Theorems 2 and 3 and at present I cannot obtain best possible estimations,
or even estimations which are likely to be anywhere near being best
possible.
Now we can prove Theorem 3. Put r = Toci(w/m)1'3> l~=[m]. By
/(n)
the corollary to Lemma 1 we first of all obtain that for all but $>{ V
graphs (x/n> their chromatic number is greater than
logr
>C3(m) ('^ m) • (1°)
if c3 is sufficiently small. (Lemma 1 applies since we can assume that
r>c6, for if not then m > 10_6wc6-3 and for sufficiently small c3 (1')
becomes trivial.)
I wl/3
Secondly, by the corollary to Lemma 2 I since m > 4, r $; v„ . and
Lemma 2 applies] for all but -¾( j') graphs £?,<"> the chromatic number of
all their subgraphs having at most u vertices is ^ 3 for
u $; \Qr*wr~* = m, (11)
101 COLORING OF GRAPHS

On circuits and subgraphs of chromatic graphs
175
(10) and (11) implies that for m > m0 at least | ( ? ) of the graphs satisfies
(1'), which completes the proof of Theorems 3 and 2.
To conclude I just wish to remark that from (4) one can deduce a
much stronger result than is obtained by putting ra = 4 in Theorem 3.
References.
1. Blanche Descartes, "A three colour problem ", Eureka. (April 1947). Solution March
1948. See also J. Mycielski, " Sur le colorage des graphs ", Colloquium Math.,
3 (1955), 161-162.
2. J. B. Kelly and L. M. Kelly, " Paths and circuits in critical graphs ", American J. of
Math., 76 (1954), 786-792.
3. P. Erdos, " Graph theory and probability ", Canadian J. of Math., 11 (1959), 34-38.
4. and G. Szekeres, " A combinatorial problem in geometry ", Comp. Math.,
2 (1935), 463-470.
5. , " Graph theory and probability (II) ", Canadian J. of Math., 13 (1961), 346-352.
6. and A, Renyi, " On the evolution of random graphs ", Pub. Mat. Inst. Hung.
Acad., 5 (I960), 17—61. See also the paper quoted in [3] and [5],
University College,
London, W.C.I.
(Received on the \'Sth of November, 1962.)
102 GRAPH THEORY

COLLOQUIUM MATHEMATICUM
XVI DED1E A M. FRANCISZEK LEJA 1967
SOME REMARKS ON CHROMATIC GRAPHS
BY
P. EEDOS (BUDAPEST)
A graph G is said to berk-chromatic if its vertices can be split into Jc
classes so that two vertices of the same class are not joined (by an edge)
and such a splitting into Jc — 1 classes is not possible. The chromatic
number will be denoted by H(G). A graph is called complete if any two
of its vertices are joined. Denote by K(G) the number of vertices of the
largest complete subgraph of G. The complementary graph G of G is defined
as follows: G has the same vertices as G and two vertices are joined in G
if and only if they are not joined in G. A set of vertices of G is called
independent if no two of them are joined. 1(G) denotes the greatest
integer for which there is a set of 1(G) independent vertices of G. We
evidently have
H(G) ^ K(G) = 1(G).
Throughout this paper Gn will denote a graph of n vertices, cr,ct, ...
will denote positive absolute constants. Vertices of G will be denoted
by Xr, X2, ..., G—Xl — ... — Xr will denote the graph G from which
the vertices Xr, ..., Xr and all the edges incident to them have been
omitted. G(Xr, ..., Xm) denotes the subgraph of G spanned by the
vertices Xr, ..., Xm.
Tutte and Ungar (see [2]) and Zykov [10] were the first to show
that for every I there is a graph G with K(G) = 2 (i.e. G contains no
triangle) and H(G) = I. I proved [6] that for every n there is a Gn with
K(Gn) = 2 and H(Gn) > en112/log n. On the other hand, it easily follows
from a result of Szekeres and myself [7] that if K(G„) = 2, then
H(GH) < cinll\
It is an open and difficult problem to decide if for every n there is
Gn with K(Gn) = 2 and H(Gn) > c2w1/2 (P 573).
In the present note we prove the following
Theorem. For every n there is a Gn satisfying
H(Gn) c3n
' K(Gn) (log/*)2'
103 COLORING OF GRAPHS

254 r. E It DOS
But, on the other hand, for every Gn ire hare
H(G„) eAn
K(Gn) (log;;)2
It seems likely that
(3)
Urn (max(^)/-!L_)
exists (P 574), but I have not been able to prove (3). By the methods
of this note it would be easy to prove that
(10^2)2 -,- ■ fl a lH(Gn)\ I «
5¾ lim inf I max —
inf Lax(£^/ « )
„ \ on \K(Gn)J/ (logn)2/
I iH(Gn)\ I n \
lim suplmax——)/———)<(log2)*
»^oo \ on \K(Gn)l/ (log?;)2/
First we prove (1). It is known [5] that for every n > nQ there is
a graph Gn so that
2 log ii _2 log n
From the definition of the chromatic number we immediately
obtain that for every graph Gn
n n
(5) H(Gn)^ = =^---
K ' { n> ^ I(Gn) K(0n)
The proof of (5) is immediate since the vertices of Gn can be
decomposed into H(G,i) independent sets or n <: H (Gn)I(Gn).
(4) and (5) immediately imply (1).
To prove (2) Ave first prove two simple lemmas.
Lemma 1. Let (wJ[v) > k. Then uv > es(logii)2.
Without loss of generality we can assume u ^ v. We then have
(6)
-<(v)«C;)<^<(^-)'
uv > c-(logn)2 follows from (6) by a simple computation.
In faet, with somewhat more trouble we could prove the following
stronger result:
If (" + l] > », then
(7) min (OT) = [l]p±i],
where t is the smallest integer for which I..,,) > n.
104 GRAPH THEORY

CIIHOM.ITIV GRAPHS 255
From (7) we obtain by a simple computation
W^(1+0(1))(logl)2-
Lemma 2. Let n 5= in > X. Assume that for every m and every
subgraph G(Xlf ..., Xm) of Gn we have l(G{Xl, ..., Xm)) 5¾ I. Then
H(Gn) <~+-V-
Let X[1}, ..., XlP be a maximal system of independent vertices
of Gn (»! = I(GH)). .Yf, ..., A*jf is a maximal system of independent
vertices of Gn — X{^ — ... — X£>; X^\ ..., A'l,3) — a maximal system of
independent vertices of GH-X{^- ... -X^-X^ - ... -X\^ etc. We
continue this process until
r
^V > //-AT.
By our assumption nt ^ Z for all t, 1 < i < r. Thus r < «/L The
Xp, 1 < j < ??,-, 1 < i < r <; ?(/?, are the vertices of the i-th. colour and
the remaining fewer than X vertices all get different colours. Thus Lemma
2 is proved.
Now we are ready to prove (2). It is known [7] that
(K(6m)+K(Gm)-2\
(8) > in .
\ K(Gm)-l J
Thus by Lemma 1
(9) K(Gm)K(Gm) >c5(logm)2.
From (9) we infer that if in ^ w/(logn)2, then for every subgraph
G(X1, ..., Xm) we have
(10) /G(Xj, ..., Xm)\ > — —^--°—-' > _6-L_^ ]_.
K ' [ ' " ' m" K{G{X1,...,Xm)) K(Gn)
Now apply Lemma 2 with jV = /?/(log»)2, I = e6(logn)2jK(Gn). We
then obtain
c6(logn)2 (log;))2
and (2) immediately follows from (11). This completes the proof of our
theorem.
Finally we state some more problems. Denote by G(n;in) a graph
of n vertices and in edges. It is easy to see that if H(G(n; in)) = fc, then
m > („) and if in = LI, then n = k, i.e. we have the complete graph
105 COLORING OF GRAPHS

256 P. ERDOS
of k vertices. Determine the smallest integer f (I, k) for which there exists
a graph G having /(/, fc) edges and satisfying K (G) ^ I, H (G) = k. As
we just stated,/(fc, k) = L) and Dirac showed that/(A —1, k) = I "t 1—5
(see [.3] and [4]). It seems to be very difficult to determine /(2, k). The
graph constructed in [6] shows that /(2, k) < r7P(logfc)3 and it is easy
to see that /(2, k) > eak3. Perhaps
lim — = (7 < oo
exists (P575).
Denote by g(n; I) the smallest integer for which there is a G[n ; </(n; Z))
satisfying l[G(n;g(n;l)))=l. Turan [9] determined g(n;l) for every
n and Z. Let g(n;k,l) be the smallest integer for which there is a
G[n ; g (n ; k, 1)) satisfying
l(G(n;g(n;k,l))) =1, K{g{h ; g(n; k, I))) = k.
By (8) we must have I 7, J ^ w. I have not succeeded in
determining g(n;k,l) (P 576) .
.E.E.F.E.E.EJVC.ES
[1] Blanche DescarteR, j-1 iAree colour problem, Eureka, April 1947 and
(solution) March 1948.
[2] — Solution to advanced problem 4526, The American Mathematical Monthly
61 (1954), p. 352.
[3] G. Dirac, A theorem of 11. L. Brooks and a conjecture of H. Hadwiger,
Proceedings of the London Mathematical Society 7 (1957), p. 161-195.
[4] — Map colour theorems related to the Heawood colour formula II, The Journal
of the London Mathematical Society 32 (1957), p. 436-455.
[5] P. Erdos, Some remarks on the theory of graphs, Bulletin of the American
Mathematical Society 53 (1947), p. 292-294.
[6] — Graph theory and probability II, Canadian Journal of Mathematics 13
(1961), p. 346-352.
[7] P. Erdos and G. Szekeres, A combinatorial problem in geometry, Cora-
positio Mathematica 2 (1935), p. 463-470.
[8] Jan Mycielski, Sur le coloriage des graphes, Colloquium Mathematicum 3
(1955), p. 161-162.
[9] P. Turan, On the theory of graphs, ibidem 3 (1954), p. 19-30.
[10] A. A. 3hhob, 0 ueKomopax ceoucmeax .luneunbix KoMn.ieKcos. MaTeMaTH-
'lecKHft CoopiiHK 24 (66) (1949). p. 163-188.
Eecu par la Eedaction le 28. 8. 1965
106 GRAPH THEORY

Reprinted from Jolrnal or Combinatorial Theory Vol. 5, No. 2. September 1968
All Rights Reserved by Academic Press, New York and London
On Coloring Graphs to Maximize
the Proportion of Multicolored /(-Edges
Paul Erdos and Daniel J. Kleitman*
Nemetvolgit 72c, Budapest XII, Hungary
and Mathematics Dept. MIT, Cambridge, Massachusetts 02139
Communicated by Gian-Carlo Rota
Abstract
The following results and some generalizations are obtained. Consider all colorings
of the n vertices of a A:-graph G into I colors. Then, if k is sufficiently large (k > k0(r, 1)),
at least a proportion r of the A--edges of G will contain vertices colored in every color
for any r < 1.
It is possible to color the points of any graph G with two colors so that
less than half of the edges in G have endpoints of the same color. Further
results of this kind have been obtained by one of the authors [ 1 ].
In this note we consider an analogous problem for A>graphs. Let Gk be
a A:-graph (a collection of A>element sets of points which we will call
A>edges) and suppose we color the points of Gk in / colors. We seek the
maximum over all colorings of Gk of the proportion of Ar-edges in Gk
which contain at least one point of every color.
Let p(Gk , /) be the maximum just described and let m(n, k, /) and
m(k, /) be, respectively, the minimum value of p(Gk , /) over all A:-graphs Gk
on n points, and the minimum over all finite A:-graphs.
Below, we evaluate m(n, k, I) by showing that the graph which minimizes
p(Gk , I) for each n is the complete fc-graph on n points, Ski„; i.e., the
A>graph consisting of all A:-edges, We also provide a simple direct
evaluation of m(k, I).
Our results imply, for example, that
lim m(k, /)=1
for all /, so that for sufficiently large k there exist colorings of any A--graph
which make most of its A>edges contain all colors.
* This research was supported in part by NSF Contract GP6165.
164
107 COLORING OF GRAPHS

COLORING GRAPHS
165
Our results can be divided into the following three parts:
Theorem 1. For any finite graph
m{n,k,l)>S^l\
where S2(k, /) is a Stirling number of the second kind, i.e., the number of
partitions ofk elements into exactly I indistinguishable parts.
Theorem 2. For any (n, k, I), the minimum value ofp(Gk , I) is achieved
if Gk is the complete k-graph on n points.
Corollary. For any (n, k, I) such that I divides n, we have
m(n,k,l)= Z(-l)s(')(
(MnV^
I
k
and
s(i<, i) n
m{k, I) =
lk
Proof of Theorem 1: There are ln distinct colorings of a graph on
n points into I colors. Of these, lnkR colorings will color a given fc-tuple
in exactly I colors, where R is the number of ways of coloring k points in
exactly I colors. Moreover, R is clearly /! times the number of partitions
of k into exactly I non-empty parts, which latter number is S2(k, I). The
proportion of all colorings which will give rise to all /-colors in any fc-tuple
is then
/!
Let Gk have q(Gk) fc-edges, and let r(C, Gk) be the number of edges of
Gk colored in all I colors under the coloring C. Let 6(C, E) be 1 if the edge
£is colored in all colors by C and 0 otherwise.
We then have
r(C, Gk) = X 0(C, E).
EeGk
By the remarks above, for any edge E, the average value of 8(C, E) over
all colorings is given by
/!
-w £>2,(k, I).
108 GRAPH THEORY

166
ERDOS AND KLEITMAN
Averaging the expression above for r over all colorings into / colors then
yields
<r(C,Gfc)>c = X <V(C,E)>
EeGt
= q(Gk)^S2(k,l).
This equation means that, for any graph Gk , the average proportion of
edges colored in all colors, over all possible /-colorings, is
Since p(Gk , I) is the maximum of r(C, Gk)/q(Gk) over all /-colorings we
have
p(Gk,l)>^S2{k,l)
for any finite graph Gk , and we can conclude that
m(k,I) >jis2(M).
Proof of Theorem 2: For a coloring C of a graph Gk on n points
and an element g of the symmetric group Sn on the n points of Gk, let Cg
be the coloring obtained by performing the permutation g before the
coloring C. For any A:-edge E, let Eg be the A:-edge whose elements are the
images of the elements of E under g, and let Gkg be the fc-graph whose
members are of the form Eg for each E in Gk .
For any coloring and any edge we have
0(C, E) =-- 0((¾. Eg)
hence
r{C, Gk) = r(Cg, Gkg).
Using these facts, we find, upon averaging r(Cg, G!c) over all g in S„ , that
<r(Cg, Gk)\eSn = <r(C, Gkg-*)\eSn
since each edge £ lies in G^g-1 for exactly Mq(Gk)/Q) elements g of Sn ■
109 COLORING OF GRAPHS

COLORING GRAPHS
167
This latter expression tells us that
(r(Cg, Gk)\eSJq(Gk)
is independent of Gk and in fact is given by r(C, Sk.„)/(£) since, for Gk = Skt„,
the averaging is trivial.
Now, if p(Gk , I) < p(Sk_n , /), there must be a coloring C such that
r{C,Gk)lq(Gk)^r{C,Sk.n)IQ.
But by the result immediately above there must then be some g e Sn such
that
r(Cg,Gk)/q(Gk)^r(C,Sk,„)/Q;
thus, for all Gk , p(Gk , 1) > p(Sk._n , /).
Proof of Corollary1. For the complete graph one can easily verify
that any coloring C which assigns exactly [n/l] or [n/l] + 1 points to each
color will satisfy r(C, Sk-n) = m(n, k, I). The value of m(n, k, I) can
immediately be deduced from this fact; by the principle of inclusion-
exclusion, we obtain
»<«.*. /)=t(-. or-, "")/©•
For large values of n this expression is asymptotic to and always greater
than
£(-!)•(!)(/-W-
An elementary identity for the Stirling numbers,
s2(k, /) = (-1)1 y -V, J ' ,,
implies that the upper limit for m(k, /) obtained here is the same as the
bound obtained in Theorem 1.
Our Theorem 2 above is capable of wide generalization. In fact, the
method of proof used above applies to any situation in which we seek to
minimize over a class of graphs the maximum over any class of colorings
that is symmetric under point permutation, the proportion of the edges
possessing any property which is invariant under point permutation.
This generalization can be expressed as the following theorem.
110 GRAPH THEORY

168
ERDOS AND KLEITMAN
Theorem 3. Let Gk be a k-graph and Cl a class of colorings of the
points or, for m $; k, of the m edges of Gk . Let tt be a property of colored
k-edges. {We say an edge E has property tt for coloring C if 6(E, C) = 1.)
Let p{Gk , Ci) be the maximum over all colorings in Cl of the proportion
of k-edges in Gk which have property tt. Let m{n, k, Ct) be the minimum
value of p(Gk , C;) over all k-graphs on n points. Then, if the class Ct is
symmetric under point interchange, and the property tt is invariant under
simultaneous permutation of coloring and edge (so that Sn g e Sn
d{E, C) = d(Eg, Cg)\
m(n, k, Q) = p(Sk.n , Q
where Sk_n is the complete k-graph on n points.
The proof of this theorem is the same as that of Theorem 2.
We can apply the generalization to obtain answers to the following
modifications of our original problem.
(a) Let
\M(n,k, /,/•))
I m(n, k, I, r)\
be the minimax, that is, the minimum over A:-graph of the maximum over
all colorings of points in I colors, of the proportion of edges colored in
tat least) .
.. \ r colors,
(exactly!
(b) Let m(n, k, I, Si,..., st) be the minimax over graph colorings of the
proportion of edges such that s,- points per edge are colored in color/
(c) Let m(n, k, I, Si ,..., Si , f1,..., t[) be the minimax over graph colorings
in n points for which tt edges are colored in color j, of the proportion of
edges such that ss points per edge are colored in color/
The possibilities of further variation in this problem are obviously
great. In each case above we can conclude that the minimum-maximum
occurs for the complete graph Sk-n , for which to answer to each of these
questions becomes a routine computation.
The conclusion that lim^^ m(k, I) = 1 has been used by C. Jockusch [2]
in proving the following result in recursion theory, nomenclature for
which will not be defined here.
The sets with the property that some set of lesser degree of unsolvability
is strongly hyper-hyperimmune have zero measure. This result implies
that the measure of the family of sets, each one of which is Turing equiva-
111 COLORING OF GRAPHS

COLORING GRAPHS
169
lent to some member of a given downward closed family C, is the same as
the measure of the family whose members each have some member of C
recursive in it. A downward closed family here means a family of infinite
sets such that every infinite subset of every member is also a member.
References
1. P. Erdos, On The Structure of Linear Graphs, IsraelJournal ofMath 1 (1963), 56-60.
6. C. Jockusch, (to be published).
112 GRAPH THEORY

PROBLEMS AND RESUITS IN CHROMATIC GRAPH THEORY
P. Erdbs
MATHEMATICAL INSTITUTE
HUNGARIAN ACADEMY OF SCIENCE
BUDAPEST, HUNGARY
AND
UNIVERSITY OF COLORADO
BOULDER, COLORADO
1. Introduction
In this note we shall discuss mostly without proofs, some recent results
on graph theory. It will be almost entirely restricted to problems on chromatic
graphs and to those problems on which I worked myself. Many of the
questions which we shall discuss are joint work with Hajnal.
First we introduce some notations. G„ denotes a graph of n vertices,
G(n\ t) denotes a graph of n vertices and t edges, and %(G) denotes the
chromatic number of G (i.e., the smallest integer [or cardinal number] such
that the vertices of G can be colored by %(G) colors and such that two vertices
which are joined never have the same color). 1(G) denotes the maximum
number of independent vertices of G (i.e., the largest number of vertices of
G no two of which are joined by an edge). K(G) denotes the number of the
vertices of the largest complete graph contained in G. G(xt, ..., xn) denotes
the subgraph of G spanned by the vertices xu ...,xn. G — R is the graph
from which the edge R has been omitted. The number of edges of G are
denoted by R{G) the number of vertices of G by v(G). Kn will denote the
complete graph of n vertices, Cr will denote a circuit of r edges and K(u1 u2)
will denote the complete bipartite graph with ut(i' = 1,2) vertices of each
color, where any two vertices of different color are joined.
We evidently have
X(G) > v(G)II(G) (1)
In the second section we will discuss problems and results on chromatic
graphs and in the third we will mention a few miscellaneous problems.
* Dedicated to the memory of Jon Folkman.
27
113 COLORING OF GRAPHS

28
P. ERDOS
2. Problems and Results on Chromatic Graphs
G. Dirac [2] calls the ^-chromatic graph G critical if for every edge R
of G, x(G — R) < /(C). For k = 3, the critical graphs are the odd circuits,
but for k = 4 it seems hopeless to characterize all the 4-critical graphs.
Denote by fk{n) the largest integer for which there exists a G(n;fk(n))
which is fc-chromatic and critical. I asked whether fk(n) > ckn2. Dirac [2]
proved
f6(4n + 2) > 4n2 + 8« + 3 (2)
To prove (2) consider a graph, G(4n + 2; 4«2 + 8« + 3) which consists of
two different C2n + \, any two vertices of which are joined by an edge. It is
easy to see that our graph is 6-chromatic and critical. Perhaps in (2) the sign
of equality holds.
Dirac's construction easily generalizes to give
f3k(k(2n + 1)) > Qj(2« + 1)2 + k(2n + 1). (3)
Perhaps the equality sign holds in (3) also. I conjecture
lim f3k{n)/n2 = lim f3k + 1(n)/n2 = lim/3lk + 2(/i)//i2 = *(1 - l/k),
but could not even prove f6(n) = (i + o(\))n2.
A plausible guess cannot even be made about limn^ xf4(n)/n and
lim,,,, x fs{n)/n . The inequality/4(n) > In + O(l) is known [19].
A well-known theorem of de Bruijn and myself [1] states that if k < K0
then every ^-chromatic graph contains a finite subgraph which is also k-
chromatic and from this result it is easy to see that if / = K0 then every
/-chromatic graph contains an /-chromatic subgraph which is denumerable.
A first guess might be that every graph which has chromatic number X,
contains a subgraph G of power Kx with /(G) = X,. Hajnalandl proved [15]
that this not true. We show that for every k < X0 there is an X,-chromatic
graph each subgraph of which, of power less than Xfc, has denumerable
chromatic number. In [15] several unsolved problems are mentioned. Here
we mention only two of the simplest ones. Is it true that there is an X2-
chromatic graph of power X2 such that every subgraph of power Xx has
chromatic number X0? Is there a graph of power Xw + 1 of chromatic number
Xx such that every subgraph of power Xw has chromatic number X0? For
other problems of this nature we refer to [14, 16].
Let G be a graph whose vertices form a well-ordered set and for which
x(G) > X0 . Babai proved (oral communication), using the theorem of de
114 GRAPH THEORY

CHROMATIC GRAPH THEORY
29
Bruijn and myself [1], that one can find a subsequence xx, x2 , ■ ■. of type co
of the vertices of G so that x(G(xx, ...)) = K0.
It was first thought that this theorem could be generalized to higher
cardinals, but Hajnal and I showed that this is not true. We construct a
graph G with %(G) = Kx whose vertices are a well-ordered set of type co12,
but every subgraph whose vertices are of type less than m2 has chromatic
number less than or equal to K0 . Let the vertices of G be {xx, y^}, 1 < a < a>x;
1 < P < cox ordered lexicographically ({xx, y^} < {xai, yfi} if a1 > a or if
a = <xx and fix > /?). This is a set of type a>2. Let {xxi, yfii} < {xX2, y^J.
These vertices are joined if and only if a1 < a2 and a2 < Pi- It is not difficult
to see that x(G) = Kx but for every subgraph G\ whose vertices have order
type less than co12 we have x(Gx) < K0.
On the other hand we can not decide the following questions: Does there
exist a graph G whose vertices form a set of type a>22, x(G) = K2 , and for every
subgraph G' of G whose vertices form a set of lesser type we have x(G') < K0 ?
We can not solve this question with %(G) = Kt and %(G') < K0. (The case
X{G) = K2, X(G') < Kx can be solved easily by an obvious modification of the
previous example.)
Tutte [25] and Zykov [27] were the first to show that for every k there is
a G with x(G) = ^ which contains no C3 .
Denote by g,(n) the largest integer for which there exists a graph G with
K(G) < I (i.e., not containing a K,) satisfying v(G) = n and x(G) = g,{n).
The term g(l\n) denotes the largest integer for which there is a G not
containing a Cr for 3 < r < I and for which v(G) = n, x(G) = g('\n). Clearly
g3(n)=g<3\n).
Graver and Yackel [20] proved that
g,(n) <cx(n log log «/log nf" 2)('_,) (4)
In fact Graver and Yackel [20] proved that if v(G) = n and K(G) < I then
1(G) > c(n log «/log log n)lK1~ *> ; (5)
but (4) is an easy consequence of (5).
I proved by probabilistic methods [4] that there is a G„ with K(Gn) < 3
for which I{Gn) < cn1/2 log n, hence by (1)
c «1/2
<73(«)>T • (6)
log n
Very likely the same method will give for / > 3
c n(>-2)l(>-<)
(log n)Ci
but the details of a proof of (7) seem to be formidable.
115 COLORING OF GRAPHS

30
P. ERDOS
The estimation of g,(n) is clearly connected with the estimation of the
Ramsey numbers, i.e., the smallest integer n for which every G„ either contains
a K, or I(G„) > k. We do not discuss here these problems but refer to [4, 20].
It is not at all obvious that
lim g('\n) = oo (8)
n-* oo
holds for every I. Equation (8) was proved by probabilistic considerations
[4] and in fact the stronger result was shown that there is a Gn which does
not contain a Cr for 3 < r < I and for which
I(Gn)<ni-^ (9)
By Eq. (1), Eq. (9) implies
g°\n) > rf" (10)
It is easy to see that if Gn does not contain a Cr for 3 < r < 21 + 1 then
I{Gn)>c1n1-U(, + u (11)
Equation (11) gives
<7<2,+ 1)(«)<c2«,/(l+". (12)
Equation (12) is almost certainly close to being the best possible since the
method used in [4] will probably give that there is a Gn which contains no
Cr for 3 < r < 21 + 1 and for which
/(G„)<«'-1/(l+1,(log«r or <7(2'+1)(«)>c4«l/(l+1)/(log«r. (13)
The technical difficulties of a proof of (13) seem to be great.
A good guess for an asymptotic formula for log#(2!)(«) is not at hand.
Undoubtedly
logg(2,)(n) = (c + o(l))log« where 1/(1 + 1) < c < l/l, (14)
but I cannot even prove (14) for 1 = 2, and I have no good guess for the value
of c (if it exists).
Hajnal and I generalized (8) for set systems [16]. We also used
probabilistic methods. Lovasz [21] proved our results by an ingenious but
complicated direct construction; thus he showed (8) by a direct construction.
His method does not give (9) and does not even seem to give /(<?„) = o(n)
instead of (9). I would like to call attention to the following question: I
proved by probabilistic methods [3] that there is a Gn satisfying
2 log n 2 log n
K(G«^lo72-' /(C^l^gT" <15)
116 GRAPH THEORY

chromatic graph theory
31
It would be desirable to prove (15) by a direct construction. I cannot even
construct a G„ for which
max (K(G„), /(G„)) < en1'2.
It would also be interesting to determine
lim min max (K(G„), /(G„))/log n. (16)
I cannot even prove that the limit in (16) exists. We have trivially x(G) > K(G).
I proved [10] that
Cl«/(log n)2 < max x(G„)/K(G„) < c2 «/(log n)2.
Very likely
X(Gn) (log n)2
lim max
n^ao G„ K{ G„) tl
exists, but this I have not been able to prove.
In [5, 6] it is shown that for every k there is a ck such that if Gn has no
circuit of length less than ck log n, then its chromatic number is less than k.
Also, this is the best possible apart from the value of ck. Thus we have the
result that there is a c such that if Gn has no circuit of length less than c log n
then it is at most three-chromatic. One could have guessed that if Gn does not
contain an odd circuit of length less than q log n, then G„ is three-chromatic,
but Gallai [19] constructed a Gn which is four-chromatic and the smallest
odd circuit of which has length greater than «'/2. Gallai and I conjectured
that for every k there is a Gn with %(G„) = k the smallest odd circuit of which
has length greater than c1nll<-k~2); but that there is a c2 such that if the smallest
odd circuit of Gn has length greater than c2 n1/(k~2), then %(G„) < k. I proved
this for k = 4 (unpublished).
Rado and I proved [18] that for every m > K0 there is a G satisfying
/(G) = v(G) = m which does not contain a triangle. Hajnal and I proved
[16] that for every I there is a graph with %(G) = v(G) = m, the smallest odd
circuit of which has more than 2/+1 edges. On the other hand we proved
[16] that every graph with /(G) > K0 contains a C4, in fact it contains for
every n a K(n; Kx). We also show that every G with /(G) > K0 must contain
a two-way infinite simple path. On the other hand, we show that there is a G
with %(G) > K0 which does not contain a A^(K0 , K0). Several further results are
proved in [16] and there are many unsolved problems; here we state only
two of them. Is it true that, for every G with %(G) = K1; there is an n such that,
for every I > n, G contains a C, ? We can prove this if we assume /(G) > K2 ■
Is it true that every graph with %(G) = K0 satisfies £ \jnr = oo where
«! < «2 < • • • is the sequence of integers n for which G contains a C„ (perhaps
117 COLORING OF GRAPHS

32
P. ERDOS
it even follows that the nt have positive upper density)? A finite form of the
above problem may be stated as follows. Put
u(/c) = min £ l/«;
G
where the minimum is extended over all graphs for which x(G) = k. Is it true
that u(k) -> oo as k -> oo ?
As an application of one of the results given [16] Hajnal and I observed:
Let G be a graph whose vertices are the points in «-dimensional Euclidean
space. Let S be any countable set of real numbers. We join two points in G
if their distance is in S. We show %(C) < K0 .
For simplicity's sake we assume n = 2. We show that G does not contain
a K(3, Kx) and thus by our theorem with Hajnal [16] we obtain /(G) < K0 .
Let us assume that there are three points in the plane xu x2, x3 and a set
{yx} so that d(xt, yx), is always in S [_d(x, y) is the distance between x and y~\.
Since 5 is denumerable, only denumerably many y's can lie on the three lines
joining the x's. Let yx be a y not on these lines. There are only countably
many choices for d(yx, xt) — d(yx, x2) and for d{yx, xx) — d(yx, x3). Thus
the y's are the points of intersection of countably many pairs of hyperbolas,
or {yx} is countable, which proves our assertion.
A similar but slightly more complicated argument shows that in the case
of the ^-dimensional space, G does not contain a K(n + 1, Kx) and thus as
shown by Erdos and Hajnal [16] y(G) < K0 .
Hajnal and I [17] proved that for every c < \ there is a G with %(C) = K0
such that for any set xl5 . .., xn of vertices of G we have
I(G(x1,...,xn))>cn. (17)
We can show that there is a G satisfying (17) for c < \ with /(G) = K1;
but we could not show the same for c < \.
If for every choice of the vertices I(G(x1, ..., x„)) > n/2, then trivially
/(G) < 2. We conjectured that if for every n and every choice of xlt ..., xn
I(G(Xl, ...,xn))>{n- fc)/2,
then x(G) < k + 2. We did not even prove this for k = I.
An interesting conjecture is due to M. Kneser. Let 5 be a set, \S\ = In + k,
to each S\ c S, ISJ = n make correspond a vertex. Two vertices are joined
if the corresponding sets are disjoint. Is it true that this graph has chromatic
number k + 2 (trivially it is less than or equal to k + 2)1
Another problem of Hajnal and myself states: Let G be a graph with
/(G) = X0 ■ Is it true that G has a subgraph G' with %(G') = K0 such that G'
contains no C3 (or more generally no Ck for 3 < k < n) ?
118 GRAPH THEORY

CHROMATIC GRAPH THEORY
33
A finite version of this problem may be stated as follows. Is it true that to
every k there is an f(k) such that if /(G) >f(k) then G contains a subgraph
G'/(G') > k, and G' contains no C3?
One final problem of Hajnal and myself: Is it true that for every / > 3
and k > 2 there is a graph G not containing K, + 1 such that if we color its
edges with k colors there is a K, all of whose edges have the same color ?
Folkman [26] settled this conjecture for k = 2. Hajnal and I recently
observed that the following question seems to be relevant here. Let Gn be a
graph whose edges can be colored by two colors such that there is no C3
all of whose edges have the same color. What can be said about I(G„)? It is
very easy to show that I(Gn) > en113; but perhaps I{Gn) > cnui3)+3 also holds.
3. Miscellaneous Problems in Graph Theory
A well-known theorem ofTuran [24] states that every G(n; [«2/4] + 1)
contains a triangle. Turan raised the following questions, which seems very
difficult.
Denote by g(n; k, I) the smallest integer such that if in a set of n elements
there are given g(n; k, I) fc-tuples, then there are always I elements all of
whose
Q
A:-tuples are among the given ones. Turan determined g(n; 2,1) for every I,
but the problem is unsolved for every 3 ^ k < I. It is easy to see that
lim g(n; k, l)jnk = ck ,
exists for every k and I. Turan [24] proved c2 , = -i(l —1/(/ — 1)), but for
k > 2 the value of ck<, is unknown.
Turan also conjectured that
#(2«;3, 5) =n2(n- 1) + 1. (18)
The proof of (18) is probably not easy; there are several ways of
constructing n2{n — 1) = triples of 2« elements so that there are no five elements
all of whose triples are chosen.
I proved [8] that there is a constant c3 such that every G(n; c3n3/2)
contains a subgraph of 7 vertices x1; y\, y2, y3; zlt z2, z3 with the edges
(*i> yd, (*i> y2), (■*!> Js), (zi, yd, (^i, y2), (z2» yd, 0?2. y3), (z3, yd, (z3, yd-
119 COLORING OF GRAPHS

34
P. ERDOS
More generally, I conjectured that for every k there is a ck such that every
G{n\ ckn3/2) contains a graph having
i+Hi)
vertices
Xilyi, ■■■ ,yk\ Zi, ■■■, Zfk\
where x1 is joined to all the y's, each z is joined to two y's, and no two z's
are joined to the same two y's. I have not proved this even for k = 4.
Moon and Moser [22] posed the following question: Let Gn be a graph
of n vertices. Denote by g(n) the maximum number of different sizes of
cliques that can occur in a Gn. They proved
n — [log n/log 2] — 2 log log n < g(n) <, n — [log «/log 2].
I improved the lower bound to n — (log ft/log 2) — H{n) + 0(1) where H{n)
is the smallest integer for which logH(n) n < 1 (logr n denotes the /--fold
interated logarithm). I expect that the lower bound is essentially the best
possible, but I cannot even prove that g(n) < n — (log »/log 2) — C for every
C if n > n0 (C) is sufficiently large [9].
Recently several papers have been published on extremal problems in
graph theory (see [7, 11-13, 23]. Here I would like to mention one such
question. Posa proved that every C(«; 2n — 3) contains a circuit with at least
one diagonal and that 2« — 3 is the best possible here. I had thought that
every G(n; kn — k2 + 1) contains a circuit one vertex of which is the end point
of at least k — 1 diagonals. Using Posa's idea I proved this for k = 3 and
k = 4, but Lewin proved (oral communication) that in general the conjecture
is incorrect. I do not have any plausible conjecture to replace my original one.
References
1. N. G. DE Bruijn and P. Erdos, A Color Problem for Infinite Graphs and a Problem in
the Theory of Relations, Nederl. Akad. Wetensch. Indag. Math. 13 (1951), 371-373.
2. G. Dirac, A Property of 4-Chromatic Graphs and Some Remarks on Critical Graphs,
J. London Math. Soc. 27 (1952), 85-92.
3. P. Erdos, Some Remarks on the Theory of Graphs, Bull. Amer. Math. Soc. 53 (1947),
292-294.
4. P. Erdos, Graph Theory and Probability, II, Canad. J. Math. 11 (1959), 34-38.
5. P. Erdos, Graph Theory and Probability, I, Canad. J. Math. 13 (1961), 346-352.
6. P. Erdos, On Circuits and Subgraphs of Chromatic Graphs, Mathematika 9 (1962),
170-175.
120 GRAPH THEORY

CHROMATIC GRAPH THEORY
35
7. P. Erdos, External Problems in Graph Theory, in Theory of Graphs and Its Applications
(M. Fiedler, ed.), pp. 29-36, Publ. House Czech. Acad. Sci., Prague, 1964.
8. P. Erdos, On Some Extremal Problems in Graph Theory, Israel J. Math. 3 (1965),
113-116.
9. P. Erdos, On Cliques in Graphs, Israel J. Math. 4 (1966), 233-234.
10. p. Erdos, Some Remarks on Chromatic Graphs, Colloq. Math. 16 (1967), 253-256.
11. P. Erdos, Some Recent Results on Extremal Problems in Graph Theory, in Theory
of Graphs {Symp. Rome, 1966), pp. 117-123, Dunod, Paris; and Gordon and Breach,
New York, 1967.
12. P. Erdos, On Some New Inequalities Concerning Extremal Properties of Graphs, in
Theory of Graphs {Symp. Rome, 1966), pp. 77-81, Dunod, Paris; and Gordon and
Breach, New York, 1967.
13. P. Erdos, Extremal Problems in Graph Theory, in A Seminar on Graph Theory
(F. Harary, ed.), pp. 54-60, Holt, New York, 1967.
14. P. Erdos and A. Hajnal, On a Property of Families of Sets, Acta Math. Acad. Sci.
Hangar. 12 (1961), 87-123.
15. P. Erdos and A. Hajnal, On Chromatic Number of Infinite Graphs, in Theory of
Graphs {Proc. Colloq. Tihany, Hungary, September 1966) (P. Erdos and G. Katona,
eds.), pp. 83-98, Academic Press, New York; and Publ. House Hungar. Acad.,
Budapest, 1966.
16. P. Erdos and A. Hajnal, On Chromatic Number of Graphs and Set-Systems, Acta
Math. Hungar. 16 (1966), 61-99.
17. P. Erdos and A. Hajnal, On Chromatic Graphs (in Hungarian), Mat. Lapok 18
(1967), 1-4.
18. P. Erdos and R. Rado, A Construction of Graphs without Triangles Having Pre-
assigned Order and Chromatic Numbers, J. London Math. Soc. 35 (1960), 445-448.
19. T. Gallai, Kritische Graphen, Publ. Math. Inst. Hungar- Acad. Sci. 8 (1963), 165-192.
20. J. E. Graver and J. K. Yackel, Some Graph Theoretic Results Associated with
Ramsey's Theorem, J. Combinatorial Theory 4 (1968), 125-175.
21. L. Lovasz, On Chromatic Number of Finite-Set Systems, Acta Math. Acad. Sci.
Hungar. 19 (1968), 59-67.
22. J. W. Moon and L. Moser, On Cliques in Graphs, Israel J. Math. 3 (1965), 23-28.
23. S. Simonovits, A Method for Solving Extremal Problems in Graph Theory, Stability
Problems, Israel J. Math. 3 (1965), 279-319.
24. P. Turan, Eine Extremalaufgabe aus der Graphentheorie (in Hungarian), Mat.
Fiz. Lapok 48 (1941), 436-452. See also P. Turan, On the Theory of Graphs, Coll.
Math. 3 (1954), 19-30.
25. Blanche Descartes (pseudonym of Tutte), A Three Color Problem, Eureka, April
1947, Solution March 1948.
26. J. Folkman, Proc. Santa Barbara Symp., November 29-December 2, 1967, 1969
(to appear).
27. A. A. Zykov, Mat. Sbornik, 24/66 (1949), 163-188.
121 COLORING OF GRAPHS

Imbalances in k-Colorations
P. Erdfis
Hungarian Academy of Sciences
Budapest, Hungary
J. Spencer
Department of Mathematics
University of California
Los Angeles, California
1. INTRODUCTION
The following problem is due to Paul Erdos [1]. Color the
edges of a complete graph K on n vertices red and blue. What
is the largest t such that we may always find a complete subgraph
in which |# red edges — # blue edges| >_ t?
We need a more precise and more general formulation. For
any set V, define
k i i
V = {w : W C V, |W| = k}. (1)
2 k
Note that V is the complete graph generated by V. V is called
the complete k-graph generated by V. The elements of V are
called k-edges. We color the k-edges. A coloring of a set A,
|A| = n, is given by a map
gk : Ak ■> {+1, -1}. (2)
The values +1, -1 may be thought of as Red and Blue. The
subscript k indicates a function on k-edges and will be dropped
when there is no confusion. The function g, induces another
k
function, also denoted by g , on the subsets of A given by
K
g (B) = I g (W) (3)
K WCB K
|w|=k
Set
H (n) = min max |g (B)| (4)
gk BCA
Networks, 1: 379-385
© 197 2 by John Wiley & Sons, Inc. 379
122 GRAPH THEORY

380 ERDOS AND SPENCER
where A = {l, ..., nl and g ranges over all functions satis-
K
fying (2). Clearly H (n) is the t required in the opening
paragraph. Erdos [1] showed — <_ H (n) <^ en . We prove
Theorem: For k >_ 1, and n sufficiently large
Ckn(k+V/2 <_Hk(n) <_Ckn(k+1)/2 (5)
where the C, 3 CJ are positive absolute constants.
2. THE PROOF
The case k = 1 is trivial, H (n) = {—}. We sketch the
proof of the upper bound.
Fix B C A, |b| = b. Letting g be random, g (B) is the
sum of ( ) values g (W). The values g (W) are +1 with proba-
K K k
bility —, -1 with probability — and independent. Thus the
distribution of g (B) may be approximated by a normal curve of
k
b V2 k/2
mean 0 and a = (, ) /2 < n . Thus
k —
t, v. rl ,„,l (k+l)/21 -c2n/2
Prob [ g. (B) > en ] < e . (6)
1 k ' — —
As there are 2 choices of B
2
^ v. r I ,■n^ \ (k+l)/2 n -c n/2
Prob [max |g (B) | >_ en ] <^ 2 e . (7)
BCA k
For c = /2 log 2 the right hand side of (7) is less than unity
so there does exist g, such that max g, (B) < en A
k ' k ' —
more careful proof, using that fact that most |b| — will show
the upper bound of (5) for
/log 2
k 2(k+l)/2k!l/2
Let us define
^\^22 ••• C^ =I^k(W) (9)
123 COLORING OF GRAPHS

IMBALANCES IN k-COLORATIONS 381
where the sum is over all W C A, |w| = n, |w H B.| = a. for
1 <^ i <^ t. This shall only be defined when the B. are disjoint
and I a . = k.
j=l D
Now we give a quick proof of the lower bound (5) for k = 2.
Applying the methods of [3] we find sets B , B C A with
g2(BlV - Cn3/2 (10)
for some absolute constant c. But
g2(B1) + g2(B2) + ^2^^2) = g2(B1 ub2) (ii)
so
i i c 3/2
|g2(V) | >_ - n for V = B, B2, or B;L U ^2 ■ (12)
Now we prove our theorem for all k > 2.
Lemma 1: Fix k > 2. Then there exists cL, ... . d-, > 0. t .
— 1 k o
such that for t > t and A . pairwise disjoint, \A . | = t,
1 <_ 3 <_ k we have
| KB,, ..., B.) : B.C. A., \g.(B_ ... B.)\ > ^/2}| > d.2t%
(13)
for all g.y 1 <_ i <_ k.
We shall first require
Lemma 2: Fix e, > 0. There exists cn > 0, t , such that t > t
1 2 ' o — o
implies that for any choice of real x ., 1 <_ j <_ t, satisfying
Ix.l > 1 for 1 < 3 < c^t, we have
1 3 ' - - " - 1
\ I x .\ > /t (14)
3'eV °
for at least c 2 choices of Vc. {I, ..., t}.
Proof: For VC {l, ..., t} set «/?(V) = £ x.,V=vH{j:l
jeV D
< j < c1t}, V2 = V - V1. Then «/J(V) = tpiV^) + ^(V^ so
(14) does not hold if «/j(V ) e [p(V ) - /t, <p(V2) + /t] . By a
124 GRAPH THEORY

38 2 ERDOS AND SPENCER
theorem of Erdos [2] for V fixed this holds for at most
r ti [cit]
I (LC1 J) < (1 - c2)2 (15)
i cifci r r
|r —| <_ /t
values of V , where c is a positive constant dependent only on
c . Summing over all V yields Lemma 2. Q.E.D.
Proof of Lemma 1: We use induction on i. For i = 1 set x. =
g. ({j}) and apply Lemma 2. Now assume Lemma 1 holds for i - 1.
Any point a e A generates, with any g., a coloring g._, on
A - {a}. The coloring is given by
g[^(W) = g±(W U {a}) .
Set
V = {((B1, ..., B ), a) : B. C A., a e B ,
We double count
|v| = I |{(B , ..., Bi_1) : ((B , ..., Bi_1), a) e V}| (16)
I |{a : ((Bx, ..., Bi_1), a) e V}|. (17)
t(i-l)
Bl Vl
By induction the inner summation (16) is at least d._ 2
Thus |V| >_ td 2 . The sum (17) has 2 addends, each
bounded by t. Thus for at least (d._ /2)2 choices of
(B , ..., B ) we have |{a : ((B , ..., B ) a) e V}| >
d._ t/2. Fix such a (B , ..., B._ ). Set
g. (B. ... B. {a})
_ l 1 l-l
x — —. .. .„ , a e B. .
a (1-1)/2 l
By assumption |x | >_ 1 for at least d. t/2 of the a. By Lemma
2 there exists c such that
125 COLORING OF GRAPHS

IMBALANCES IN k-COLORATIONS 383
aeB.
l
|g(Bn ... B. B.)| = | T g(Bn ... B. Aa})
aeB.
l
= t(i-1)/2|I xj
^ti/2
for C 2 choices of B.. As this is true for at least
(d /2)2 1 choices of (B , ..., B._ ) we may show (15) for
d. = d. ,c_/2, completing the induction. O.E.D.
l l-l 2
Now let g = g be any coloring on A, |a| = n. For t = [—]
find disjoint A , ..., A c A, |A.| = t. From the proof of
1 K 1
Lemma 1 we find, and fix B , ..., B and 6 > 0 such that
A. K —J.
|{a : |g(B ... B]<_1 • {a})| ^t(k_1>/2}| >_26t. (18)
Either fit a's have g(B ... B • {a}) > t 5/ or fit a's
1 K —1
have g(B ... B • {a}) <_ -t . By symmetry (between
g and -g) assume the former. Set B = {a : g(B ... B • {a})
(k-l)/2, mu k 1 k-l
> n }. Then
g(B1 ... Bk) = I g(B1 ... B^ ■ {a})
k
(k+l)/2
aeB,
k
>_ 5 t
^en(k+1)/2 (19)
where e ~ 5/k > 0, independent of n.
To prove our result we first need a result in polynomial
approximations. If G is a polynomial in, say, s variables we
set IgI = the maximum absolute value of a coefficient of G and
| | g| I = max {G(x, ..., x ) : 0<_x. < 1 for 1 < i < s}.
Lemma 3: There exists e = z(s) > 0 such that if G is a
polynomial in s variables with degree at most s then
IIGII > e |G| (20)
126 GRAPH THEORY

384 ERDOS AND SPENCER
Proof: Set T={G: |g| =l}. With the |*| metric, T is compact,
I I • I I is continuous, non zero, so there exists e, |G| = 1 =>
| | G | | >_ e. But any G = | G | G , G e T, so | | G | | = | G | | | G | | >_
e | g| .
It should be noted that by other methods explicit bounds
on e (s) may be found.
Proof of Theorem: We need transfer the imbalance (19) of a
product into the imbalance of a set. For 1 <_ i <_ k let W. range
over all subsets of B., W. = [x.Ib.I], where 0 < x. < 1 will
i'i1 i'i1 — l —
be determined later.
a a
g(Wn U ... U W, ) = T g(w/ ... W, K) (21)
1 k L 1 k
where the summation ranges over all nonnegative integers a.,
k
y a. = k. Fix the a.. Set
i=l "
v(V . ..., V , W . ..., W, ) = 1 if V. C w. for 1 < i < k
1 kl k l— l — —
0 otherwise.
Then from (9) the expected value
a a
E[g(W1 ... Wk )] = E[£ g(Vx U ••• U Vk)v(Vx, ..., Vfe, V*1 , ..., W^
(the summation over all V. CB., V. = a.)
l—i ' l1 l
= I g(V1 U... U Vk)E[v(V1, ..., Vk, W1, ..., Wk)]
= J g(V, ii ... U V, )Prob[V. C W., 1 < i < klv. C B. , Iv.l = a.]
L ^ 1 w k i— l — — ' i— l ' l1 l
= n x.^g^U.-.UV^
i=l
(an approximation valid as k is fixed and n sufficiently large)
a a k a.
= g(Bl ... Bk ) n x.1
i=l
127 COLORING OF GRAPHS

IMBALANCES IN k-COLORATIONS 385
Setting
f^1 *\ / (k+D/2
Can...av = g(Bl '•■ \ )/n
1 k
we have, using (22) in (21)
E(g(W m ... UW )) = n(k+1)/2 I c Xl ' ••• ^
J. n a, . • . a,
1 k
where by (19) , |cn n | >_ e. By Lemma 3 we find, and fix,
(k+l)/2
x., , . . . , x, so that
1 k
E(g(W1 U ... U W )) = en'
where |e,| ^_ |c |e (k) >_ e e(k) which depends only on k.
By the definition of expected value we find, and fix,
(k+l)/2
W' ..., W, W! = [x. B. ], such that
1 n'l' i'i'
g(W" U... U W) > E(g(W U ... U W )) > en
1 n — 1 n — 1
proving our theorem.
REFERENCES
1. Erdos, P., "Ramsey es Van der Waerden tetelevel Kapcsolatos
Kombinatorikai Kerdesekrol," Mat Lapok 14, 1963, pp. 29-37,
(in Hungarian).
2. Erdos, P., "On a Lemma of Littlewood and Offord," Bulletin
Amer. Math. Soo. 51, 1945, pp. 898-902.
3. Spencer, J. H., "Optimal Ranking of Tournaments," Networks,
Vol. 1, No. 2.
Any views expressed in this paper are those of the authors.
They should not be interpreted as reflecting the views of
The RAND Corporation or the official opinion or policy of
any of its governmental or private research sponsors.
Paper received May 15, 1970.
128 GRAPH THEORY

Chapter 5
Extremal Graph Theory
This topic has been one of Erdos' main interests over the past decade.
In 1940 Paul Turan (see "On the theory of graphs," Coll. Math., 3
(1954)19-30) found the maximal graph on n points that did not contain a
complete subgraph of k points. Here "maximal" refers to the number of edges
and n and k are parameters. Turan's theorem is generally considered the
start of extremal graph theory. More generally, one can ask for the maximal
graph on n points having a certain property. Most of the papers in this section
fit into that general rubric.
Paper [434] was extended by two papers of Bollobas in Acta. Math.
Acad. Sci. Hungar. (1964). In particular, the problem on page 1108 was
settled.
The conjecture on the top of page 252 of [452] was settled negatively
in [473].
In [437], Erdos and Hajnal have since noted (unpublished) that the
conjecture m —> (m, mi)2 contradicts the existence of a Souslin tree.
The conjecture on page 17 of [366] was proven by M. Simonovits
(University of Budapest, unpublished).
Papers in Chapter 5
[360] On a theorem of Rademacher-Turan
[366] Uber ein Extremalproblem in der Graphentheorie
[378] Remarks on a paper of Pdsa
[389] On the number of complete subgraphs contained in certain graphs
[408] On the number of triangles contained in certain graphs
[418] Extremal problems in graph theory
[434] (with A. Hajnal and J. W. Moon) A problem in graph theory
[437] (with A. Hajnal) On complete topological subgraphs of certain graphs
[445] On extremal problems of graphs and generalized graphs
[ii] Extremal problems in graph theory
129 EXTREMAL GRAPH THEORY

[452] On an extremal problem in graph theory
[460] A problem on independent r-tuples
[467] On the construction of certain graphs
[471] (with M. Simonovits) A limit theorem in graph theory
[473] (with A. Rcnyi and V. T. Sos) On a problem of graph theory
[490] Some recent results on extremal problems in graph theory
[504] On some new inequalities concerning extremal properties of graphs
[538] (with L. Moser) An extremal problem in graph theory
[544] (with M. Simonovits) Some extremal problems in graph theory
[545] (with V. T. Sos) Some remarks on Ramsey's and Turan's theorem
130 GRAPH THEORY

Reprinted from Illinois Jovr-val of Mathematics
Vol. 6, No. I, March 19()2
ON A THEOREM OF RADEMACHER-TURAN
Dedicated to Hans Rademacher
on the occasion of his seventieth birthday
BY
P. Erdos
A set of points some of which are connected by an edge will be called a
graph G. Two vertices are connected by at most one edge, and loops (i.e.,
edges whose endpoints coincide) will be excluded. Vertices will be denoted
by a, 13, • ■ • , edges will be denoted by el , e2 , • • ■ or by (a, /3) where the edge
(a, /3) connects the vertices a and /3.
G — e-i — ■ ■ ■ — ek will denote the graph from which the edges eY , ■ ■ ■ , e,:
have been omitted, and G — ai — ■ ■ ■ — <%- denotes the graph from which the
vertices a.i , ■ ■ ■ , ak and all the edges emanating from them have been omitted;
similarly G + e-i + • • • + ek will denote the graph to which the edges e,x , • • • , ek
have been added (without generating a new vertex).
The valency v(a) of a vertex will denote the number of edges emanating
from it. G{uy will denote a graph having v vertices and u edges. The graph
G\f) (i.e., the graph of fc vertices any two of which are connected by an edge)
will be called the complete /c-gon.
A graph is called even if every circuit of it has an even number of edges.
Turan1 proved that every
G&, 7 = ^_(„'-^) + 0
for n = (k — l)t + r, 0 S. r < k — 1, contains a complete fc-gon, and he
determined the structure of the Grn),s which do not contain a complete fc-gon.
Thus if we put, f(2m) = m2, f(2m + 1) = m(m -\- 1), a special case of Turan's
theorem states that every (?/"^)+i contains a triangle.
In 1941 Rademacher proved that for even n every G/"n)+i contains at least
[n/2] triangles and that [n/2] is best possible. Rademacher's proof was not
published. Later on I simplified Rademacher's proof and proved more
generally that for t ^ 3, n > 2t, every Gy"n)+j contains at least t[n/2] triangles.
Further I conjectured that for t < [n/2] every G/*n)+l contains at least t[n/2]
triangles. It is easy to see that for n = 2m, 2m > 4, the conjecture is false
for t = n/2. To see this, consider a graph GL^+m whose vertices are
Received March 20, 1961.
1 P. Turan, Matematikai 6s Fizikai Lapok, vol. 48 (1941), pp. 436-452 (in Hungarian);
see also On the theory of graphs, Colloq. Math., vol. 3 (1954), pp. 19-30.
2 P. Erdos, Some theorems on graphs, Riveon Lematematika, vol. 9 (1955), pp. 13-17
(in Hebrew with English summary).
122
131 EXTREMAL GRAPH THEORY

ON A THEOREM OF RADEMACHER-TURAN
123
«i , • ■ ■, aim and whose edges are
(a, , a,), 1 ^ i ^ m + 1 < j ^ 2m,
and the m + 1 further edges
(at, ai+i), I ^ i ^ 7)1, and (¾ , am+1).
It is easy to see that this graph contains m2 — 1 triangles (for 2m = 4 an
unwanted triangle (¾ , a-i , o3) enters and ruins the counting, and in fact it is
easy to see that for 2m = 4 the conjecture holds for t = m = 2). For odd
n = 2m + 1 perhaps every G/^m+n+f , t ^ 2m — 2, contains at least tm
triangles. But here is a GytX+n+zm-i , 2m + 1 =¾ 9, which contains fewer
than m(2m — 1) triangles. The vertices of our graph are ai , • • • , a2m+1 ,
the edges are
(¾ , a,-), 1 ^ i ^ m + 2 < j ^ 2m + 1,
and the following 2ni + 1 edges:
(«i , oik), {ai,ak), (a3,a4), (0:3,0:5), (0:3,0:5),
3 ^ fc ^ m + 2.
It is easy to sec that this graph contains 2m' — m — 1 < m{2m — 1) triangles.
For 2»i + 1 = 5 wc must have t ^ 4, and it is easy to see that the conjecture
holds for all these t. For 2m + 1 = 7,t ^ 9, and by a little longer argument
one can easily convince oneself that the conjecture holds for all these t.
In the present paper we are going to prove the following
Theorem. There exists a constant cx > 0 so that for t < cxn/2 every G/"i)+(
contains at least t[n/2] triangles.
First we need three lemmas.
Lemma 1. Every G$"n-1)+2 which is not even contains a triangle.
Lemma 1 was found jointly by Gallai and myself. (The lemma was also
found by Mr. Andrasfai independently.)
Let G be a graph with n vertices which is not even and contains no triangle.
Let ai , • • • , otik+i be the vertices of the odd circuit of our graph having the
least number of vertices. We can assume 3 < 2fc + 1 5£ n. The subgraph
of G spanned by a-i , • • • , a^+i can have no other edges; otherwise our graph
would contain an odd circuit having fewer than 2fc + 1 edges. Let /3i , • • • ,
/3;>-2A-—1 be the other vertices of G. Any of the /3's can be connected with at
most two of the a's, for otherwise G contains an odd circuit of fewer than
2/,- + 1 edges. Finally by Turan's theorem the subgraph of G spanned by
ft , • • • , /3„_2a_i can have at most f(n — 2k — 1) edges. Thus the number
of edges of G is at most
2¾ + 1 + 2(n - 2k - 1) + f(n - 2k - 1) ^ f(n -1) + 1
132 GRAPH THEORY

124
P. ERDOS
by a simple calculation (equality only for Ik + 1 = 5). This completes the
proof of our lemma.
Our proof in fact gives that a graph G of n vertices whose smallest odd
circuit has 2k + 1 vertices, k > 1, has at most 2« — 2k — 1 + f(n — 2k — 1)
edges, and the following simple example shows that this result is best possible.
Let the vertices of G be
<*i, ■ •■ ,<*v , ft , • ■ • , /3„ , Ti» ■ ■ • » T2t+i,
n - 2k — l"
The edges of G are (a, , /3,), (yi , a,), (73 , on), 1 ^ i ^ v, (72 , /3,), (74 , /3,),
1 ^ z ^ w, further the edges (y, , 7,+1), 1 ^ ?'^ 2fc, (7! , 72t+i).
Lemma 2. T/jere exists a constant c2 > 0 so £/«a£ every G}"2)+i contains at
least [c2 tj] triangles having a common edge (ai , a2) (i.e., a?? the edges (oti , /3,),
(a2 , /3,), (a! , a2), 1 ^ z ^ [c2 w], are z'w G}[\o+i)-
Let (a, , ^,-, 7i)> 1 = * = r> be a maximal system of disjoint triangles of our
graph G{f"n)+i ■ In other words if we omit the vertices a; , ft , 7;, 1 ^ i ^ r,
the subgraph of G/"n\+i spanned by the remaining n — 3r vertices contains no
triangle and has therefore at most/(tj — 3r). edges (by Turan's theorem).
Denote by G(n, i) the graph G/J^+i — Sy-i (a/ + ft' + 7j)> and let
v{^(al), v(t)(ft), v(l)(7i) be the valencies of a, , /3,-, 7,- in G(n, z). Now we
show that for some z, 1 ^ z 5£ r, we must have
(1) v(i)K) + v(i)(/3,) + v{i)(y,) > n(l + 9c2) - 3z\
For if (1) would not hold for any z, 1 2s z 5= r, then the number of edges of
G/"n)+i would be not greater than
(2) EU (n(l + 9c2) - 3i) +/(w - 3r) </(n)
by a simple calculation for sufficiently small c2 . But (2) is an evident
contradiction since Gy"n)+i has by definition/(tj) .+ 1 edges.
Thus (1) holds for say i = i0 . Then a simple computation shows that
there are at least 3[c2tj] vertices of G(tj, z'o) which are connected with more
than one of the vertices <*,•„ , /3,0 , 7,0 . Therefore there are at least [c2tj] of
them which are connected with the same pair, i.e., G(tj,z0), and therefore
G/(l)+i , contains the configuration required by Lemma 2, which completes
the proof of the lemma.
By more careful considerations we can prove that every G/"b)+i contains
tj/6 + 0(1) triangles (ai , a2 , ft), 1 ^ z ^ tj/6 + 0(1), and that this result
is best possible.
Lemma 3. Let 5 > 0 be a fixed number. Consider any graph
G{u"\ u > /(w) - (71/2)(1 - S), n > n0(8),
133 EXTREMAL GRAPH THEORY
n — 2k — 1
2

O.N A THEOREM OF R.\DEMACHEK-TUHAX
125
which contains a triangle. Then (?,"' contains an edge (¾, a2) and [c3n] + l,c$ —
C;,{8), vertices ft , • • • , ft , r = [c3n] + 1, so /AaZ a// £/«c triangles (cti , a2 , /3,),
1 2= i 2= r, arc m &!,'".
By assumption G„"' contains a triangle (ai , a2 , <%)■ Assume first tha"
(3) K«i) + f(a2) + w(a,) > «(1 + 9c3) + 9.
Then as in the proof of Lemma 2 we can show that G[n) contains the required
configuration.
If (3) is not satisfied, then G" — ai — a2 — a3 has n — ?■> vertices and at
least u — n(l + !)c3) — 9 edges. But if c;i < 5/18, then for n > n0
i/-?i(l+ 9c;i) - U
> /(w) - (n/2)(l - a) - n(\ + 0c3) - 0 > f(n - 3).
Thus by Lemma 2, Gi,'° — ax — a2 — a3 , and therefore Ci"', contains the
configuration required by Lemma 3, which completes the proof of Lemma 3.
Now we can prove our Theorem. Let there be given a G/"rl)+( , t < Ci w/2.
Assume first that after the omission of any r = fo n/2c3], c3 = c3(j) (5=¾ in
Lemma 3), edges the graph will still contain a triangle. For sufficiently small
c1 , Ci/2c3 < -j; thus it will be permissible to apply Lemma 3 during the
omission of these edges.
By Lemma 3 (or Lemma 2) there exists an edge e1 which is contained in at
least [c3n] + 1 triangles of G$*n)+t ', again by Lemma 3 in G;[ll)+t — e,\ there
exists an edge c2 which is contained in [c3w] + 1 triangles of G/<n»)+j — ey.
Suppose we have already chosen the edges e-i , ■ ■ ■ , e, each of which is con-
taincdinatleast [c3 n] + 1 triangles. By our assumption G;^)+t — c-i — ■ ■ ■ — er
contains at least one triangle; thus by Lemma 3 there is an edge er+i in
Gf'(n)+t — e-i — ■ ■ ■ — e,r which is contained in at least [c3w] + 1 triangles in
this graph. These triangles incident on the edges el , ■ ■ ■ , er+i are evidently
distinct; thus Gf[%+t contains at least (r + l)([c3w] + 1) > c^n/2 > tn/2
triangles, which proves our Theorem in this case.
Therefore we can assume that there are I 5= r < w/4 edges e.\ , • • • , ei so
that G = Gf(rU+t — ei — • • • — e; contains no triangle, and we can assume that
I is the smallest integer with this property. By I ^ r < w/4, G has
f(n) + t - I > f(n) - re/4 > f(n -1) + 1
edges. Thus by Lemma 1, G is even.
By Turan's theorem, / 5^ t. Assume first I = t (it is not necessary to treat
the cases I = t and / > t separately, but perhaps it will be easier for the
reader to do so). Then G has /(re) edges, and by Turan's theorem G is
of the following form: The vertices of G are (¾ , • • • , a[ni±] , ft , • • •, ft,_[„/2] ,
and the edges are (a, , ft,), 1 ^ i ts [n/2], 1 g j g » - [n/2]. A simple
argument shows that the addition of every further edge introduces at least
134 GRAPH THEORY

126
P. ERDOS
[re/2] triangles and that these triangles are distinct. Thus (7}(",!)+, contains
at least t[n/2] triangles, and our Theorem is proved in this case too.
Assume finally I = t + w, 0 < w < n/4 (since I < n/4). It will be more
convenient to assume first that n is even. Put n = 2ni. Since G is even, it
is contained in a graphG(E, u) whose vertices are oti , ■ ■ ■ , a,„_„ , ft , ■ • • , fS,„+u
and whose edges are (a,■ , /3,), 1 ^ i ^ m — u, 1 ^ j ^ in + u (since 0 has
more than /(2m) — m/2 = m2 — m/2 edges, wc have 0 ^ u < 0»/2)' 2).
Clearly every one of the edges d , • • • , et connect two a's or two /3's. For
if say i\ would connect an a with a /3, then
(f/(;!)+f — Ci — • • • — Ci_i — c!+i — • • • — et
would be even, and hence would contain no triangle, which contradicts the
minimum property of L
By our assumption G is a subgraph of G(E, u). Assume that G is obtained
from G(E, u) by the omission of x edges. Then we evidently have
I = x + u" + t (or w = x + u ),
and Gf"l)+t is obtained from 67 by adding I edges e-i , ■ ■ ■ , et which are all of
the form (an , a,,) or (J3ll , /3/2). Put e, = (/3U , /3,,), and let us estimate
the number of triangles ((3^ , /3,2 , aj) in $(#, u) + e, . Clearly at most x of
the edges (J3n , aj), (/3,, , a/) are not in G(E, u); thus (7(/2, w) + e, contains
at least
m — u — x
triangles (if e, connects two a's, then G(E, u) + c, contains at least in + u — .r
triangles), Tor different c's these triangles arc clearly different; thus
«};?„+, = 0 + ^+---+0,
contains at least
(4) (m — u — x)l = (m — u — x)(x + u~ + 0 = tin = t(n/2)
triangles. (4) follows by simple computation from I = u + x + f < m:2.
(4) completes the proof of our Theorem for n = 2in. For n = 2m + 1 the
proof is almost identical and can be omitted. Thus the proof of our Theorem
is complete.
It seems possible that a slight improvement of this proof will give the
conjecture that every Gf'(%+i , t < [n/2\ contains at least l[n/2] triangles, but I
have not been successful in doing this.
I have not succeeded in formulating a reasonable conjecture about the
minimum number of triangles a G')"n)+t must contain if [n/2] ^ t ^ (■_'') — f(n).
It is easy to see that if t is close to (2') — J(n), then Gfl'l)+t contains more
than t[n/2\ triangle.s, and it would be easy to obtain a best possible result in
this case. But I have not investigated the range of t for which this is possible.
I just remark that G{nJ_t ,1^2, contains at least (3') — l(n — 2) triangles
135 EXTREMAL GRAPH THEORY

OX A THEOREM OF RADEMACHER-TURA.N 127
and that G[»J-3 contains at least (3n) -3(^-2) + 1 triangles, and that
these results are best possible. The simple proofs are left to the reader.
Turan's theorem implies that every G3nV+1 contains a complete 4-gon. As
an analogue of the theorem of Rademacher I can prove by very much more
complicated arguments that it contains at least n2 complete 4-gons; this
result is easily seen to be best possible.
Budapest, Hungary
1-36 GRAPH THEORY

Uber ein Extremalproblem in der Graphentheorie
Reinhold Baek zum 60. Geburtstag
Von
P. ErdoS
G\n^ sei ein Graph mit n Knotenpunkten und I Kanten. Mehrfache Kanten und
Schlingen werden nicht zugelassen. v(G) wird die Anzahl der Kanten, n(G) die An-
zahl der Knotenpunkte von G sein. Knotenpunkte von G werden mit den Buchstaben
x, Xi,yi, Kanten mit den Buchstaben e, ei bezeichnet. Die Kante, die die
Knotenpunkte x\ und X2 verbindet, wird mit (xi, xz) bezeichnet. 'fix), die Valenz von x,
ist die Anzahl der mit x inzidenten Kanten.
Ein Graph heiBt vollstandig, wenn je zwei seiner Knotenpunkte durch eine Kante
verbunden sind. Ein vollstandiger Graph mit n(G) = n kann also alsCr'S geschrieben
I n"\ \"'
werden. Ein Graph G'f0 mit I > ~ soil immer den vollstandigen Graphen (?®
bedeuten. V ' W
Ein Dreieck ist ein vollstandiger Graph mit drei Knotenpunkten. Dreiecke werden
mit d, di bezeichnet. Das Dreieck, das durch die Kante e und den Knotenpunkt x,
der nicht mit e inzidiert, bestirmnt wird, wollen wir mit [e, x] bezeichneti.
Der Graph (G — x± — ■■■ ~ xr) wird erhalten, indem wir die Knotenpunkte
xi, ... , xr und alle mit ihnen inzidierenden Kanten von dem Graphen G weglassen.
Sonst werden wir die in der Graphentheorie iiblichen Bezeichnungen beniitzen.
Tuban [1], [2] bestimmte fiir jedes n und r die kleinste Zahl fr(n), so daB jeder
Graph Gy^L + 1 einen vollstandigen Teilgraphen G{,rr\ enthalt. Er bestimmte auch (die
iibrigens eindeutig festgelegte) Struktur des Graphen Gf)n), der kein G(J\ enthalt.
(2)
Weiter stellte er audi folgende Frage: Was ist die kleinste Anzahl von Kanten I,
die ein Graph (¾7° enthalten muB, damit gewisse Konfigurationen im Graphen Crj"'
sicher vorkommen sollen ? Vor kurzem erschienen verschiedene Arbeiten iiber dieses
Thema [3], [4]. Insbesondere bewiesen Posa und ich, daB fiir
n ^ 24¾ und 1^ (2¾ - 1)»- 2k2+ k + 1
jedes (?{"> k unabhangige Kreise (d. h. k geschlossene Kantenziige, die paarweise
keinen gemeinsamen Knotenpunkt haben) enthalt. Wir zeigten weiter, daB fur
n^24& und I = (2k - 1) n — 2k2 + k
Cr}"' mit einer einzigen Ausnahme cbenfalls k unabhangige Kreise enthalt. Der Aus-
nahme-Graph hat 2 & — 1 Knotenpunkte x\, ... , x^-x mit "f (xt) = n — 1, die ande-
137 EXTREMAL GRAPH THEORY

Vol. XIII, 1962 (Jber em Extremalproblem in der Graphentheorie
223
ren n — 2k + 1 Knotenpunkte haben Valenz 2¾ — 1; dies bestimmt die Struktur
des Graph en G\n) eindeutig.
In seiner oben zitierten Arbeit [1] zeigt Turan, daB fur n 2: 3
/3W=[t
ist. Vor einiger Zeit zeigte ich, daB fiir »S;4 jedes G%n+n zwei unabhangige Dreiecke
enthalt, und daB n2 + n die kleinste Zahl mit dieser Eigenschaft ist. Hajnal fragte
mich, ob man die kleinste Zahl hk(n) bestimmen kann, so daB jeder Graph 0^n) + 1
k unabhangige Dreiecke enthalt (also k Dreiecke, die paarweise keinen gemeinsamen
Knotenpunkt haben) ? In dieser Allgemeinheit kann ich die Frage nicht beantworten,
kann aber hk(n) fiir jedes k und n > no(k) bestimmen.
Zunachst einige Definitionen. Es sei
+ (k-l)n-(k-l)2 + (k~
l(n,k) =
Besteht keine Gefahr von MiBverstandnissen, so wollen wir I anstelle von l(n, k)
schreiben. 5?(™' sei folgender Graph: Er hat k—1 Knotenpunkte Xi,...,xic-i mit
'f (Xi) = n — 1 (der Knotenpunkt X{, 1 :£ i ^ k — 1, ist also mit alien anderen
Knotenpunkten verbunden). Die iibrigen n — k + 1 Knotenpunkte von ^n) sind
in zwei Klassen von „ und „ Punkte verteilt, so daB je zwei
Knotenpunkte in verschiedenen Klassen mit einer Kante verbunden sind. Offenbar
gilt
jr(S?j">) = w, v(&ja)) = l(n,k)
und die Struktur von @\n) ist eindeutig festgelegt. Jedes Dreieck von @\n) enthalt
mindestens einen der Knotenpunkte xi, 1 <I i <^ k — 1, daher kann ^;n) nicht k
unabhangige Dreiecke enthalten.
Nun beweisen wir den folgenden
Satz 1. Es sei n 2: 400k2, l\ 2: l(n, k) = I. Dann enthalt jedes G\n^ k unabhangige
Dreiecke mit der einzigen Ausnahme: wenn l\ = I und G\n^ eben unser S^™' ist.
Korollar. Fiir n Jg 400¾2 ist hk(n) = l(n, k).
Bevor wir unseren Satz beweisen, wollen wir zunachst zeigen, daB er nicht fiir alle
n 2: 3 k rich tig ist. Wir definieren einen Graphen G^in wie folgt: X\ ist nur mit x2
( 2 j + 1
verbunden, der Graph /G'9., — x{\ ist vollstandig. Offenbar kann dieser Graph
\ (2)+1 J
nicht k unabhangige Dreiecke enthalten. Also gilt
hk(3k)>(3k2 l) +l>l(3k,k) fiir k^3.
/3 ^ ]\
Es ist leicht zu sehen, daB h(3k, k) = „ + 1 ist; wir iiberlassen aber den
Beweis dem Leser. Das obige Beispiel zeigt noch, daB eine absolute Konstante c > 0
existiert, so daB fiir n < 3k(I + c) hk(n) > l(n, k) gilt.
138 GRAPH THEORY

224
P. Erdos
ARCH. MATH.
Satz 1 ist fiir einen Induktionsbeweis nicht gut geeignet, darum wollen wir ihn
etwas umformen. Denselben Kunstgriff wandten wir in unserer Arbeit mit Posa an.
Satz 1'. Es sei n = n fiir n < 400&2, n = 400¾2 fiir n 2g 400¾2 und
l'(n,k) = V = j("-^+J)i| + {k _ 1)n _ {k _ 1)2 + |* - l j + [^ {400P _ ^] _
.Fw w 2: 3¾ und l[ 2: Z' enthdlt jedes <?}"' ^ unabhangige Dreiecke mit der einzigen
Ausnahme: wenn n S; 400¾2. l[ = l' = 1 und G\n) eben unser ^n) ist.
Da fiir n ^> 400 k2 I'— I ist, folgt Satz 1 aus Satz 1'. Es wird also geniigen, Satz 1'
zu beweisen.
Wir werden Satz 1' fiir jedes k durch Induktion nach n beweisen. Fiir n :¾ 20 k
ist (?}"' offenbar ein vollstandiger Graph wegen I' > I _ , also ist Satz 1' fiir
3¾ < n < 20k richtig. Es sei also n > 20k und wir nehmen an, daB Satz 1' fiir alle
3¾ ;S m < n gilt und wir wollen ihn fiir n beweisen. Wenn uns dies gelungen ist,
haben wir Satz 1' durch vollstandige Induktion bewiesen.
Es sei also ein Graph <?}"' mit n > 20 k, l[ ;> V gegeben. Es sei t die maximale
Anzahl unabhangiger Dreiecke von G<™> und d±, ..., dt ein maximales System
unabhangiger Dreiecke. Wenn t 5: k ist, so ist nichts zu beweisen. Wir konnen also
annehmen, daB t < k ist. Wir werden zeigen, daB dies nur dann der Fall sein kann,
wenn n > 400k2, l\ = I und Gf unser gf > ist.
Lemma 1. Jeder Knotenpunkt von <?(") fiat Valenz grofier als 5k— 1.
Wenn "f" (x) ;S 5 k — 1 ist, so hatten wir die Ungleichung
(1) v(G^-x) ^l'(n,k) - 5k + I > l'(n - l,k),
da, wie eine leichte Rechnung zeigt, (wegen n > 20 k)
l'(n,k)- l'(n — l,k)^5k
ist. Wegen n (G\f —x) = n — 1 folgt aus (1) und aus unserer Induktionsvoraus-
setzung, daB (<?}"' — x), also auch (?}"', k unabhangige Dreiecke enthalt, was unserer
Voraussetzung widerspricht.
Lemma 2. (xi, x2) sei eine Kante von G(£). Dann gilt
(2) -r(xi) + -T(x2)^n+k- 1.
Eine leichte Rechnung zeigt
(3) l'(n, k) - l'(n — 2,k) ^n — k + 2 k — 2 — 1 = n + k — 3 ,
wo in (3) das Gleichheitszeichen nur fiir n < 400¾2 gilt. Wenn daher (2) falsch ware,
wiirde aus (3) folgen, daB
(4) v(Gp - xi - x2) ^ l[ - n - k + 3 ^ l'(n, ¾)~w~¾+3^ l'(n - 2, k)
ist. Das Gleichheitszeichen gilt nur fiir l[ = I' (n, ¾) und n < 400¾2. Aus unserer
139 EXTREMAL GRAPH THEORY

Vol. XIII, 1962 Uber ein Extremalproblem in der Graphentheorie
225
Induktionsvoraussetzung folgt daher, daB (0\^ — xi — x2) k unabhangige Dreiecke
enthalt (fur n > 400 F wegen v(OJ^ — xi — x2) > V (n — 2, k) und fur n g 400¾2
wegen V(n — 2, k) > l(n — 2, k)). Dies aber widerspricht unserer Voraussetzung.
Lemma 3. Jede Kante von (?}"' kommt in mindestens k—1 Dreiecken vor.
Aus Lemma 2 folgt sofort, daB mindestens k—1 Knotenpunkte mit beidenKnoten-
punkten der Kante verbunden sind.
Lemma 4. (?{"' enthalt mindestens k-\-2 unabhangige Kanten, die keinen gemeinsamen
Knotenpunkt mit dem maximalen System unabhangiger Dreiecke d\,... ,(¾ haben.
Sei ei, ... , er ein maximales System unabhangiger Kanten, die mit d\, ... , dt
keinen gemeinsamen Knotenpunkt haben. Ist r^i| 1, so sei x ein Knotenpunkt
von (?}"', welcher nicht mit den (¾ und ey inzidiert (wegen t < k, r rg k + 1 und
n > 20¾ muB es einen solchen Knotenpunkt geben). Wegen 'f (x) > 5k (Lemma 1)
folgt aus t < k und r ;S k + 1, daB eine Kante (x, y) existiert, so daB y weder mit
den di noch mit den ey inzidiert. Dies aber widerspricht der Maximalitat von r.
Jetzt konnen wir Satz 1' beweisen. Auf Grund des TuRANSchen Satzes ist unser
Satz fur & = 1 fur jedes h ^3 richtig. Also geniigt es, den Fall k Sj 2 zu betrachten.
Nach Lemma 3 enthalt (?)"' mindestens ein Dreieck, daher ist also t Sj 1. Es seien
ei, ... , ek+2 die nach Lemma 4 existierenden unabhangigen Kanten, die auch von
den di, 1 <, i ^ t, unabhangig sind. Nach Lemma 3 gibt es zu jedem j, 1 g j <,
;S k + 2, mindestens k—1 Knotenpunkte x^\ 1 r=i s <^ k — 1, die mit ey ein
Dreieck bilden (x^1' kann mit x^ zusammenfallen, wenn ji 4= j2 ist). Wegen der
Maximalitat von t sind die x^> fur jedes s und j mit den (¾ inzident.
Wir teilen nun die Kanten ey,l:£j:£&+2, in zwei Klassen. In der ersten Klasse
sind die Kanten e;-, fur welche es ein Dreieck dpy;, 1 :£ F (j) ;S t, gibt, so daB
mindestens zwei der Knotenpunkte x(J\ 1 ^ s :£ k — 1, mit dpy) inzidieren. Die Anzahl
der Kanten der ersten Klasse kann aber hochstens t sein. Wenn doch t + 1 Kanten
in der ersten Klasse waren, so wiirde durch SchubfachschluB folgen, daB zwei Zah-
len ji und j2 mit -F(ji) = F(j2) = I existieren (1 < I 5£ t). Dann gibt es aber zwei
unabhangige Dreiecke [e^, yi\ und [eya, 2/2], wo yi und t/2 Knotenpunkte von di =
= dp{j) = dpyj sind. Offenbar sind dann die t + 1 Dreiecke d±, ... , di-±, di+i,
...,dt, [e;-t, y{], [eya, t/2] unabhangig, dies aber widerspricht der Maximalitat von t.
Fur die Kanten der zweiten Klasse gehoren die x\'\ 1 5£ s :¾ & — 1, zu lauter ver-
schiedenen Dreiecken (¾, 1 ^i f^ t. Aus Lemma 4 und der eben bewiesenen Tat-
sache, daB die Anzahl der Kanten der ersten Klasse hochstens t — also hochstens
k-^1 — ist, folgt, daB mindestens drei Kanten, z. B. ei,e2,e3, in der zweiten
Klasse sind. Schon aus der Tatsache, daB es eine Kante in der zweiten Klasse gibt,
folgt, daB t ^zk — 1, also wegen t < k, t — k — 1 ist.
Zu jeder Kante ey, 1 <,j <3, gehoren k—1 Knotenpunkte x^>, l<s <k — l = t, so
daB xW mit ds inzidiert und [ey, x^>] ein Dreieck von (?}"' ist. Wir konnen annehmen,
daB fur jedes s x^1' = Xj2) = Xj3) ist. Wenn dies nicht der Fall ware, wenn also z.B.
xsJ' =•= xfJ ist, sind die t -\-1 = k Dreiecke
di,...,ds-i, dSt+1,...,dt, [ei,4i']. fo,^']
offenbar unabhangig, was der Maximalitat von t widerspricht.
Archiv der Mathematik Xlll 15
140 GRAPH THEORY

226
P. Erdos
ARCH. MATH.
Setzen wir nun x^ = xf' = x[s) — ys;y$ inzidiert mit ds, l<s!^t = k—I.
Wir behaupten nun die entscheidende Tatsache, daB jedes Dreieck von (?}"' mit einem
der Knotenpunkte y±,..., yk-i inzidiert.
Um dies einzusehen, sei d ein Dreieck, das mit keinem der Knotenpunkte 2/1, - - - ,2/4-1
inzidiert. d kann nicht mit alien drei Kanten ei, ei, ¢3 einen gemeinsamen Punkt
haben, da sonst d,dlt ... , dk~\ unabhangig waren, was unmoglich ist. Nehmen wir
also an, daB d und e± keinen gemeinsamen Knotenpunkt haben. Dann sind aber die
k Dreiecke d, [ei, yi], 1 < i 5j k — 1, unabhangig, was unmoglich ist. Hiermit ist
unsere Behauptung bewiesen.
Jetzt ist es aber ganz leicht, den Beweis von Satz 1' zu beenden. Aus dem TurAn-
schen Satz folgt
(5) v(G\y-yi yk-i)^
da ((?}"> — t/i — ■" — 2/fc-i) kein Dreieck enthalt. Die Anzahl der Kanten, die mit
2/i, • • • > Vk-i inzidieren, ist aber hochstens
k-l
2
(6) (k—l)n-(k—l)* +
Aus (5) und (6) folgt
(7) l[<l(n,k)
und Gleichheit in (7) ist nur moglich, wenn sie auch in (5) und (6) gilt. Dann ist aber
offenbar (?}"' eben unser ^f1 und es muB n 2g 400¾2 gelten. Damit ist Satz 1' und
auch Satz 1 bewiesen.
Mit einem etwas komplizierteren Beweis konnten wir zeigen, daB eine absolute
Konstante C existiert, so daB Satz 1 schon fiir n > C • k gilt, und es ware leicht
eine grobe Abschatzung fiir C zu erhalten. Die Induktion miiBte aber sowohl nach n
wie nach k gefiihrt werden. Ich sehe aber nicht, wie man den genauen Wert von C
bestimmen konnte. Es ist leicht zu zeigen, daB fiir k = 2 Satz 1 fiir alle n 5: 7 richtig
ist, aber fiir n = 6 falsch ist. Den Beweis iiberlassen wir dem Leser.
Wir wollen noch ohne Beweis folgenden Satz vom ZAEANKiEVicz-DrBACschen
Typus1) aussprechen:
Satz 2. Es sei (?<») ein Graph und k = n (mod 2). Wir nehmen an, da/3 fiir jeden
Knotenpunkt x von (?<"> "V (x) 22 —-z— ist. Dann enthalt (?("> fiir n > no (k) k unab-
hangige Dreiecke.
Der Beweis ist ahnlich aber einfacher als der Beweis von Satz 1.
Vielleicht gilt aber Satz 2 fiir jedes k Si -^-, 7? = k (mod 2). Fiir n — 3k ist dies
ein vor einigen Monaten bewiesener Satz von Corradi und Hajnal [6].
Weiter gilt noch der folgende
x) In den Satzen vom TuRANSchen Typus folgern wir die Existenz gewisser Teilgraphen aus der
Anzahl der Kanten, in den Satzen vom ZARENKlEvicz-DlRACschen Typus ist die untere Grenze der
Valenzen der Knotenpunkte gegeben. Siehe noch die Arbeit [4] von Qallax und mir.
141 EXTREMAL GRAPH THEORY

Vol. XIII, 1962 tiber ein Extremalproblem in der Graphentheorie 227
Satz 3. Es sei n > c • k, wo c eine geniigend grofie absolute Konstante ist. Dann ent-
halt jedes G\'% i k Dreiecke, die paarweise keine gemeinsame Kante haben.
m+*
Wir vollen hior Satz 3 nicht beweisen. Der Beweis ist almlich dem Beweise, den
ich in meiner Arbeit [7] benutzt habe.
Literaturverzeichnis
[1] P. Turan, Eine Extremalaufgabe aus der Graphentheorie (ungarisch mit deutscher Zu-
sammenfassung). Mat. Fiz. Lapok 48, 436^452 (1941).
[2] P. Turan, On the theory of graphs. Colloquium Math. 3, 19—30 (1954).
[3] P. Ebdos and A. H. Stone, On the structure of linear graphs. Bull. Amer. Math. Soc. 52, 1087
bis 1091 (1946).
[4] P. Erdos and T. Gallai, On maximal paths and circuits of graphs. Acta Math. Acad. Sci.
Hungar. 10,337-356 (1959).
[5] P. Erdos and L. Posa, On the maximal number of disjoint circuits of a graph. Erscheint
in Publ. Math., Debrecen.
[6] Corradi und Hajnal. Erscheint in Acta Math. Acad. Sci Hungar.
[7] P. Erdos, On a theorem of Rademacher-Turan. Illinois J. Math. 6, 122 — 127 (1962).
Eingegangen am 20. 2. 1962
Anschrift des Autors:
P. Erdos
Abonyi u 8
Budapest XIV
Ungarn
142 GRAPH THEORY

REMARKS ON A PAPER OF POSA
by
p. erdos
This note will use the terminology of Posa's paper. G\nl will denote
a graph of n vertices and I edges and G^fHk) denotes a graph of having n
vertices I edges and every vertex of which has valency :> k. Ore [2] proved
that if I > + 2 then every CJ"> is Hamiltonian. and he showed that
n — 1
2
the result is false for I
general
Theorem. Let 1 ^ k < n/2. Put
yn - t
2
+ 1. Now I prove the following
move.
h
1 + max
t-
(1)
1 + max
n — k
9
+ k\ I
n - 1
2
2
\
) +
n — 1
2
2
TAe» eyen/ #[">(£) is Hamiltonian. There further exists a G!|"l1(fc) which is not
Hamiltonian.
First of all observe that by the theorem of Dirac (see the preceding
paper of Posa) if every vertex of O has valency 2i n/2 then O is Hamiltonian,
thus the condition 1 ^ k < n/2 can be assumed without loss of generality.
Next a simple computation shows that
2
T2 decreases for
)/3 < f < n/2, which proves the
1 5£ < g± (n — 2)/3 and increases for (h
second equality of (1).
Now we are ready to prove our Theorem. If our Gin)(k) is not Hamiltonian
then by the Theorem of Posa there exists a t, k ii t < n/2 so that G(n)(k)
has at least t vertices xx, . . . , xt of valency not exceeding t. The number of
edges of G("\k) which are not incident to any of the vertices clearly
ln f\ _ in i
at most (i. e. if the vertices of ©'"'(fc) ave xY xn we obtain
edges if every two of the vertices x and .r t < jx < j2 rg n are connected
227
143 EXTREMAL GRAPH THEORY

228
KKJXJ:
by an edge). The number of edges incident to one of the vertices xlt . . . , x,
is at most t1 (since each of them has valency SL t). Thus our G{n){lc) has at
most + t- edges for some k <L t < n/2 i. e. it can have at most L — 1
\ 2 i
edges which proves (1).
To complete our proof we show that (1) is best possible. Let the vertices
of G\"\{t) be X', xn. Its edges are :
{x]v xj2) . I < /j < /,, <L n and (x,, Xj) , 1 g> 2 ^ /: < } gL 2/ < » .
A simple argument shows that our 6¾¾^) is not Hamiltonian (it clearly has
l( — 1 edges). It is easy to see that every G^jA^t) which is not Hamiltonian has
this structure. (If t -— (n — 1)/2 (n odd) by Posa's theorem we can assume
that there are t -f 1 — (n. -f 1)'2 vertices of valency f£ * but by '
r2
vertex)
t \
2
t(t > 1) we do not obtain a better result by utilising this (t-\- l)-st
Tt is easy to see that the argument of Posa's paper gives the following
Theorem. Let 67("> be a graph and assume that for every 1 ^ k < (n — l)/'2
G{n) has at most k vertices of valency <£ k. Then G(n) has an open Hamilton line.
The theorem is best possible.
The proof can be left to the reader of Posa's paper. Using this result
we obtain by the same argument as used in this paper that every G{^(lc)
with
(-/-11
t*k
max
n- I
£/<-2-
+ t(t + })
has an open Hamilton line. The theorem is best possible. Finally we mention
that by the method of this paper we can prove the following sharpening of
Lemma (3.2) of | 1]. Let G("> be a graph with the vertices ,r1; . . . , xn and
2 <; k < n/2. Assume that v(xY) <i k and that there is a circuit containing the
vertices x2, x3, . . . . xn. Then if Gin) has > lk edges it is Hamiltonian. The.
result is best possible. We leave the simple proof to the reader.
(Received August 2. 1962)
REFERENCES
[1) Erdos, P.—Gallai, T.: On maximal paths and circuits of graphs." Acta Math.
Acad. Sci. Hung. 10(1959)337--3.56.
[2] Ore. O.: ,,Arc coverings of graphs." Ann&li di Mathemutica Para ed Applicatn.
IV. 55(1961)315—321.
[3] Posa, L.: ,,A. theorem concerning Hamilton lin^s." Publications of the Math. /«s(.
7(1952) A, 225—226
144 GRAPH THEORY

ON THE NUMBER OF COMPLETE SUBGRAPHS
CONTAINED IN CERTAIN GRAPHS
by
P. erdOs
#(«) will denote a graph of n vertices, Gt a graph of I edges and G\n)
a graph of n vertices and I edges. Loops will not be permitted and two vertices
can be connected by at most one edge. In the complete graph (3/m of n vertices,
every two vertices are connected by an edge. A complete graph (333) of three
vertices is called a triangle. The complementary graph G\n^ of G\n^ is defined
as follows: The two graphs have the same vertices and two vertices are
connected by an edge in G\n) if and only if they are not connected by an edge in
(?Jn). In other words a graph Crjn) and its complementary G\") gives a splitting
of the edges of the complete graph Gm\ jnto two disjoint classes. G\n^ can be
written as (?</£,, but of course this in general does not determine its structure
uniquely since the number of vertices and edges does not determine the
structure of the graph.
The vertices of G will be denoted by x,xv . . . ,yv . .. . The graph
(G — xY — . . . — xr) will denote the graph from which the vertices xv . . ., xr
and all the edges incident to them have been omitted. G(xlt ..., xk) will
denote the subgraph of G spanned by the vertices xlt . ,.,xk. The valency
v(x) of x is the number of edges incident to it. v(G) will denote the number
of edges of G, and n(G) the number of its vertices.
Ck(G) will denote the number of complete subgraphs G%\ of O. Recently
A. Goodman [1] proved that
(1)
min(C3(G<">) + C3(G(">))
if n = 2u
3
u(u— 1) {4u 4- 1) if
4m + 1
— u(u + 1) (4U — 1) if w = 4m+3
where the minimum is to be taken over all graphs GW having n vertices.
A simpler proof of (1) was later given by A. SattvIs [2],
Goodman asked if the sign of equality in (1) can hold if Cr3(Gf(n)) = 0, i.e.
if G(") contains no triangle. His answer was affirmative for even n. For odd
n I showed [2] that the answer is negative for n > 7 and it is easily seen to be
affirmative for n <L 7.
459
145 EXTREMAL GRAPH THEORY

460
ERDOS
G. Lobden [3] proved the following stronger result:
Assume that Cd(G^n)) = 0. Then for all even n and odd n > 9
(2)
minC3(G(">)
w + 1
(i.e. (?("> runs through all graphs whose complement contains no triangle).
Louden further determined all cases where there is equality in (2).
Goodman also raised the problem of determining
mm(Ck(G^) + Ck(G^)),
but this seems difficult even for k = 4.
I will prove by probabilistic arguments the following
Theorem 1. For every k ^ 3 and every n
n
2
mbi (Ck(G^) + Ck(G^)) <
It is surprising that a crude probabilistic argument gives a result which
for k = 3 is so close the correct one. This phenomenon ean often be observed
in this subject [4]. Theorem 1 seems to show that Goodman's problem will
be much more difficult for k > 3 then for & = 3, since it does not seem easy to
find graphs which give values of C4(67(">) + Ci(G^n'>) which are as small as
n
n
32. The construction analogous to the one of Goodman gives only 3 I
which is much bigger. It seems likely that
(3)
n= «
k
. Ck(G<">) + Cfc(G"n>)
lim mm —
in
1
2V2-'
(3) follows from (1) for k = 3. I can not prove it for k > 3. I will only
j12 k — 2 I
outline the proof of the crude estimate | | \~ ^
w
. CJG<">) + Ck(G~W)
hm mm -^ L-L—^ 1 >
Jfc!
t(t— 1) . . . (t— I + 1)
The following further problems might be of interest. Determine
min Ck{G<n)) = /(ra, k, I)
where GC) runs through all graphs of n vertices for which (?("> does not contain
a complete graph of I vertices.
The result of Lobden gives that for all even n and for odd n > 9
/(", 3,3)
n'
2
3
\ J
1 +
'■n + 1
2
3
146 GRAPH THEORY

01ST THE NUMBER OF COMPLETE SUBGRAPHS CONTAINED IN CERTAIN GRAPHS 461
I can not at present determine/(w, k, I) for any other values of k and I. Perhaps
for n > nJk, I) ,r . ■-
' I n + i
1-2
(5) _
1=0
in'
l-\
k
f(n,k,l) = 2
1=0
The simplest special case which I can not do is f(3n, 3,4) = 3 J
Hanani and I proved the following
Theorem 2. Let I =
taken over all graphs having I edges)
(6) max Gk(G,)
k
Finally we prove
Theorem 3. Let l> k. We have
+ r, 0 < r ^. t. Then (the maximum is to be
9(1)-
+
k- 1
(?)
n + ir
I- 1
h(n,l,k)
max Ck(Gb») = V JJ
0<,h<...<iic<,l-2 r=\
where the maximum is taken over all graphs having n vertices which do not contain
a complete l-gon (i.e. a G,i\\.
Theorem 3 is probably connected with the conjecture (5). (See [8].)
Proof of Theorem 1. The number of graphs G(,7) having the labelled
i")
vertices xlt ■ ■ ., xn clearly equals 2^2 . A simple argument shows that the
number of graphs G(n) for which either G(,7) or Cr('7) contains the complete
(n)-(k)
subgraph having the vertices xit .. ., xik is 2 - 2w V2'. Thus summing over
A;-tupIes
all the
(8)
k
2{Gk(G^)+Gk(G^))
- 2G)-0-
k
(a)
where the summation is extended over all the 2^' graphs Cr(n). (8)
immediately implies
(9)
min(Gk(GW)+Gk(GW))
<
2$)
The sign of inequality in (9) follows if we observe that if (x(,7) is the
complete graph of n vertices, then
An
Ok(G^)
>
2(a)
\GPn\ is the graph without edges). Thus for at least one of the 2« summands
(8) we have the inequality sign in (9), which completes the proof of Theorem 1.
147 EXTREMAL GRAPH THEORY

462
brdOs
Now we prove (4). A well known theorem of Ramsey [5] asserts that
for t=2k-2\
k- l j
(10) Ck(G<-») + Ck{G<f>) ^ 1.
(10) implies that if xtl, ..., xu are any t vertices of G!(n) then
(11) Ck(G(xk xit)) + Ck(G(xh xi,)) ^ 1.
From (11) we have by a simple argument (every i-tuple gives at least
one complete k-gon of G*7' or G(n) and the same &-tupIe occurs in exactly
In — k\ .
^-tuples)
It — k
Ck(G<»>) + Ck(GM)
> —
n(n — l) ... {n — k+1)
In — k
\t-k,
which easily implies (4).
It would not be difficult to show that
t(t ~ 1) ... (t — k + 1)
lim min
Ck(GW) + Ck(Qto)
exists, but I can not determine it (its value was conjectured in (3)).
Now we outline the proof of Theorem 2. max Ck(Gt) >. g(l) is trivial.
-f(0
ra)
It suffices to consider the complete graph G,]\ and an extra vertex connected
with r vertices of Gn\- The Theorem is trivial for I gL
r(D
sides of (6) are 0, and for I =
. For I <
Ik)
both
both sides are 1. We shall now use induction
and assume that Theorem 2 holds for all V < I and then prove it for I
+ r,
0 < r gL t, k 5i t. We clearly must have n(G,) >, t + 1 since I > . Assume
first that Gt has a vertex xx of valency < I. Clearly Gt contains at most
v(x ) \
K 1 complete ^-graphs one vertex of which is xv Thus clearly
k — 1
Ck(G,)
<
+ Gk(G, — x1) and v(G, — x±) = I — v(x1) .
Hence by our induction hypothesis and a simple computation (v(&J < i)
■ v{x1) \
Ck(G,) S
k
+ g(l-v{x1))g±g{l).
148 GRAPH THEORY

OlST THE NUMBER OF COMPLETE SUBGRAPHS CONTAINED IN CERTAIN GRAPHS 463
If all vertices of Gt have valency ^ t, then from I 5i , n{Gt) ^
^. t -\- 1 we easily obtain
t+ 1
2
^(£,) = t + 1
But then Ck{G,)
t+l\
k
g(l), which completes the proof of Theorem 2.
We prove Theorem 3 by induction with respect to n. (7) holds for all
k if n < k (for n <k both sides of (7) are 0 and for n = k they are both 1).
Assume that (7) holds for every m < n and every k. Since G^n) does not contain
-.(0
a G,K by a theorem of Zarankiewtcz [6] it must contain a vertex x of
valency not greater than
n + I-
I- 1
N.
By our induction hypothesis
(12) Ck{Gw-x)<h(n-l,l,k).
Denote by ylt . . ., y,; t= v(x) 51 N the vertices of GM connected to
x by an edge. Clearly the graph G(ylt . . .,yt) contains no GHz\{, thus by our
induction hypothesis it contains at most h(t, I — \,k — 1) subgraphs G$~i\ .
Hence the number of subgraphs Gfk\ of (r(n) one vertex of which is x is at most
(13) h{t,l—l,k-l)^h{N,l-l,k-l).
From (12 and (13) we easily obtain by a simple argument
(14) max Ck{GW) < h{n — 1,1, k) + h{N, l—l,k—l) = h{n, I, k) .
To show that in (14) the sign of equality holds it suffices to consider
the graph of TubAn [7] where the vertices are split into I — 1 classes, the i-th
n + i — 1]
class has
I- 1
vertices and no two vertices of the same class are
connected, but every two vertices of different class are connected by an edge.
Thus the proof of (7) and Theorem 3 is completed.
Finally I would like to state the following conjecture which is a
sharpening of (7): Put
F(n, I)
2
n + ix
I
1
n + i2
I
1
F(n, I) is the number of edges of Turan's graph, by his theorem [7] for every
G^$i,/)+i contains a G,}\. I believe that
(15)
max Ck(GF(n^) = h(n, I, k)
15*
149 EXTREMAL GRAPH THEORY

464
erdOs
where the maximum is taken over all GF(nA^ which do not contain a Guy
(15) would imply (7) since by the theorem of Tuban just stated a graph G(n^
which contains no G,}\ has ^ F{n, I) edges.
(Received October 8, 1962)
REFERENCES
[1] Goodman A. W.: "On sets of acquaintances and strangers at any party." Amer.
Math. Monthly 66 (1959) 778-783.
[2] Sauve, L.: "On chromatic graphs." Amer. Math. Monthly 68 (1961) 107—111.
[3] Lorden, G.: "Blue-empty chromatic graphs." Amer. Math. Monthly 6 (19G2) 114 —
120.
[4] See Erdos, P.: "Some remarks on the theory of graphs." Bull. Amer. Math. Soc.
53 (1947) 292 — 294 and also "Graph theory and probability I and II." Canadian
Journal of Math. 11 (1959) 34-38 and 13 (1961) 346-352.
[5] ErdSs, P. and Szekeres, G.: "A combinatorial problem in geometry." Gompositio
Math. 2 (1935) 463-470.
[6] Zabankiewicz, K.: "Sur les relations symmtftriques dans les ensembles fini." Ooll.
Math. 1 (1947) 10—14.
[7] Toban, P.: Matematikai es Fizikai Lapok 48 (1941) 436 — 452 (in Hungarian), see
also "On the theory of graphs." Ooll. Math. 3 (1954) 19-30.
[8] Moon, I. W.—■ Moser, L.: "On a problem of Turan." A Magyar Tudomdnyos
Akade'mia Matematikai Kutato Inte'zetenek Kbzleme'nyei 7 (1962) A, 311 — 314.
150 GRAPH THEORY

ON THE NUMBER OF TRIANGLES
CONTAINED IN CERTAIN GRAPHS
P. Erdos
(received October 11, 1963)
Let G(n;m) denote a graph of n vertices and m edges.
Vertices of G will be denoted by x , .
; edges will
be denoted by (x,y) and triangles by (x,y, z). (G - x - x
- . . . - x ) will denote the graph G from which the vertices
x x and all edges incident to them have been omitted.
1 k
G - (x, ,x.) denotes the graph G from which the edge (x,,x.)
has been omitted.
A special case of a well known theorem of Turan states
21
+ 1) contains a triangle and that there is
4 J
that every G(n;
a unique G(n;
proved that every G(n;
) which contains no triangle [3]. Rademacher
21
+ 1) contains at least — triangles
(Rademacher' s proof was not published). I gave a very simple
proof of Rademacher' s theorem [l] and recently proved that if
21
+ k) contains at least k —
triangles and that k — is best possible [2]. In [2] I conjectured
that this holds for k < | — j .
k< en then every G(n;
1
n
4 J
n
4 J
) contains a
Recently I observed that if a G(n;
triangle it contains at least —1 - 1 triangles. More generally
we shall prove the following:
Canad. Math. Bull, vol.7, no. 1, January 19&4
53
151 EXTREMAL GRAPH THEORY

THEOREM 1. Let I > 0. If a G(n;
n
4 J
I ) contains
a triangle then it contains at least — - 1 - 1 triangles.
The theorem is trivial if 1 > — - 2; thus we can
henceforth assume
(1)
»-<'i[§-3-
First we show that the theorem is best possible. Let
21
- 1) be x ,...,:
the vertices of our G(n; — - l ) be x , . . . , x,, ,.,-,-,,"
L(n+l)/2]
y ...-,¼ ,. Its edges are: (x, x.^); (x^y.) for 2<i<|—^-J,
1 < j < [-^] ; (x ,y.) for l<j<[^]-i - 1- Clearly the graph
J[" 2]
has n vertices, — - 4 edges, and contains only the
If] " ■* _1 triangles (x1-x2-y.). 1£J<[f]~i - i-
Now we prove theorem 1. First we need the following
simple lemma.
LEMMA. Assume that G has n vertices x ,..., x
1 n
and that it contains the triangle (x , x , x ). Further assume
12 3
that there are n+r edges each incident to at least one of
x , x and x . Then G contains at least r triangles
(x., x,, x ) with
i J k
(2)
1 < i < j < 3 < k
We prove the Lemma by induction with respect to r.
For r = 1 the lemma is evident, for in this case there are
n - 2 edges connecting x , . . . , x with x , x , x and thus
4 n 12 3
at least one x , k > 3, is adjacent to two of the x , j < 3;
k J
thus there is at least one triangle of the form (2).
54
152 GRAPH THEORY

Let now r > 1 and assume that the lemma holds for r- 1.
Just as in the case r = l, there is at least one triangle (x , x , x )
i J k
satisfying (2). In the graph G - (x.,x ) there are n+r-1 edges
incident to x , x and x , so by our induction hypothesis it
contains at least r-1 triangles. G contains further the r-th
triangle (x.,x.,x ). Hence the proof of our lemma is complete.
i J k
Now we prove Theorem 1. The theorem is trivial for
Z
r
n < 5. By the assumption of Theorem 1 our G(n;
n
14 J
i )
contains a triangle, say (x ,x , x ). Assume first that
21
(G(n;
n
4 J
- i )
V
has not more than
least
(n-3)
edges. In this case there are at
n
4 J
(n-3)
--FJ
edges incident to x ,x and x . Thus by our lemma there are
at least —1 -2.-1 triangles in our graph which satisfy (2).
Together with (x , x , x ) this gives the required — - I - 1
triangles and hence our theorem is proved in this case.
Assume next that
(G(n;
n
L4 J
- i )
X2 ' V
has more than
(n-3)
edges. By Rademacher' s theorem it
then has at least
m
triangles and together with (x , x , x )
55
153 EXTREMAL GRAPH THEORY

we obtain that G(n;
St ) has at least
Prht!
triangles (I > 0). This completes the proof of Theorem 1.
Our proof used Rademacher' s theorem, but the latter
■would be quite easy to prove by our method. In fact an induction
argument easily gives the following theorem.
THEOREM 2. Consider a graph G(n;
n
L.4 J
+ I ) and
assume that if i < 0 it contains at least one triangle. Then
it contains at least — + t - 1 triangles.
For I < 0 this is our Theorem 1; for 1 = 1 it is
Rademacher' s theorem; for I > 1 [2] contains a sharper
result.
We suppress the details of the proof since they are
similar to those of Theorem 1. For I < 0 there is nothing
to prove. If 1 > 0 by Turin' s theorem our graph contains at
least one triangle. We now use induction from n- 3 to n and
the proof proceeds as the proof of Theorem 1.
REFERENCES
1. P. Erdos, Some theorems on graphs. Riveon lematematika,
10(1955), 13-16 (in Hebrew).
2. P. Erdos, On a theorem of Rademacher-Turan. Illinois
Journal of Math. 6(1952), 122-127.
3. P. Turan, Eine Extremalaufgabe aus der Graphentheorie.
Mat. Fiz. Lapok 48 (1941), 436-452 (in Hungarian). See
also, P. Turan, On the theory of graphs. Colloquium
Math. 3 (1954), 19-30.
University of Manitoba
56
154 GRAPH THEORY

EXTREMAL PROBLEMS IN GRAPH THEORY
P. ERDOS
In the present paper @(n; l) will denote a graph of n vertices and I edges. Kp will
denote the complete graph of p vertices © I p; i | | and K(p, p) will denote the
complete bipartite graph, more generally K(plt ..., pr) will denote the complete
achromatic graph with pt vertices of the i-th colour, in which every two vertices of different
colour are adjacent. C„ will denote a circuit having n edges.
In 1940 Turan [l] posed and solved the following question: Determine the smallest
integer m(n, p) so that every @(n; m(n, p)) contains a Kp. Turan in fact showed that
the only ®(n; m(n, p) — l) which contains no Kp is K(m0, .... mp„2) where the m;
are all as nearly equal as possible, i.e. for 0 ^ i ^ p — 2 m-, = [(n + i — l)j(p — l)].
Thus a simple computation gives that if n = r (mod p — l) then
Turan further asked: How many edges must a graph contain that it should certainly
have subgraphs of a prescribed structure? In particular he asked: Determine the
smallest h{k, n) so that every @(n; h(k, n)) should contain a path of length k. Gallai
and I [2] and Andrasfai [2] investigated these and related questions and solved
them nearly completely. In the present paper we shall try to investigate as
systematically as possible the following question: What is the smallest integer /(n; k, I) for
which every graph ©(«;/(«; k, 1)) contains a ®(k; I) as a subgraph? These problems
become very much more difficult, but in my belief also more interesting, if we also
consider the structure of the graphs @(k; I). We now define three functions/;(n; k, I),
1 ^ i ^ Z.f^n; k, l) is the smallest integer for which every @(n; j\(n; k, 1)) contains
at least one @(k; l). f2(n; k, l) is the smallest integer for which there is a ®(k; l) of
given structure so that every @(n;/2(n; k. 1)) contains this ®(k; I). /3(n; k, I) is the
smallest integer so that even the Qb(k; l) which requires most edges occurs in
@(n;/3(n; k, 1)). Clearly ®(n;/3(n; k, 1)) contains all the graphs of k vertices and I
edges. Trivially
fi{n;k,r)Zf2(n;k,l)£f3(n;k,l).
It is easy to see that in general/^/?; k, l) < f2(n; k, l), since it is not hard to see that
29
155 EXTREMAL GRAPH THEORY

for n > 7 ft(n; k, [k2/4] + 2) = [n2/4] + 2 but @(n; [n2/4] + 2) does not have to
contain any @(/c; [k2/4] + 2) of any given configuration. Further it is easy to see
that in general/2(n; k, l) </3(n; k, I). Further I recently proved that
(1)
/3(»; k, i) = h U u, fu\\
where | ) < I <
2) \ 2
Now we will try to determine as systematically as possible the values of fjji; k, l)
for fixed k as I increases from 1 to I J, as far as possible we will investigate /,(«; k, l)
too (for 2 5£ i ;£ 3), in other words we will investigate structural problems too.
We will give no proofs in this paper; if no reference is given to a result then it is
not yet published.
Assume first I < k. If I ^ \k then trivially (where there is no danger of
misunderstanding we write f(n; k, l) for/^n; k, 1))
(2) f(n; k,l)=l.
If |/c < I < k then it is easy to see that
(3) f(n; k, I) = f(n; 21 + 2 - k, 21 + 1 - k) .
Finally
(k - 2) n
(4)
f(n; k, k - 1)
1
+ 1
The structural problems are very much more difficult: Gallai and I proved [3]
that every
/2*2",)+l.(fe-l)n-(fe-l)a +
+ 1
(5) ®(n; e(k, n)), e(k, n) = max
contains k independent edges and that this result is the best possible. The proof is not
easy. Trivially every @(n; [|(/c — l) n] + 1) contains a star of valency k. Further
Gallai and I [2] proved that every ®(n; [i(k — l) n] + l) contains a path of length k.
V. T. Sos and I conjectured that every @(n; [|(k — l)«] + l) contains all trees
having k edges and that every ®(n; e(k, n)) contains all forests (i.e. graphs all whose
components are trees) of k edges, but we did not succeed to prove any of these
conjectures.
For I = k there is a sharp jump in the behaviour of/(n; k, I) since/(n; k, k)jn -* oo
for every fixed k as n ->■ oo. Before we continue our investigations for general k, we
30
156 GRAPH THEORY

discuss as completely as possible the cases k = 3, 4 and 5. For k = 3 there is only
one graph ©(3; 3), the triangle, and by Turan's theorem [1]
/(»; 3,3)
+ 1
For k = 4 there are two graphs @(4; 4), the square and the triangle, with an edge.
A simple argument shows that for n ^ 4
/3(»; 4, 4)
+ 1 .
Fig. 1.
On the other hand the determination of f^n; 4, 4) = /2(1; 4, 4) seems to be a very
difficult problem (i.e. how many edges does a graph of n vertices have to have in
order to contain a square?).
E. Klein and I proved [4] that (the c's denote suitable absolute constants)
Ci«3/2 </i(«;4, 4) <c2n3'2 .
The sharpest estimates at present are due to Reiman [5]; he proved
1
(5)
2^2
lim/(n; 4, 4)/n3/2
< i
= 2 '
It seems likely that the limit in (5) equals -^2 but it is not even known whether the
limit in question exists.
It is easy to see that f(n; 4, 5) = [«2/4] + 1 [6], there is only one graph ©(4; 5):
K^ minus an edge. More generally Dirac and I [7] proved (independently) thai
every @(n; m(n, k)) already contains a Kk + l from which at most one edge is missing.
f(n; 4, 6) is given by Turan's theorem.
It is not difficult to see that for n > n0,f3(n; 5, 5) = [n2/4] + 1. The graphs ©(5; 5'
are in Fig. 1 and c gives for n > n0, /j(n; 5, 5) = /2(n; 5, 5) = f^n; 4, 4).
31
157 EXTREMAL GRAPH THEORY

Cavallius [8] obtains an upper bound for fi(n; 5, 6) = f2(n; 5, 6) (more
generally he gives an upper bound for u(n; 2, k) where u(n; 2, k) is the smallest integer
for which every ®(n; u(n\ 2, k)) contains a K(2, k)).
Besides K(2, 3) the other graphs ©(5; 6) are given by Fig. 2 and it is easy to see
that if n > n0 all of them appear in an ®(n; [n2/4] + 1), thus/3(n; 5, 6) = [n2/4] +
+ 1.
Fig. 2.
There are four types of graphs ©(5; 7), Fig. 3. Dirac and I showed (independently)
that for n > n0 every @(n; [«2/4] + l) contains graphs of types a and fa, i.e.
/i(«; 5, 7) = /2(n; 5, 7) = [n2/4] + 1. I showed that every @(n; /„), /„ = [n2/4] +
+ [n/4] + [(n + 1)/4] + 1 contains also subgraphs of the type c and it is easy to
determine all the graphs ®(n; ln — l) which do not contain graphs of the above type.
Fig. 3.
As already stated, Dirac and I showed that every @(n; [n2/4] + 1) contains d, in
fact it even contains a @(5; 9).
There are two types of ©(5; 8), Fig. 4. a is settled by the sharpening of Turan's
theorem due to Dirac and myself, Dirac and I further showed (independently) that
every @(n; /„) contains fa.
@(5; 9) and ©(5; 10) present no new difficulties.
I did not carry out a similar discussion for graphs having 6 vertices. I only state
one result which seems interesting:
Mn; 6, 12) > - + c3n3<* , /2(n; 6, 12) < - + c^2
4 4
32
158 GRAPH THEORY

and in fact every @(n; [n2/4 + c^n312]) contains as subgraph an octahedron. I cannot
prove that lim^n; 6, 12) - n2/4)/n3/2 exists.
Now we return to the discussion of/^n; k, I) for general k. First of all I proved
that for k >J 3
(6) ft(n; k, k) <c'kn1 + UW2K
It seems likely that
(7) f^n; k, k) > c''n1 + 1/W
but I can prove (7) only for 3 f£ k ^ 5. I can
prove the weaker result a t>
(8) f1(n;k,k)> n1+£" Fi8- 4-
for a certain ek > 0.
I can also prove that every ®(n; [Cfc"n1 + 1/,:]) contains a C2t; the proof is more
difficult than the proof of (6).
For large values of k our three functions /,(n; k, I) do not suffice to describe
completely the many problems, since there are very many graphs @(/c; I), but we have
not succeeded in solving or even in classifying the many problems which can be posed
here; I will try to state here all the results which are known.
I showed that for n > n0(k), every ®(n; [n2/4] + 1) contains a C2k + i; in fact
there exists an absolute constant c so that every @(n; [n2/4] + l) contains every Cm
for 3 ^ m ^ en. The proof is not trivial. K(\nj2\, [(n + 1)/2]) shows that [n2/4] + 1
is best possible [13].
Now we investigate the range k < I ^ /c2/4. Kovari, the Turans [9] and
(independently) I proved that every @>(n;[ckn2~1/kJ) contains a K(k,k). It seems
likely that this result is best possible and in fact we conjectured
/1(n;2/c,/c2)>a,n2-1/fc;
but this is proved only for k = 2, and we could not even prove that
lim/(n;6, 9)/n3/2 = oo .
n— go
Further I proved that every ®(n; [Pkn2~1/k]) contains a K(k + 1, k + l) from
which one edge is perhaps missing (the structure of this graph is uniquely determined).
In the range k < I f£ [fe2/4] I do not have good estimates for/(n; k, I), I cannot
even prove that for fixed k and sufficiently large n.f(n; k, I) is a strictly monotone
function of I.
33
M M
159 EXTREMAL GRAPH THEORY

I proved that for every e > 0 there is an n = n(e) > 0 so that for k > k0(n) and
n > n0(k, e, rj),
(9) /(n;fc,[(l +r,)k] <n1+£
(the opposite inequality, with a different e, follows from (8)).
Further for every k ^ 2c and n > n0(fc)> ' < cfc>
(10) /(n;fe,0<"2"£
where e depends only on c. In fact the following stronger result holds: Every
@(n; [«2_c]) contains a subgraph of K([kj2], [fe/2]) which has at least ck edges for
every 2c < k ±£ n if n > n0{k) and e = e(c).
On the other hand, to every e > 0 there is a C = C(e) so that
(")
f(n; k, Ck) > n2
Instead of (11) the following sharper result can be proved: Let e' < e and C =
= C(e) be sufficiently large. If n > n0(e, e', C) then there exists a ®(n; [n2_£])
which does not contain a subgraph ®(/c; Ck) for every k < nc'. This result is nearly
the best possible, since it is not hard to show that every @(n; [n2_£]) contains
a ®(fe; Ck) for some k < Qn' where Ci = C^e) is sufficiently large.
I would like to state here one further result which can be proved by probabilistic
methods [10]: Let e > 0, C > 1 be arbitrary. There is a graph ®(n; Cn) so that
every subgraph of it spanned by m < tjn vertices has fewer than m{\ + e) edges;
r\ = n(z, C) could easily be estimated explicitly.
It is not hard to show that [11] for a > 1 every ®(n; [a«]) contains a circuit of
length < /Hog n, where /? depends only on a. Probably every ®(n; [an]) (a > 1)
contains a subgraph ©([/J, log n]; [/?, log n] + t) where /?, depends only on t.
Now we give a very short discussion of I > [fe2/4]. Dirac and I showed
independently that every @(n; [n2/4] + 1) contains, for every k f£ n, a ®(/c; [fe2/4] +
+ 1). In fact Dirac proved a more general theorem.
Considerably more difficult is the proof of the following result: To every k there
is an n0(k) so that for every n > n0(k) every (3(n; [n2/4] + 1) contain sa K(k, k)
with an extra edge (the structure of these graphs is uniquely determined) [13].
It is not hard to show by complete induction that for [(k + 1)/4] 5; u,
(12)
/i
(n;k,
r/c2i
—
_4_
\
+ u =
J
[n2l
—
_4_
= — + u .
It is easy to see that (12) no longer holds for u > [(k + l)/4], but the discontinuity
is not very sharp since it is easy to see by induction that if n S: k then
(13)
f h;k,
rvi
—
_4_
+
~k - r
2
\
=
/
r«2i
—
_4_
+
~n - r
2
34
160 GRAPH THEORY

and now there is a sharp discontinuity since it is not difficult to show that
(14) fln;k,
-fc2~
_4_
+
~k + r
2
\ ^
^
/
[n2l
—
_4_
J
+ /(
V
n
—
2_
fe, fe I ^ — + n
l+£k
by (8). We can determine the values of f(n; k, [k2/4] + I) for [(fc + 1)/4] S / <
< [(/c — l)/2], but the formulas are complicated and we omit them.
By very complicated arguments I can show that every ®(n; [n2/4 + n1 + 1/[,:/2]])
contains a K(k, k) and a circuit whose vertices are all the vertices of one of the /c-tuples.
In 1946 Stone and I proved [12] that for every e > 0 and every k if n > n0(e, k),
®(n; [(«2/4)(l + e)]) contains a K(k, k, k); by a refinement of our method I can in fact
show that for sufficiently large C every @(n; [n2/4 + Cn2~1/k~\) already contains
a K(k, k, k).
I do not pursue the investigations of/(n; k, I) further since a complete analysis is
hopeless at present and so far I have succeeded to find no new phenomena in the
//A
interval /c2/3 ^1½
Before completing the discussion of/((n; k, I) I would like to mention two further
problems: It is not hard to show that
/("; 7, 15)= -
r«2i
—
_4_
+
n + r
3
+ 1.
But I cannot decide the question, how many edges must a graph of n vertices have
in order that it contain a K(l, 3, 3)? Perhaps [«2/4] + n + 1 edges suffice for this
purpose. It is easy to see that [«2/4] + n edges are not sufficient. (Added in proof:
I succeeded in proving this conjecture.) Very many other such problems could be
stated.
Turan asked in a conversation to determine the smallest number of edges that
a graph of n vertices must have in order that it contain the various regular bodies.
For the tetrahedron the answer is m(n, 3) by [1], the octahedron has already been
discussed. The problem of the cube seems difficult. I can show that for sufficiently
large c every @(«, [cn3/2]) contains a hexagon and a vertex joined to three non
adjacent vertices of the hexagon but I cannot decide whether it contains a cube. The
icosahedron, dodecahedron and higher dimensional cubes have not been investigated
so far.
Before completing the paper I would like to state a few related results which cannot
be described in terms of our functions /;(n; k, I). Posa and I [11] proved that for
n > 24k, every @(n; (2k — 1) n — 2k2 + k + l) contains k vertex independent
circuits (i.e. k circuits which pairwise have no common vertex). But we have not
succeeded in solving the extremal problem for 3/c ^ n f£ 24/c, except for a few special
values of k.
35
161 EXTREMAL GRAPH THEORY

Posa proved that every ®(n; In — 3) contains a circuit with at least one diagonal
and that the result is false for @(n; In — 4). Czipszer found a very simple and
ingenious proof of this result; by his method one can easily show that for a certain c
and n > n0{k) every @(n; kn + c) contains a circuit with at least k — 1 diagonals
emenating from a vertex. It is easy to see that c 5: 1 — k2. Perhaps c = 1 — /c2?
For k = 2 this is Posa's result, and I can prove it for k = 3 and k = 4 also.
Finally I proved that for every e and r there is an n0 = n0(e, r) so that for every
n > n0(e, r) every @(n; [n2/4] + n{\ + e)) contains a circuit and r vertices not on
this circuit each of which is adjacent to every vertex of the circuit. It is easy to see
that this result does not hold for every @(n; [n2/4] + n).
References
[11 P. Turdn: Eine Extremalaufgabe aus der Graphentheorie (written in Hungarian). Mat.
es Fiz. Lapok 48 (1941) 436—452. See also P. Turdn: On the theory of graphs. Coll. Math.
3 (1954) 19—30.
[2] P. Erdos and T. Gallai: On the maximal paths and circuits of graphs. Acta Math. Hung.
Acad. Sci. JO (1959) 337—356. B. Andrdsfai: On the paths, circuits and loops of graphs
(written in Hungarian). Matematikai Lapok 13 (1962) 65—107.
[31 See P. Erdos and T. Gallai, [21, p. 354—356.
[41 P. Erdos: On sequences of integers no one of which divides the product of two others and
on some related problems. Mitt. Forschungsinst. Math. u. Mech. Tomsk 2 (1938) 74—82.
[5[ I. Reiman: Ober ein Problem von K. Zarankiewicz. Acta Math. Acad. Sci. Hungar. 9
(1958) 269—279.
[61 P. Erdos: Some theorems on graphs (in Hebrew). Riveon Lematematika 9 (1955) 13—17.
[71 G. Dirac: Extensions of Turan's theorem on graphs. Acta Hung. Acad. Sci. 74(1963)417—422.
[81 H. Cavallius: On a combinatorial problem. Coll. Math. 6 (1958) 59—65.
[91 T. Kovdri. V. T. Sds and P. Turdn: On a problem of K. Zarankiewicz. Coll. Math. 3 (1954)
50—57.
[101 P. Erdos: On circuits and subgraphs of chromatic graphs. Mathematika 9 (1962) 170—175.
The result in question is not proved explicitly in this paper, but can easily be deduced by the
method used in the proof of Lemma 2, p. 173.
[Ill P. Erdos and L. Posa: On the maximal number of disjoint circuits of a graph. Publ. Math.
Debrecen 9 (1962) 3—12, see p. 11 — 12.
[121 P. Erdos and A. H. Stone: On the structure of linear graphs. Bull. Amer. Math. Soc. 52
(1946) 1087—1091.
[131 p- Erdos: On the structure of linear graphs. Israel Journal of Math. 1 (1963) 156—160.
36
162 GRAPH THEORY

A PROBLEM IN GRAPH THEORY
P. ErdOs, A. Hajnal and J. W. Moon, University College, London and
Math. Inst, of the University of Budapest
A graph consists of a finite set of vertices some pairs of which are adjacent,
i.e., joined by an edge. No edge joins a vertex to itself and at most one edge joins
any two vertices. The degree of a vertex is the number of vertices adjacent to it.
The complete k-graph has k vertices and (|) edges.
We shall say that a graph G has property (n, k), where n and k are integers
with 2^k^n, if G has n vertices and the addition of any new edge increases
the number of complete ^-graphs contained in G. For example, let Ak{n) denote
a graph with n vertices and n(& —2)-(4¾1) edges which consist of a complete
(k — 2)-graph each vertex of which is also joined to each of the n — (k — 2)
remaining vertices. At(n) contains no complete ^-graphs but it is easily seen that with
the addition of any new edge a complete &-graph is formed. Hence, Ak(n) has
property (n, k).
163 EXTREMAL GRAPH THEORY

1108
MATHEMATICAL NOTES
[December
We wish to determine the "minimal (n, k) graphs," i.e., those graphs with
property (n, k) and with the minimal number of edges. We prove the following
result.
Theorem 1. For every pair of integers n and k, with l^k^n, the only minimal
(», k) graph is Ak(n).
We will apply Theorem 1 to prove a conjecture of Erdos and Gallai (see
[l]). A set of vertices is said to represent the edges of a graph if each edge
contains at least one of these vertices. A graph G is said to be edge p-critical if the
maximal number of vertices necessary to represent all the edges of G is p, but
if any edge is omitted the remaining edges can be represented by p— 1 vertices.
For example the complete (£+l)-graph is edge ^-critical. In [l] it is conjectured
that an edge ^-critical graph can have at most ("J1) edges. Theorem 1
immediately implies this conjecture. In fact we prove
Theorem 2. Every edge p-critical graph has at most ("J1) edges and the only
edge p-critical graph with ("J1) edges is the complete (p-\-l)-graph.
Finally we would like to state a conjecture. A bipartite graph (k, I) is a
bipartite graph having k green and / blue vertices. A complete bipartite graph
{k, k) is a graph where all green and blue vertices are adjacent. We now say
that a bipartite graph (n, m) has property (n, m, k, k) if any new edge increases
the number of complete bipartite (k, k) graphs in our graph (we assume k^n,
fcrgra).
Problem. Is it true that every (n, m) graph with property (n, m, k, k) has
at least (fe—l)(rc+m —fe + 1) edges?
A weaker conjecture would be that every bipartite graph (n, m) which
contains no complete bipartite (k, k) but which loses this property when any new
edge is added has at least (k — \){n-\-m — fe+1) edges.
One of the difficulties of proving these conjectures may be that the obvious
extremal graphs are certainly not unique, which fact may make an induction
proof difficult. One can easily formulate the analogous conjecture for property
(«, m, k, I), but we leave this to the reader.
Proof of Theorem 1. We first show that Ak(n) is a minimal (n, k) graph and
then we show that it is the only one. We begin by establishing the inequality
(1) fk(n) ^ fk{n - 1) + (k - 2), for n = k + 1, k + 2, • • • ,
where fk{n) denotes the number of edges in a minimal (n, k) graph.
Let G be any minimal (n, k) graph where n^k + i. There exist nonadjacent
vertices in G, say p and q, as the complete w-graph is clearly not a minimal (n, k)
graph. Since G+(p, q), the graph obtained from G by adding an edge joining
p and q, contains at least one more complete &-graph than G, it must be that p
and q are both adjacent to all the vertices of some complete (k — 2)-graph. Hence,
if we let G* denote the graph obtained from G by removing q and then joining
164 GRAPH THEORY

1964]
MATHEMATICAL NOTES
1109
p by an edge to every vertex which originally was adjacent to q but not to p,
it follows that G* has at least k — 2 fewer edges than G. We may assert that G*
has property (n — 1, k). For if a and b are nonadjacent vertices in G*, both
different from p, then the addition of the edge (a, b) still forms at least one new
complete &-graph since none of the complete ^-graphs formed by adding (a, b)
to G could have contained both p and q and in G* the vertex p can serve wherever
q was required before; in the remaining cases the addition of a new edge to G*
forms the same new complete ^-graphs as were formed by the addition of the
same edge to G. Since G* contains at least /i(w —1) edges, inequality (1) now
follows.
It is obvious thatfu(k) = (|)~ 1. This combined with (1) implies that
(2) h{n) ^( ) - 1 + (n - k)(k - 2) = n(k -2)-
for n = k + 1, k + 2, • • • .
But Ak(n) is an example of a graph having property (n, k) and with only
n(k — 2) — (4^1) edges. Therefore, it must be that Ak(n) is a minimal (n, k) graph
and that equality holds throughout in (1) and (2).
We now use induction to show that Ak{n) is the only minimal (n, k) graph.
For any fixed admissible value of k this is certainly the case when n = k. Assume
that the assertion is valid whenever k^n<m, for some integer m, and consider
any minimal (m, k) graph G. From the fact that equality holds in (1) it is not
difficult to see that G*, constructed as before, must be a minimal (m — 1, k)
graph. Hence, we may suppose that G* is the same as Ak(m—\).
If in G* the vertex p, using the same notation as before, is one of the k — 2
vertices adjacent to every other vertex in G*, then in G it must be that q is
adjacent to all the other k — 3 such vertices and to one of the remaining vertices.
This is so that the addition of the edge (p, q) to G will form at least one new
complete &-graph. Each of the other m — k vertices is adjacent to either p or q but
not both for otherwise p and q would be mutually adjacent to more than k — 2
vertices and G would contain more than ft(m) edges. We may suppose that one
such vertex h is not adjacent to p. But it is now easily seen that the addition
of the edge (p, h) would not form a new complete &-graph in G, contradicting
the definition of G. The only alternative is that p is one of the vertices of degree
fe — 2 in G*. From the definition of G* it now follows that G differs from G* only
by the presence of the vertex q of degree k — 2 which is adjacent to the same
k — 2 vertices as in p. This implies that G is the same as Ak{m) which completes
the proof of the theorem.
We may restate the above theorem in the following slightly weaker form:
Of all graphs with n vertices which contain no complete fc-graphs, where
2^k^n, but which lose this property when any new edge is added, the graph
Ak(n) and only that graph has the minimal number of edges. This statement
cr>
165 EXTREMAL GRAPH THEORY

1110
MATHEMATICAL NOTES
[December
could be considered as the dual of the theorem of Turan in [2], which treats the
corresponding problem of determining those graphs with this property which
have the maximal number of edges.
Proof of Theorem 2. If G has n vertices and is edge ^-critical then it is easy to
see that the maximum number of vertices, no two of which are adjacent, is
n — p, i.e., the complementary graph of G does not contain a complete (n — p +1)-
graph. But, since G is edge ^-critical, if we add any edge to the complementary
graph it will contain a complete (n — £ + l)-graph. Hence, by Theorem 1, the
number of edges in G is at most
on*-'- >-cr)]=cr)
with equality only for the complete (j> + l)-graph, which proves Theorem 2.
References
1. P. ErdOs and T. Gallai, On the minimal number of vertices representing the edges of a
graph, Publ. Math. Inst. Hung. Acad. Sci., 6 (1961) 181-203, see conjecture (3.3) p. 195.
2. P. Turan, On the theory of graphs, Colloq. Math., 3 (1954) 19-30.
Reprinted from the American Mathematical Monthly
Vol. 71, No. 10, December, 1964
166 GRAPH THEORY

ON COMPLETE TOPOLOGICAL SUBGRAPHS OF CERTAIN
GRAPHS
By
F. ERDOS and A. HAJNAL
Mathematical Institute of the Hungarian Academy of Sciences and
Department of Analysis I. of the Eotvos Lorand University, Budapest
(Received September 6, 1963)
Let G be a graph. We say that G contains a complete /c-gon if there are k
vertices of G any two of which are connected by an edge, we say that it contains
a complete topological /c-gon if it contains k vertices any two of which are
connected by paths no two of which have a common vertex (except endpoints).
Following G. Dirac we will denote complete /c-gons by </c> and complete
topological /c-gons by </c>,. G(k, I) denotes a graph of k vertices and I edges
I k 11'
k,\ | ]. P. Turan |1] proved that everv
the complete /c-gon is thus G
(0) G
n.
k-'l
2(/c-l)
(n--/*)-
r (mod/c- 1), 0 ^ r
contains a «i/c> and showed that Wiis result is best possible. Trivially
evened, n) contains a <3>, and G. Dirac [2] proved that every G(n, 2n—2)
contains a «=4>, and gave a G(n, 2n — 3) which does not contain a <4->,.
It has been conjectured that every G{n, 3n — 5) contains a <5>, but this has
never been proved and in fact it is not known if there exists a c so that every
G(n, [en]) contains a <5> ,. Denote by ti(k, n) the smallest integer so that every
G(n, h(k, n), contains a </;>-, It is easy to see that
(1)
h(k. n) > c1k'-n.
cl7 c2» • • • denote positive absolute constants (not necessarily the same if there
is no danger of misunderstanding).
To show (1) it will clearly suffice to show that the complete pair graph
(l, l) does not contain a complete"
for then if we consider
I
disjoint
copies of our (I, I) we obtain a graph of s= 2n vertices,
/- edges which contains
167 EXTREMAL GRAPH THEORY

144
P. ERDOS AND A. HAJNAL
no \|_4/-J/j. Choosing now I to he the greatest integer for which [_4r2J^A:
we clearly obtain a proof of (1).
Let xlt . . ., x„ yv ..., y, be the vertices of our (I, I). If it would contain
/T 11\ f 1]
anv AVl J/, we can assume that at least [2/- Jof its vertices are x,'s. To connect
any two with disjoint paths we clearly need more than /v,'s but there are only
/ of them, hence (1) is proved.
Perhaps
(2) h(k, n) -:- c2k2n
holds uniformly in k and n. Thus in particular any (J(n, c3n-) perhaps contains
/r --|\ i
a \Lc4h- j/(. We can prove this only if c3 ^ —. In fact we shall prove
6
1 / !"|\
Theorem 1. Let rg2,c3 -- . 7hen everv 0(n, c^n'1) contains \ c,n' J/<
2r + 2
where ci depends on c3.
We postpone the proof, but deduce the following
Corollary. Split the edges of a graph -=rc=»- into two classes, then at least
/\ J."]\
one of them contains a \Lc5«aJ/(.
The corollary follows immediately from Theorem I since at least one of
the classes contains — 1--=- — > — edges.
2 1 2 I 5 6
Denote by f(k, I) the smallest integer so that if we split the edges of an
--/(/c, /)> into two classes in an arbitrary way, either the first contains a <&==-
or the second an </>. Trivially f(k, 2) == k, /(2, I) = 1. Further it is known
|3] that
2^/(/^)^(^
9 I k
,^ .k + l-2
(3) c7\
k— 1
< f{k, 0=-, ,
i k ■ l
c8 fc* (loRfc) "--=/(*, 3) *=[
k+l
9
The exact determination or sharper estimation of i{k,l) seems a difficult
problem.
Denote further by f(k„ 1,) the smallest integer for which if we split the edges
of an </(/c(, /,)-=- into two classes in an arbitrary way either the first class
contains a </c=-( or the second class an </>-,. Finally f(k, 1,) denotes the smallest
integer for which if we split the edges of an </(/<;, 1,) into two classes in an
arbitrary way, then either the first class contains a -k-- or the second class
contains a -</>,. Trivially
168 GRAPH THEORY

ON COMPLETE TOPOLOGICAL SUBGRAPHS
ur>
KK <3>() -
k+\
~2 "
since unless 0 is a tree it contains a <3>, and the vertices of every tree can
be split into two sets none of which contains an edge. It seems likely that
(4) f(kt, it)<c,k and perhaps even f(k, lt)<c,k
but we can not prove (4) if />4. For I =- 4 both inequalities of (4) easily
follow from Dirac's result according to which every G(n, In —2) contains a <4=»r
(2), or the weaker conjecture: h(k, ri)-~c]'n would easily imply (4).
We shall prove
Theorem 2.
(<>
('0
(HI)
c9k- - f(kt,lt) < cwk-
^-(^0^-/(3,/,)5
/ + 1
c12/c3 (log k) 1 - /(/c, fc,).
In a paper of P. Erdos and R. Rado [4] the following partition symbol is
introduced-.
//(— (rn, in)'2 denotes the statement that if we split the edges of a complete
graph of power //; into two classes in an arbitrary way then there exists a
complete subgraph of power rn all whose edges belong to the same class, in^(m.m)-
denotes the negation of this statement.
We introduce the symbols iii-(m„ in,)2, m-*(m. //(,)2 which have a self
explanatory meaning. (Similarly as the notations f(k,, lt), f(k, I,).)
Theorem 3. Let m be any infinite cardinal. Then
m — (//;,, m,)-.
Remark. W. Sierpinski |5j proved 2^-^(^, tfj)'2.
Very likely m — (m, //(,)2 also holds for every insx,u. We can prove this only
in case rn is singular and is the sum of t*0 ca'dinals less than m: using a
theorem of a forthcoming triple paper with Rado [6j. We will not give the details.
Now we turn to the proofs of our Theorems.
Proof of Theorem 1. We need the following
and let Av
As be subsets of a set
Lemma. Let s be an integer, u =■
s
S satisfying
\S\ — n 'A,\ -' un (i ~ I s).
S denotes the number of elements of the set S). Then for some I-;</^s
'AiOAj. - nisu- I)
169 EXTREMAL GRAPH THEORY

14'; P. ERDOS AND A. HAJNAL
The proof follows immediately from the obvious inequality
sun =s V',4,. s n + v \AfnAr.
Let now cs > and let there be given a G(n, c3n'2) = G. A simple arsni-
2r + 2
ment shows that our G contains at least ci3n vertices of valency ^cun where
c13is an arbitrary number satisfying c14 -- 2c13 and c13 could easily be de-
r+ 1
termined explicitely as a function of c14). Denote these vertices of our G by
xx, ..., xp (p - [c13«]). To each x,- l^i^p we make correspond the set A,
which consists of the vertices connected to x, by an edge. Aj\s=cun. Thus
by our lemma among any r+ 1 A's, say Ailt ..., ^41, + 1 there are two say Ain -
and Ain for which
(5) Alhr\Aijv>cun.
Define now a graph G* spanned by the vertices xv ..., xp a* follows:
Two vertices x,- and Xj are connected in G* by an edge if A,- and A-s satisfy
(5). By (5) the maximum number of independent vertices of G* is at most r.
Hence by the second inequality of (3) G* contains a complete graph of at least
q vertices; yv . . .,yq, q & i\6nr Let now s = U',«r J, Q < cis sufficiently small.
A simple argument shows that the vertices yv ..., yh form a <s>„ in fact
any two vertices yl and yy can be connected by disjoint paths of lenght 2.
To see this observe that if we want to connect y-( to y,- /=- i by a path of length
two, by (5) there are clbn possible vertices we can use for this purpose and at
most
s 1
~-cisn (if c4 is sufficiently small) have been used up- this proves
that Vj ... }'s is an -<f>, in G and hence completes the proof of Theorem 1.
Proof of Theorem 2. The upper bound of the first inequality of Theorem 2
is just a restatement of the Corollary of Theorem 1. We only outline the proof
of the lower bound. It is well known and can be shown by simple probabilistic
arguments [8] that the edges of the graphs </o can be split into two classes
in such a way that if AA.-(log/f)-1 —°° then every subgraph of </;> of Ak
vertices contains \- o{I)\Af edges of both classes. Let now c, < - and consi-
■ 4 ' 4
der such a splitting of the edges of a complete graph of [c9k2] vertices. We
show that neither class contains a -=;/;>,. For if say the first class would
contain a </<>-, say x^ ...,x,,., then I t-o(l) fc- of the edges (x,, xy) \^i<j^k
is in the second class. Thus these — +o(l)\k- pairs of vertices (x,xy) have to
he connected by disjoint paths of length at least two (using edges of the first
170 GRAPH THEORY

ON COMPLETE TOPOLOGICAL SUBGRAPHS
147
class). But for this purpose we need at least | +0(1)
k- =- c9/c2 vertices (if k
=- k0(cg) is sufficiently large). Thus the first inequality of Theorem 2 is proved.
The upper bound of the second inequality of Theorem 2 is the upper bound
in the third inequality of (3). The lower bound follows easily from the lower
bound of the third inequality of (3). From this inequality it follows that we
can split the edges of an -<??=- into two classes so that the first class does not
/\ L ,\
contain a triangle and the second class does not contain a XLc^??2 log ti \/.
Thus it follows from (1) by a simple computation that for sufficiently large
/ :! 'TX
\ f18n4 (log n)2 J/ contains more than n edges of the first class. Thus
lc18"4(logn)-J/
if the second class would contain a \|_c18?/4 (Iogn)~J/( we would need more
than n vertices for the necessary disjoint paths — this completes the proof of
the second inequality.
Now we outline the proof of the third inequality of Theorem 2.
Split the vertices of a complete graph of n vertices into|_n;i (log?;)3 J - p
classes C,, each having nearly the same number of vertices (i. e. each C, 1 ?_ isp
contains [ns (log n):t ] or [ ?;•'(log ??)■'J + I vertices). Two vertices which are
in different C,'s are connected by an edge of the first class. The edges of each
C. we divide amongst the two classes in such a way that every complete
.subgraph of /?(?/) vertices of C, satisfying /?(?/) log n-3 --^ contains [n* (log n)'
edges of both classes. See p. 146 of this paper. A simple argument (used
already in the proof of Theorem 2) gives that the first class does not
/r 1 Ll\
contain a \U19??-! (log nY \/ and the second class does not contain a
/\ " I]\
\U'i9« ' (log nY \/t if cn, is sufficiently large.
We do not know how far the third inequality of Theorem 2 is from being
best possible, since we have no satisfactory upper bound for f(k, k,). we can
i
onlv show lim f(k, k,)k - I and we do not give the details since probablv this
h- -v
estimation is very poor. It seems possible that every G\n. en2}) contains a
'---[tj0c)>, any two vertex of which are connected by disjoint paths of length
one or two. This result if true would be very useful in deducing good upper
bounds for our function f(k, /:,) but we have not been successful in deciding it.
Proof of Theorem 3. Consider a set S of power m and assume that the
edges of the complete graph spanned by S are split into two classes 1 and II.
Define a two valued measure on the subsets of S so that all sets of power ■=??/
have measure 0. Without loss of generality we can assume that there is a
subset S' of measure I so that if x6 S' the set of vertices connected with x in class
I is of measure I. But then a simple argument by transfinite induction shows
that any two vertices of S' can be connected by disjoint paths of length two,
whose edges belong to class I (since if xe S', y € S' the set of vertices Z for which
171 EXTREMAL GRAPH THEORY

148
P. ERDOS AND A. HAJNAL
the edges (x, z) and (y, z) both belong to class 1 is of measure 1 and therefore
of power m). This completes the proof of Theorem 3.
In connection with Theorem I we can put the following problem: Let
c21< I, c22<c2i be two constants r^2 and let there be given n sets of measure
&c21 in (0, I) and determine the largest integer f(n, r, c21, c22) = / so that there
are always at least /sets any r of them have an intersection of measure -(½.1
One can easily obtain lower bounds for /(«, r, c21, c22) by Ramsay's theorem which
are not too bad for r — 2, in fact as in the proof of Theorem I we obtain from
the second inequality of (3) that
(6) /(n,2,c21,c22) > n
■'(cai , c22)
where e(c2], c22) depends only on c21 and c22. For r> 2 the lower bounds obtained
for /(n. r. c,,v c22) by Ramsay's theorem are probably very poor, quite possibly
f(n, r, c21,f22) > n'
(r, c2j, C2Z)
Finally we show that (6) is not very far from being best possible. We shall
show that
(7) fin, 2, —, c\-- nf{c)- for c--- —
■ ! 2 I 4
where /(c) is a function which we could easily determine explicitely.
Recently G. Katona proved the following conjecture of Chao-Ko, R.
Rado and P. Erdos. [7] (Katona's result is not yet published.) Let \S\ = 2m
and {At)j , 2U be a family of subsets of S so that for everv l^i</^u
'A,C\A]^2k. Then
(8)
(8) will easily imply (7). We define a graph as follows: Let \S\ -= 2m and let
the vertices of our graph be the subsets of S containing m or more elements.
We connect two vertices by an edge if the corresponding sets have fewer than
2cm elements in common \c -~ —
I 4,
maximum number nf independent vertices is
(9) 2 I 2m i "= (22'")1-*
rs.->cm\rn + r!
where « depends only on c (the inequality in (9) is well known and follows by
a simple computation). Our graph has =-2^1-1 vertices. Now make correspond
to the i-th element of S the interval I , — and to a subset the union of
I 2m 2m I
the intervals corresponding to the elements. An independent set of vertices
gives a collection of sets any two of which have an intersection of measure
at, but if two vertices are connected their intersection has measure <c, hence
(9) implies (7).
1 A well-known result of (9| states thet f(n, r, c51) u,)^r if n-na(r, i\it c,2).
. By the theorem of Katona stated above the
172 GRAPH THEORY

ON COMPLETE TOPOLOGICAL SUBGRAPHS
14!i
It is easy to see that if c -= — then our i^rapli contains no triangles, hence
6
our construction gives a simple example of a graph of n vertices which contains
no triangle and for whicli the maximum number of independent vertices is
less than n1 \
It is well known that /(x0, r, c21, c22) = x0 and it is not hard to prove that
if there are given m sets of measure >c21 there are always m of them so that
the intersection of any x„ of them has measure -»c21.
References
[11 P. Turan, On the theory of graphs, Colloquium Math., 3 (1954), 19-30, see also Mat.
es Flz. Lapok, 48 (1941), 436-452 (in Hungarian).
[2] O. Dirac, In abstrakten Orapheti vorhatidene vollstandige 4-Graphen und ihre Unter-
teilungen, Math. Nachr., 22 (I960), 61-85.
f3] P. Erdos and 0. Szekeres, A combinatorial problem in geometry, Comp. Math., 2 (1935),
463-470; C. Frasnay, Sur des fonctions d'entiers so rapportant au theoreme de
Ramsay, C. R. Acad. Sci. Francais, 256 (1963), 2507-2510; P. Erdos, Some remarks on
the theory of graphs, Bull. Amer. Math. Soc, 53 (1947), 292-299; P. Erdos, Graph
theory and probability II., Canad. J. Math., 13 (1961), 346-352.
[4) P. Erd6s and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc, 62
(1956), 427-489.
[5] W. Sierpinski, Sur un probleme de la th^orie des relations, Annali R. Scuola Norm.
Sup. de Pisa, Ser. 2, 2 (1933), 285-287.
[6] P. Erdos, A. Hajnal and R. Rado, Partition relations for cardinals. This paper is
expected to appear in Acta Math. Acad. Sci. Hung.
[7J P. Erdos, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart.
J. Math., 12 (1961), 313-320.
[8 | Erd6s PAl, Ramsay 6s Van der Waerden tetclevel kapcsolatos kombinatorikai kerde-
sekr6I, Mat. Lapok, 14 (1963). 29-38 (In Hungarian).
173 EXTREMAL GRAPH THEORY

Reprinted from
ISRAEL JOURNAL OF MATHEMATICS
Volume 2, Nunber 3, 1964
ON EXTREMAL PROBLEMS
OF GRAPHS AND GENERALIZED GRAPHS
BY
P. ERDOS
ABSTRACT
An r-graph is a graph whose basic elements are its vertices and r-tuples. It is
proved that to every I and r there is an e(l, r) so that for n > n0 every r-graph
of/i vertices and n'~Hl<r)/--tuples contains/- . I vertices x(J',l fg/5£/-, 1 </ 5i /,
so that all the /--tuples (x,/1', *;2(2V-- , *;r(r)) occur in the /--graph.
By an r-graph G {r)(r ^ 2) we shall mean a graph whose basic elements are its
vertices and r-tuples; for r = 2 we obtain the ordinary graphs. These generalised
graphs have not yet been investigated very much. Gin(n; m) will denote an r-graph
of n vertices and m r-tuples; G(r)(n;( J), the complete r-graph of n vertices,
will be denoted by K(r)(n), i.e., K(r)(n) contains all the r-tuples formed from
n elements. K(r\nu ••-,«,) will denote the r-graph of £j=i "/ vertices andFJ^jn,
r-tuples defined as follows: The vertices are
and the r-tuples of our r-graph are the ]~[j = i nj »*-tuples
Thus K{2)(2,2) is simply a rectangle.
Denote by/,w(n) the smallest integer so that every G(n(n ;//r)) contains a
complete r-graph of I vertices.
As is well known Turan [5] determined/,(2)(n) for every I and n and also proved
that there is a unique G(2,(n;//2)(n) — 1) which contains no complete 2-graph
of I vertices (ordinary graphs have to be denoted as 2-graphs here). In particular
/3(2(n) = [n2/4] + l.
For r > 2 the determination of fir\n) seems to be a very difficult question
which is unsolved for all r > 2, I > r. (This question was also posed by Turan.
Turan in particular conjectured that
(1) /<» = n\n - 1).
Received August 18, 1964.
183
174 GRAPH THEORY

184 P. ERDOS [September
Vera T. Sos observed that if (1) is true then the extreme graphs are certainly
not unique, and this may be one reason why the proof of (1) is difficult.
It is easy to see that
s?s <»>/(;H"
exists, but the value of c/r) is not known for any r > 2,1 > r.
Denote by/(n; X(r:i(/],,/r)) the smallest integer so that every
G('W(n;K(,)(Z,. ■",',))
contains a K(r\lir-,lr). UH-^J^n we define: f{n\K{,\lu--,lr))= (")+!•
In particular, f (n ; K (2)(2, 2)) is the smallest integer so that every
G(2\n;f(n; K(2)(2,2)) contains a rectangle. E. Klein and I [1] proved that
(2) ■ cln3<i<f(n;K<2X2,2))<cln312.
Very likely
(3) lim f{n ■ K(2\2,2))/n3/2 = -*
■ = aj 2^/2
but it is not even known that the limit in (3) exists.
Kdvari, the Turans [4] and I showed that
f(n;K(2\l,I))<cn2-1f.
Probabby
(4) f(n;K(2\l,l))>c'n2-(1"\
but we are unable to prove (4) for I > 2.
Stone and I [2] proved that for every e > 0 and a sufficiently small ct and
n > n0(e)
(4') /(n;K(2)([c£logn],[c£logn])) <en\
It can be shown by probabilistic methods (similar to those used in [4] that for
sufficiently large c£
(4") /(„;K(2)([cElog„],[c£logn])) > (1 - e)^ )•
In the present paper we first of all shall prove the following
175 EXTREMAL GRAPH THEORY

1964] EXTREMAL PROBLEMS OF GRAPHS AND GENERALISED GRAPHS 185
Theorem 1. Let n > n0(r, I), I > 1. Then for sufficiently large C=(C is
independent of n, r, I)
(5) nr-C(/r"></(n;X(r>(/,...,0)gnr"(1/r"'.
We only prove the upper bound and will discuss the lower bound later. We use
induction with respect to r. First we prove the right side inequality of (5) for
r = 2, (this is substantially contained in [6], then we use induction with respect
to r.
Consider now the case r — 2. Denote the vertices of our graph G(2\n;t),
t^n2"11' by Xj,-••,*„, and by i>(x,) we denote the valence of xs (i.e. i;(x,) denotes
the number of edges incident to x,). Clearly
(6) Z i>(xi)^2n2-1" •
Let X]'V",Xu(xi> be those x/s which are joined to x,. Form all the /-tuples from
these vertices for all i, 1 5= i 5= n. The number of these /-tuples (each counted
with the proper multiplicity) clearly equals
i(f>).
An elementary inequality states that the sum (7) is a minimum if all the v(x,) are
equal (S"=1 f(x,) satisfies (6)11 j = 0 if y < I J. Thus by a simple computation
for n > n0(l)
l-i/i
inn ,)>•(",)■
Hence there are / vertices yit ■■■,y1 which are all joined to the same / vertices
Xy,, ••• Xj-j, which means that our graph contains a K(2\l, /) as stated.
Assume now that the right side inequality of (5) holds for r — 1, we shall prove
it for r. First we need the following
Lemma. Let S be a set of N elements yit ■■■, yw and let Ah lfgifgn, be subsets of
J". (Assume thatQ(Ai denotes the number of elements of At)
(8) i;u—•
176 GRAPH THEORY

186 P. ERDOS [September
Then ifn ^ 2l2wl there are I distinct A's, Au,---,Ait, so that
1 N
(9) P/^^-
Denote by fi(y) the characteristic function of Ax (i.e., ft(yj) = 1 if y- is in At, and
is 0 otherwise). Put
F(y) = 2 f,(y).
Clearly by (8)
(10) IFW^.
Thus from (10) we obtain by an elementary inequality that
I F(yjy
is minimal if for all; F{yj) =«= n/w, or
(11) ,?/<*>'*" (£)'•
On the other hand we obtain by a simple argument
(12) E%)'=E4n4n...n4
where the summation in (12) is extended over all the choices of »',,•■■> h{l ^ ir= n).
There are J7' = o(" — 0 = "' choices of it, ••-, i, where all the indices are distinct,
and if (9) would be false the contribution of these terms to the sum (12) would be
less than
The number of the summands in (12) where not all the indices are distinct is
easily seen to be less than /2n!_1. The contribution of each of these terms to the
right side of (12) is clearly at most N. Thus finally from (12) and (13)
177 EXTREMAL GRAPH THEORY

1964] EXTREMAL PROBLEMS OF GRAPHS AND GENERALISED GRAPHS 187
(14) if^^+zV-itf.
Now since n 2: 2/2w' (14) contradicts (11). Thus (9) must hold for at least one
choice of distinct At's, 1 fg i < I which completes the proof of the Lemma.
The Lemma is clearly not best possible, but is good enough for our purpose.
Now we are ready to prove the right hand inequality of (5) for r > 2. Assume
that it has already been proved for r — 1 if n > n0(r — 1, /), and we are going to
prove it for r if n > n0(r, I). Suppose then that we have a G(r\n; t) with
t 2: n,r_(1/!'"1). Denote by xlt ---.x,, the vertices of our G(r) and by ylt ---,^^,
N=z ( 1 J. the set of all (r - l)-tuples formed from the x;, 1 g i g n. P[r\ • • -, PJr)
denotes the r r-tuples of our G(r\n;i). To apply our Lemma denote by At the set
of all (r — l)-tuples ys such that yjUXj = P£r) for some 1 ^ k < t. We evidently
have
Thus our Lemma applies with N = I I , w = n1/r 7r'> since for
n > n0(r>') " ^ 2/½1 is clearly satisfied. We thus obtain that there are /distinct
A's Al{, ■••,Ah for which
(15) £)Ah ^ y (r _" J (r! n-1""')' > n"^^.
By (15) there are more than ,Jr~1_1'r~2 (r - l)-tuples
(16) p(r» ,-pt(rl}, r1>n,-i-i/,p-a,
so that all the r-tuples
(17) (x,. * P(;~l) x.upf"1' 1 £ s £ /, 1 g j < f,
is clearly satisfied by our construction) are one of the P^'s of our G(r\n;t).
These (r — l)-tuples define a
which by our induction hypothesis contains aK(r_1)(/, •••,/) if n > 1 + n0(r — 1, /).
By (17) this implies that our G(r\n;t) contains a K(r)(/, •••,/) which proves the
right side inequality of (5).
178 GRAPH THEORY

188
P. erdOs
[September
Theorem 1 easily implies the following
Corollary. Let n > n0 (r, I), r, ^ n, i = 1, ••-,r. Let
:W
{i •,:")."> ^T(jt
be s subgraph of K{r) (tu---,tr). Then G(r)( £,- = 11,; U) contains a K(r\l, ••-, /)
A set of fj elements can be decomposed into the (non-disjoint) union of
[f,7[(n/r)]] + 1 sets having \_n/r] elements. Hence clearly a K(r)(tlt ••-, tr) can be
decomposed into the union of at most
n
~ +1 <—n '-
K(r ([n/r], ■■-, [i/^'s (the union is non-disjoint but every r-tuple of K(r){t,, ■■-, rr)
occurs in at least one of theK(r\'[n/r],---,[n/r])'s). Thus at least one of these
K(,)([n/r],---,[n/r]'s say Kir) contains at least n,-1/'''"r-tuples. Our K({} has
'"[w/'"] ^ « vertices, hence the corollary follows from theorem 1 (the right side
inequality of (5)).
The corollary has applications in number theory, this will be discussed in a
subsequent paper.
Without much change in the proof of Theorem 1 we could show that the right
side inequality of (5) holds for every n ^ rl. But in fact the right side of (5) is
trivial if I > 2 (logn)1/r""' , for then
0
<n'-(,/,r-'>
Further we can prove the following
Theorem 2. Let a>0be any number, n > n0(a, I, r), 2 < I < a(logn),/(r_1).
Then we have for a sufficiently large absolute constant Cj
(18)
(;)"-c,/"-'</(";^a,-,/)) < (nry-if'""-
We do not prove the upper bound of (18) since it is similar to that of (5), we have
only to carry out the esiimations and the induction with respect to r a little more
carefully. The most interesting special cases are those which correspond to (4')
and (4"). For every e > 0 and a sufficiently small c(£r)
(18') /(n;K(r)([c(;>(logn)1/(r-^,--, [c^logn)1^-1'] < enr.
179 EXTREMAL GRAPH THEORY

1964] EXTREMAL PROBLEMS OF GRAPHS AND GENERALISED GRAPHS 189
(18') in fact follows from the fact that the right side inequality of (5) holds for
every n ^ Ir. Further we have for a sufficiently large c/r)
(18") /(n;K('>([Ct)(logn)1^-1>],---:,[ct)(logn)1^-1)]))>(l -e)(") .
To give the reader an illustration how to prove the lower bound of (5) and (18)
we prove in full detail (18") for e = ^. In fact we prove a stronger result. If G(r\n ;m)
is an r-graph then G(r\n;m) will denote its complementary graph i.e. the
G(r)(n;( I ~m) whose r-tuples are precisely those which do not occur in G(r\n; m).
Theorem 3. Putt = [4(logn) ^f-')] + i por every n there js a G(r\n)* so that
neither G(r\n) nor G{r)(n) contains a K(r\t,---,t).
The proof will follow very closely the method used in [3]. The total number
of r-graphs G(r)(ri) is clearly 2{"/r).The number of those r-graphs for which either
G^"\n) or G"(r\n) contains a K(r)(t, ■■■, t) having the vertices x(/\ 1 <; i S t; 1 Sj S r
clearly equals
2 • 2{",r)"r,
since the tr r-tuples of our K(r)(t, ■■-,{) either all have to belong to our G(r)(n), or
none of them belong to our G<r\n). The number of choices for our K(r\t,■■■,{)
is clearly less than n"\r\ Sin". Therefore the number of graphs G(r)(n) for which
G(r)(n) or G(r\n) contains a K(r\t,■■■,() is clearly less than
nn . 2("/r)-'r < 2("/r).
Thus there is a Gir)(n) so that neither G(r)(n) nor G^r\n) contains a K(r\t,---,t), as
stated.
The proof of the lower bound of (5) and (18) uses the same methods combined
with the methods of [4].
It is possible that
lim/(n;K('U-",/))/n'-(1/r">
n — oo
exists and is different from 0 (by (5) it is < Al), but as stated in (3) this is not
even known for r = I = 2.
References
1. P. Erdfls, On sequences of integers no one of which divides the product of two others and
on some related problems. Izv. Nauk Mat. i Mech. Tomsk 2 (1938), 74-82. (The best estimation
off(n; K2(2,2)) is due to I. Reiman, Ober ein Problem von K. Zaranbievicz, Acta Math. Hung.
Acad. Sci., 9 (1958), 269-273.
* G^(n) is an /^graph having n vertices, the number of its r-tuples is not specified.
180 GRAPH THEORY

190
P. ERDOS
2. P. Erdos and A. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc.
52 (1946), 1087-1091.
3. P. Erdos, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947),
292-294.
4. P. Erdos and A. Rinyi, On the evolution of random graphs, Publ. Inst. Hung. Acad. Sci.
5(1960), 17-61.
5. P. Turan, On the theory of graphs, Colloqium Math. 3 (1954), 19-30.
6. T. Kovdri, Sos V.T. and P. Turan, On a problem of K. Zarankiewicz, Colloquium Math.
3 (1954), 50-57.
Mathematical Institute
University of Budapest
181 EXTREMAL GRAPH THEORY

COLLOQUIUM MATHEMATICUM
VOL. XIII 1965 FASC. 2
ON AN EXTREMAL PROBLEM IN GRAPH THEORY
BY
P. EEDOS (BUDAPEST)
In the present paper G(n;l) denotes a graph of n vertices and I
edges, Kv — the complete graph of p vertices, i. e. olp; l\\\, K(pif ...
...,pr) —the complete r-ehroniatic graph with pt vertices of the i-th
colour in which every two vertices of different colour arc adjacent.
Vertices of our graphs will be denoted by or, y, ..., edges by (x,y).
The valence v(,r) of x is the number of edges adjacent to x.
Denote by m(ri;p) the smallesl integer so that every G(v; ni(n;p))
contains a Kv. Turan [t>] (comp. also [7]) determined m(n;p) and also
showed that the only G[n; m(n;p)—l) which contains no K)t is K(mu ...
..., m,,_j), where
> rrij = n and /;t, = or - +1.
Dirac [1] and 1 (independently) proved that every G(n; m(n; p))
contains a Kp+] from which one edge is missing. In fact, the following
stronger result also holds:
There is a constant cv so that every G[n; m(n;p)) contains a Kv^j
and evn vertices each of which is joined to every vertex of our Kp_i
([2], Lemma 2 (1)).
Denote by u(n;p) the smallest integer such that every G[n;u{n;p))
contains a K(pt p). The value of u(n;p) is not known and its
determination seems to be a very difficult problem. As far as I know the first result
in this direction is due to E. Klein and myself [3]; we proved
(1) a]M3/2 < u(n; 2) < a2w3/2.
(1) This lemma concerns only the case p = 3 but tlao same proof works in the
general case.
182 GRAPH THEORY

252
P. ERDOS
Probably \imii(n; 2)/w3/2 = 1/2J''2, but it is not even known that
this limit exists. The best result in this direction is due to Eeiman [5]
who among others proved that
lim supw(»?; 2)jn3t~ < —, lim infu(n; 2)//t3/2 > ——.
n^x> 2 n^co 21-2
Kovari, Hos and Turan [4] tiiid independently I proved that for
a suitable constant fin
(2) «(m;p) <Pvn2-vp.
Probably u(n; p) > fjpn2"v'', but this is known only for p = 2
(see [1]).
In this note we prove the following refinement of (2):
Theorem 1. There is a constant yp such that every G(n; [yr,n~ 3,'J)])
contains a K(p-\-l, p-\-l) from ivhich one edge is missing.
Remarks. Clearly the structure of a K(p-\-l, p-\-\) from which
one edge is missing is uniquely determined.
One could conjecture (by analogy to [1]) that every G[n; u(n;p))
contains a K(p-\-l, p-\-l) from which one edge is missing. This Mould
of course be a much stronger result than Theorem 1, but, if true, it will
be hard to prove since we do uot know the value of v(n;p) and have
no idea of the structure of the extremal graphs G[n; it (n; p)— l) which
do not contain a K(p,p).
Instead of Theorem 1 we shall prove the following sharper
Theorem 2. Let I > p be any integer. Then there is a constant yjKl
such that for n>n0(p,l) every G(n; [ylhl>i2 ~]i"]) contains a subgraph
H(p,l,l) of the following structure: the vertices of II(p, I, I) are xl, ..., xx\
yi,---,1/1 and its edges are all {%i,yj), where at least one of the indices i
or j is < p.
In other words, H(p,l,l) is K(l,l) from which the edges (.^,2/,-),
min(t,j) > p, are missing.
First we prove two Lemmas.
Lemma 1. Every G(n, m) contains a subgraph G' each vertex of which
has valence (in G') not less than [m/n].
If Lemma 1 would be false we could clearly order the vertices of
G(n; m) into a sequence x1,x2, . -., xn where for every i, 1 < i < n, xf is
joined to fewer than [m/n] vertices xu i <j < n. But this would imply
that the number of edges of G(n;m) is less than m. This contradiction
proves the Lemma.
183 EXTREMAL GRAPH THEORY

EXTREMAL PROBLEM IN UltAPH TJIEORV 253
Consider now our G(n; Xyvj,n~ 1/?']). By Lemma 1 it has a subgraph
G(X; m) each vertex of which has valence u = {yVilnl~l/p]. Now we prove
Lemma 2. Let cvl > 0 &e any constant. Then if ypX is sufficiently
large, our G(N;m) contains a K(p—l,s) with s = \cVyX n11"].
For each vertex y of G(N;m) consider all the {p — l)-tuples formed
from the vertices which are joined to y. Since by assumption y is joined
to at least u vertices, the number of these (p—1)-tuples counted for
each y separately is at least N\ __.]. Now since N < n, we obtain by
a simple calculation that for sufficiently large ypj
»1A)
(3) A'L"-.1^..^(^
Thus to some (p— l)-tuples correspond more than s — [cpX nltv~\
vertices y, i. e. (3) implies that there are p—1 vertices xx, ..., xp_t which
are all joined to the same s vertices y±, ..., ys. In other words, our graph
contains a K(p~ 1, s) and Lemma 2 is proved.
Now we are ready to prove Theorem 2. Denote by zl} • • •} ^n—p-s+i
the remaining vertices of G[N;m), i.e. those vertices which are not
included in K(p—l,s). By our assumption the valence (in G(N;m))
of each y is at least u and clearly for yv x > 2cp x and sufficiently large n,
s-{-p < w/2, hence each y is joined to more than u/2z's. Hence there
are more than ws/2 edges joining the y's with the, z's. Denote now by
v'[Zj) the number of ?/'s which are joined to Zj (1 <j < ^-p — s + 1).
Clearly
N—P—S+]
v^ us
(4) 2j *'(**>>-^-
1=1
and (£' denotes that the summation is extended only over the z-t for
which v'(Zj) > p-\-l)
(5) JV (*,-)> -~ -(p+l)(N-p-s+l) >-— -n(p+l)>-yvJctKln
for sufficiently large cpX and yp x.
Form now for every z,- satisfying ¢/(¾) ^ p + Z all the ^-tuples from
the y's which are joined to «,-. The number of these ^-tuples, counted for
each z; separately, clearly equals
(6) y
("'?')•
184 GRAPH THEORY

254
P. ERIlOS
Using (5) we obtain from an elementary inequality that the sum (6)
is minimal if all the v'(z}) are as nearly equal as possible and if their
number is as large as possible (it is < n). Thus by a simple computation
we get
(7) r(v'(;)>n({[^])>(i-P+i)(;)
for .sufficiently large ylty Formula (7) implies that the number of these
multiply counted p-tuples is larger than I — p+1 times the number of
all the ^-tuples formed from the s distinguished y's of K(p — 1, s). Hence
there are I — p+1 z's, say zlt ..., zt_p+l, satisfying
(8) *'(*)> P + h 1 <*' ^J-p + 1
(only v'(zi)^ '. will be needed) and which are all joined to the same
p y's, say to y1, ...,yv. JBy (8; we can further assume that zl is joined to
2/jj+ij ••• j Vi- Let ajj, ..., a?j,_1 be the distinguished p— 1 x's of K(p —
— l,s). Now the even graph spanned by xx, ..., xp_ i, z1, ..., zt p+1;
yi,---,yp,yv+i,---,yi is clearly an H(p,l,l), since, by Lemma 2,
x1, ..., xv_i are all joined to all the y's, ylt .--,¾ are joined to all the z;
(1 <j < I — p+1) by the argument following (7) and s1 is joined to z,
(p+1 <j < 2) by construction. Thus the proof of Theorem 2 is
complete.
REFERENCED
[1] G. Pirae, Extensions of Turdn's theorem on graphs, Acta Mathematics
Academiae Soientiarum Hungavicae 14 (1963), p. 417-422.
[2J P. Erdiis, On a theorem of Radeniachey-Turdn, Illinois Journal of
Mathematics G (1962), p. 122- 127.
[3] — On sequences of integers no one of which divides the product of two others
and on some related problems, Mitteilungen des Forsehungsin&tituies t'iir MatbematiU
und Mechanik, Tomsk, 2 (1938), p. 74-82.
[4] P. Kovari, V. T. Sos and P. Tuian, On a problem of K. Zarunkiewiez,
Colloquium Mathematicum 3 (1954), p. 50-57.
[5] I. Eeiman, Vber eiv Problem von K. Zarankiewicz, Acta MatbematSca
Academiae Scientiarmn Hungaricae 9 (1958), p. 269-279.
[6] P. Tvtran, Eine Extremalaufgabe aus der Graphentheorie, Matematikai cs
Fizikai Lapok 48 (1941), p. 436-452 (in Hungarian).
[7] — On the theory of graphs, Colloquium Mathematicum 3 (1954), p. 19-30.
Recu par la Redaction le 6. 10. 1964
185 EXTREMAL GRAPH THEORY

A PROBLEM ON INDEPENDENT r-TUPLES
By
P. ERDOS
Department of Analysis of the EOtvos Lorand University, Budapest
(Received October 2, 1964)
G(n; /) denotes a graph of n vertices and / edges. A set of edges is called
independent if no two of them have a vertex in common. Gallai and I [1]
proved that if
(1) /^maxfj2^1), (*-!)(„-*+!) + (*-!)|
then G(n; /) contains k independent edges. It is easy to see that the above result
is best possible since the complete graph of 2/c-l vertices and the graph of
vertices Xj, . . ., x,.^; yu . . ., yn-,l+1 and edges (x,, x;), 1 =/</' ==k- 1 ; (x,., yj),
l=s/=sfc—1, 1 =sy; ^, n — k+ 1 clearly does not contain k independent edges.
By an r-graph G(r) we shall mean a graph whose basic elements are its
vertices and r-tuples; for r = 2 we obtain the ordinary graphs. G(r) (n; m) will
denote an r-graph of n vertices and m r-tuples. Forr>2 these generalised graphs
have not yet been investigated very much. A set of r-tuples is called independent
if no two of them have a vertex in common.
f(n; r, k) denotes the smallest integer so that every G(r)(n; /(«; r, k)) contains
k independent r-tuples. (1) implies that
(2) f[n-2,k)= l+maxj^1], (^-1)(^-+1) + ^1^.
It does not seem easy to determine /(;;; r, k) for r^2 and every k. For k = 2
Ko. Rado and I [2] proved that for /;?^2r
(3) Kn-r,2) = \n~\\+\.
I r— I I
The case n < 2r is trivial since then no two r-tuples are independent.
Denote by g(n; r, k- 1) the number of those r-tuples formed from the
elements xt, . . ., xn each of which contain at least one of the elements
xv . . ., x, _j. Clearly f(n; r, k) =- g(n; r,k—\) and a simple computation shows
that
(4) g{n;r,k-\)=^
k-\\ui-k.+
i I r-i
(*-')r-*rli
I r— 1 I
186 GRAPH THEORY

94
P. ERDCS
where the dash indicates that i runs from 1 to min (r, k- 1).
Now we prove the following
Theorem. For n>crk (cr is a constant which depends only on r)
f(n;r,k) = 1 +g{n; r, k- 1).
The proof uses induction with respect to k. For k = 2 the result is known
[2). We assume that it holds for k— 1 and prove it for k .
Let n>crk and consider an arbitrary G(r\fi; l+g(n; r, k- 1)). Denote by
v(x,) the number of r-tuples in our G(r\n; 1 +g(n; r, k— 1)) which contain xt.
Without loss of generality we can assume that max v(x;) = v(x1). We
distinguish two cases. Assume first t3|s"
(5) r(x,)< ,+g(n;r'"'^
(k-l)r
and let /?j, . . ., Rt be a maximal system of independent r-tuples of our G(r).
We show
(6) / ^ k.
If (6) would be false our r-tuples Rv ...,/?, would contain at most (k— l)r
vertices and by (5) the number of r-tuples containing any of these vertices
is less than
l+g{n;r,k-l).
Thus our G(r\n; 1 +g(n; r, k— 1)) contains an Rl+1 which is independent of all
the /?,., l=sj=s/, which contradicts the maximality of Rv ...,/?„ hence /<k
leads to a contradiction, which proves (6) and disposes of the first case.
Now we consider the second case, that is, we assume
(7) ,^)-1^^-0
(k-l)r
Consider now the r-graph G(r) whose vertices are x2, . ., x„ and whose
r-tuples are those r-tuples of our G(r) (n; l+g(rc; r,k— 1)) which do not contain
xr The number of r-tuples of G[r) is clearly at least
(8) l+g(n;r,*-l)-
l+g(n-l,r,k- 1),
since there are at most r-tuples containing xx. Thus by our induction
: r-
hypothesis Gir1) contains at least k— 1 independent r-tuples Rlt ...,Rk_1.
The proof of our Theorem will be complete if we succeed to show that there
is an r-tuple of our G(r\n; 1 +g(n; r, k- 1)) containing x1 which does not
contain any of the (k— l)r vertices of Rlr ..., Rk-x- To see this observe that the
number of r-tuples containing xa and x,- is at most i , and therefore the
number of r-tuples containing Xj and one of the vertices of R1: ..., /?^-i is at
187 EXTREMAL GRAPH THEORY

A PROBLEM ON INDEPENDENT /--TUPLES
95
most (k- l)r i . By (7) and (4) we obtain by a simple computation that
I r— 2'
for n>crk
{k-\)r\n~2\ ^v{Xl);
\r-2)
hence there is an r-tuple of our G(r\n; 1 +g(rc; r, k - 1)) containing Xj which is
disioint from Rv ..., Rk-V as stated. This completes the proof of our
theorem.
It is not impossible that
(9) f{n;r,k)= 1 +max |(r&~ '), g(n;r, *-1)J.
For r = 2 (9) is implied by (1) and for k = 2 (9) is proved in [2], but the general
case seems elusive.
References
[1] P. Erdos and T. Gallai, On the maximal paths and circuits of graphs, Acta Math. Acad.
Sci. Hung., 10 (1959), 337-357.
[2] P. Erdos, Chao Ko and R. Rado, Intersection theorems for systems of finite sets,
Quarterly J. of Math.. 12 (1961), 313-320.
188 GRAPH THEORY

Reprinted from Journal of combinatorial theory Vol. ], No. ], June ]966
All Rights Reserved by Academic Press, New York and London Printed in Italy
NOTES
On the Construction of Certain Graphs
Denote by G(n) a graph of n vertices and by G(n; m) a graph of n
vertices and m edges. 1(G) denotes the cardinal number of the largest
independent set of vertices (i.e., the largest set xtl, ..., xn, r = 1(G)
of vertices of G no two of which are joined by an edge). v(x), the valency
of the vertex x, denotes the number of edges incident to x, Cj ... will
denote positive absolute constants.
1. Turan [8] proved that every G(n; [n2/4] + 1) contains a triangle
and that the only graph G(n; [n2/4]) which does not contain a triangle
is defined as follows: Its vertices are xv ..., x[nn]; yit ..., yun+iy2]>
and its edges are (x„ y,), 1 < / < [n/2], 1 < j < [(« + 1)/2]; in other
words if G(n; [n2j4]) does not contain a triangle then
'H"=m))-
Andrasfai [1] has investigated the following question: Let u < [(n— 1)/2].
Determine the largest integer f (n, u) for which there is a G(n; f (n, «))
which contains no triangle and for which 1(G) < u. Andrasfai
determines f (n, u) for u > [2nj5]. It is clear that f(n, u) < unjl since the
v(x) vertices joined to x must be independent (for otherwise our graph
would contain a triangle); hence v(x) < u for all vertices of G thus
G has at most unjl edges. Andrasfai [1] in fact determines all graphs
for which
f(n,u) = unjl (1)
for u> [Inj5] and gives some examples of graphs satisfying (1) for
u > n/3.
149
n -r
189 EXTREMAL GRAPH THEORY

150
ERDOS
In the present note, I will construct graphs for which (1) holds and
nr\ 1-c-mxi) 5 log 2 - 3 log 3
u =- 1(G) — n1 c+°a> , c = — . (2)
v 2 log 2 K '
Denote by g(n) the largest integer so that every graph of n vertices
which contains no triangle satisfies I(G (n)) > g(n). A very special
case of the well-known theorem of Ramsay [7] implies g(n) - * oo as
n —> oo. Szekeres and I [2] proved that g(n) > V 2n -\- 0(1) and I
showed first by a direct construction that g(n) < n1'11 [3] and later
by a "probabilistic" method that g(n) < x2n1'2 log n. I cannot at present
decide whether g(n) < c3nlri is true, in fact perhaps g(n) =■- V 2n + 0(1).
It would be of interest to construct all graphs satisfying (1) — this may
be difficult or impossible — or at least to decide if (1) is possible if
u = ni/2+p 1 cannot even show that f(n, u) = (1 — o(l)) un/2 can
hold if u = nv.Mc. The construction given here does not seem to
help to settle this problem. The construction given in [3] only yields
f(n, u) = (1 + o(l)) un/2 and not (1) for u > n1"fl.
I conjectured and Kleitman [6] proved the following result: Denote
by {At} I < i < 2" the 2" sequences of O's and l's of length n. Put
A, = (tf, ..., f<«>), (f<« = 0 or 1). Define
d(Ait A,) = S
P(i)
Let Ati, ..., Ais be a family of sequences satisfying
d(Alu, A,v) <2k, k < nj2, \ <u <v ^s.
Then
maxj S (") (3)
We have equality in (3) if the A's are the sequences having at most
k l's.
Using Kleitman's theorem we now construct our graphs as follows:
Put n — 3k -f- 1. The vertices of our graph will be the sequences
{At}, 1 < i < 2"; y4; and A/ are joined if and only if
d(At, Aj) > 2/: + 1.
190 GRAPH THEORY

ON THE CONSTRUCTION OF CERTAIN GRAPHS
151
* /3k + 1\
Our graph has 23A'+1 vertices and 2:!* £ I . I edges. It is easy to
i=0 \ ' /
see that our graph contains no triangle. To see this, observe that if it
would contain a triangle we could assume without loss of generality
that one of its vertices has all its coordinates 0, i.e., is (0, ..., 0). The
other two vertices must be sequences containing at least 2k -f- 1 ones
and hence they must coincide in at least k + 1 places, or their distance
is < 2k; thus they are not joined. In other words our graph contains
no triangle. The valency of each vertex of our graph clearly equals
k /3k
-n ^ '
On the other hand if At , ..., At is an independent set of vertices
we must evidently have d(At- , Aiv) S 2k (for if not then by definition
At and A, are joined and the set was not independent). But then by
the theorem of Kleitman
max 5= £ ( .' ) = V(X,), l</<23itl.
1-0
i
In other words, 1(G) ■-= V(x,), 1 < / < 23i-\ and thus (1) holds for
our graph. A simple computation using Stirling's formula shows that
(2) is also satisfied.
This construction could be generalized if the following generalization
of Kleitman's result would hold: Let tT>\, \<r<n, and denote by
n
{Bt}, 1 </< I] (?,. + 1), the sequences of the form (<5j, ..., dn),
0 < 3,. < tr. Let B, = (&?, ..., df), Bj = (<5<'\ ..., 6^), define
d(B„ Bj) = £ j (5<r) - <Yp\. Let /: < i 2 tT and let B,x, ..., B,s be
r=l r-1
a family of sequences satisfying
d(B,u, B,J <2k, 1 <u < v <s.
n
Then s is maximal if the Bt are the sequences satisfying 2 <5, < k.
r-\
But even if this would be true we could not improve (2) by this method.1
1 Kleitman showed that this generalization is false, but perhaps it holds if all the
/r's are equal.
191 EXTREMAL GRAPH THEORY

152
ERDOS
2. A graph is called /^-chromatic if its vertices can be split into k
classes so that no two vertices of the same class are joined, but such a
splitting is not possible into fewer than k classes. Tutte and Zykov
were the first to show that for every integer k there is a /achromatic
graph which contains no triangle. Rado and 1 [5] showed that for every
infinite cardinal m there is a graph of m vertices which contains no
triangle and which has chromatic number m.
A very simple and intuitive proof of this result could be given if
the following conjecture of Czipszer and myself would hold: Is it true
that the unit sphere of an /w-dimensional Hilbert space is not the union
of fewer than m subsets of diameter less than 2 — r. The unit sphere
of the m-dimensional Hilbert space is the set of all transfinite sequences
of real numbers {.Ya} where a runs through an index set of power m and
^a xa2 < 1 (all but denumerably many of the xa's are 0). As far as I
know this conjecture has not even been settled for m = t>t.
If the answer to our conjecture is affirmative our graph can be
constructed as follows: The vertices of our graph are the sequences {a-q},
^u xa 5~ I; where all the xa are rational and only a finite number of
them are different from 0. Clearly our graph has m vertices and the points
of the w-dimensional unit sphere defined by these vertices are dense
in the unit sphere. Two vertices are joined if their distance (in the
tridimensional Hilbert space) is greater than V 3 . Clearly this graph
contains no triangle and the diameter of any independent set is < V 3 .
Thus if the answer to our conjecture is affirmative, the vertices of our
graph cannot be split into the union of fewer than m independent sets,
i.e., our graph is /w-chromatic.
References
1. B. Andrasfai, Graphentheoretische Extremalprobleme, Acta Math. Acad. Sci.
Hangar. 15 (1964), 413-438.
2. P. Erdos and G. Szekeres, A Combinatorial Problem in Geometry, Compositio
Math. 2 (1935), 463-470.
3. P. Erdos, Remarks on a Theorem of Ramsay, Bull. Res. Council Israel Sect. Fl
(1957), 21-24.
4. P. Erdos, Graph Theory and Probability, II. Canad. J. Math. 13 (1961), 346-352.
5. P. Erdos and R. Rado, A Construction of Graphs without Triangles Having
Pre-assigned Order and Chromatic Number, J. London Math. Soc. 35 (1960),
445^48.
192 GRAPH THEORY

FAMILIES OF NON-DISJOINT SUBSETS 153
6. D. J. Kleitman, Families of Non-disjoint Subsets, Journal of Combinatorial
Theory 1, (1966) 153-155.
7. F. P. Ramsay, Collected papers, pp. 82-111; see also reference 2.
8. P. Turan, On the Theory of Graphs, Colloq. Math. 3 (1954), 19-30.
P. Erdos,
Panjab University
Chandigarh, India
193 EXTREMAL GRAPH THEORY

Suulia Sciemiaruni Maihematicarum Hungarica 1 (1966) 51—57.
A LIMIT THEOREM IN GRAPH THEORY
by
P. ERDOS and M. SIMONOVITS1
In this paper G(n; 1) will denote a graph of n vertices and / edges, Kp will denote
the complete graph of p vertices G l/>; I J J and AT,. (pl5 ...,/?,.) will denote the r-
chromatic graph with pt vertices of the i-th colour, in which every two vertices
of different colour are adjacent. tc(G) will denote the number of vertices of G and
v(G) denotes the number of edges of G. G(n: I) denotes the complementary graph
of G(n; I) i. e. G(n; I) is the G In; II — /1 which has the same vertices as G(n; I)
and in which two vertices are joined with an edge if and only if they aren't joined
in G(n; I). K (pt, ..., p,.) thus denotes the union of the disjoint graphs Kp.(i= 1,2, ..., r).
In 1940 Turan [8] posed and solved the following question. Determine the
smallest integer m(n, p) so that every G(n; m(n, p)) contains a Kfi. Turan
in fact showed that the only G (n; m(n, /7)-1) which contains no Kp is
p-i
Kp^i(mi, .... w;,_!) where y mt=:n and the mt are all as nearly equal as possible.
A simple computation shows that
(1) hm—^-^ = 1 {p--1)-
... (.j ,-,
Recently several more extremal problems in graph theory have been investigated
and in this paper we continue some of these investigations [4]. First of all we prove
the following general
Theorem 1. Let Gu ..., G, be I given graphs and denote by f(n; Gt, ..., G,)
the smallest integer so that every G{ii\ f(n\ (71? ..., G,)) contains one of the graphs
Gy, ..., Gt as subgraphs. We have
{.mf(n;G^Gl=[A
»-«• n) r
where r^ 1 is an integer which depends on the graphs G,(l s5/s/).
Theorem 1 easily follows from the following known result [3]:
1 E6t\6s L. University, Department of Mathematics Budapest.
4«
194 GRAPH THEORY

52
P. ERDOS AND M. SIMONOV1IS
For every /;>J, />1, e>0 and n^-n0(p, r, s)
(2)
\v
1
-0(1)
:/{n;Kr(p,...,p))^
U
Denote by y.(G) the chromatic number of G and put
(3) minx(G,) = /-+1.
Without loss of generality assume that *((/,) = /■-H 1.
r
Turan"s graph AT,.(w,, ...,mr) where ~^ m, = /? and the //?,- are as nearly equal
i= l
as possible is clearly /--chromatic thus by (3) can not contain any of the G{, 1 S/'s/.
A simple computation shows
in\( 1
(4) v(Kr(m„...,m,))--
1 -■
-o(l)
Put 7i((7,) = / and let t>0 be arbitrary and let/!=»//„(/,/•+ l,c). Then by (2) every
Glw;j 111 — - + ejj contains a Kr+l(t t) which by 7i(C,) = / clearly contains
G\. This together with (4) completes the proof of Theorem 1.
An unpublished result of P. Erdos states that
(5)
/{n;Kr(p,...,p))
1
1,
0(//^)
where c depends only on p and r. (5) easily implies that
/(1,-,0,,...,0,)-
12.
-0(n-~n
where /--(-1 = min *(G,) and r depends only on the graphs (7,, ..., G,. Now we prove
Theorf.m 1'. ie/ /: be an integer and //,, ..., H,„ with v(Hj)=kk given graphs.
Denote bv h(n\ H',, ..., //,„; A:) /A?e smallest integer /or which there is a graph
G(n;h(n; Hl, ..., Hm; kj) every subgraph o/ which spanned by any k vertices of our
graph (](n: /?(«; //,, ..., //„,; A:)) contains one o/ the graphs //,, ..., //„,. Then
\imh(n; //,, ..., Hm;k) ,
n
2
where t : 1 is an integer, or t = <».
Theorem 1' could also be deduced easily from (2), but we show, that it follows
from theorem I. In fact we shall show, that the two theorems are equivalent.
a) First we show that if there are given graphs /,. ..., /,, with 7:(/.,)¾A-, then
there exist graphs M„ ..., M, so, that a graph G of A- vertices contains at least one
of Ll, ..., /,, if and only if G contains none of A/,, ..., Mv.
195 EXTREMAL GRAPH THEORY

A LIMIT THEOREM IN GRAPH THEORY
53
From this of course follows that a graph Gofk vertices contains none of Ll5 ...,£,„
if and only if G contains at least one of Ml, ..., Mv which shows the symmetricity
between Ll5 ..., L^ and Mu ..., Mv.
To prove our statement we define the graphs M,-: Let Mi be those graphs, for
which n {M}) = k and M} contains none of Ll5 ..., L„. A very important property of the
set of graphs Mu ..^Mvjs that if H^>M, and n(H) = k then 7/ occurs among
Mt, ..., Mv because MjCH, further Mj contains none of Lu ..., L)r and so H does
not contain any of Lu ..., L;I.
Now, if Gz>Lh then G does not occur among M1, ..., Mv so G does not contain
any of Mv, ..., M;I. On the other hand, if G does not contain any of Lv, .-., L;I,
then G occurs among M v, ..., M , and this proves the second half of our statement.
If we have a graph F, which has/(n, Lu ..., L^) — 1 edges and does not contain
any of the graphs Lu ..., Lfl then each subgraph spanned by its k vertices contains
none of Lt, ..., L^, so each subgraph off spanned by its k vertices contain at least
one of those M1; ..., Mv which we have defined in a), moreover F has the minimal
number of edges among the graphs, each subgraph of which spanned by its k vertices
contain at least one of Mv, ..., Mv:
v(F) = h(n;Ml,...,Mv)= ^) -f(n\ L,, ..., L„) + 1.
So we can investigate a problem of the second type instead of a problem of
the first type.
b) On the other hand, if there are given Mv, ..., Mv, with n(Mt)^k, we
know, that there exist Lx. ..., Lti so, that a graph G of k vertices contains at least
one of Mi, ..., Mv if and only if G contains none of L1; ..., Lhl, or (what is equivalent
with this) a graph Gofk vertices contains none of Mu ..., Mv if and only if G contains
at least one of Lu .... Lu. Now, if H is a graph, which has h(n; Ml7 ..., Mv) edges,
and each of its subgraph, spanned by its k vertices contains at least one of Mv, ..., Mv,
then each subgraph of H spanned by its k vertices contains none of Lu ..., Lti,
moreover has the maximal number of edges among the graphs each subgraph of
which spanned by its A: vertices contain none of Lu ..., L;I:
v(H) =f(n;Llt ...,L„)-1 = [^-hin-M,, ...,Mv).
This proves in particular Theorem 1'.
Now we return to the study of our function f(n; GY, .... Gt). The proof of
Theorem 1 shows that the order of magnitude of f(n\ G1; ..., Gt) depends only on
min x(Gt). Nevertheless we show that the graphs G; of higher chromatic number
and in fact the structure of all the G,-, (lsis/) also have an influence on
f(n; GY, ..., G,). To see this let Gx be the graph consisting of a quadrilateral and
a fifth vertex which is joined to all four vertices of the quadrilateral. It is known
that [4] for «>n0
V
——
4
+
n
—
4
+
n+ 1
4
196 GRAPH THEORY

54
P. ERDOS AND M. SIMONOV1TS
But on the other hand it is easy to show that for n >»0
(7) /(njG^^^j + ^j+l.
Both (6) and (7) are easy to prove by induction and can be left to the reader.
Observe that /(n; Gj) >/(n; Gy, KA), Gx is three-chromatic and AT4 is four-
chromatic.
In every case we know the structure of the ,,extremal graphs" i. e. those
(8) G(n;/(n; d, ..., G,)-l)
which do not contain any of the graphs G,- (1 ^;'S/) these graphs are Turan graphs
^p-i(wi5 ■■■,mP-v) f°r some p to which perhaps o(n2) further edges are added.
Perhaps this is true in the general case, or at least the extremal graphs (8) contain
a very large Turan graph (with say en vertices). At present we are unable to attack
this conjecture. The simplest case where we do not know anything about the structure
of the extreme graph is the case of Kz(2, 2, 2). It is known [4] that
2 2
^-+c2n3'2</(n;A:3(2, 2, 2))<-"- + C]n3/2
but we don't know whether the extreme graphs contain a „large" Turan graph
Let "|^/<rl" ),((^2. We now prove ([4])
Theorem 2. Let n be sufficiently large. Then
(9) /(n; G(k; 1)) s/(„; G (w; ^)]) = m{n, u).
Equality only if either G(k; /) contains a Ku or if u= 3 and G(k; /) is a pentagon.
First we prove the following
Lemma 1. Let/< . Then either x(G(k;l))<u or G(k;l) lias an edge
eso that x(G(k; I) - e) < u. G—e is the graphfrom which the edge e has been omitted.*
We use induction with respect to u. It is easy to see that the Lemma holds
for u= 3. Assume that it holds for u— 1 and we prove it for u. If G has a vertex
x of valency ^u, let G* be the graph which we obtain from G by omitting x and
all edges incident to x. G* has fewer than II edges. Hence by the induction hypothesis
there is an edge e so that x(G*-e)?u-2, or x(G — e)^u— 1.
We can therefore assume that all vertices of G have valency exactly w—1,
(since the vertices of valency <«— 1 cculd simply be omitted.) Since G has at most
I 1—1 edges, we obtain that it has at most u + 2 vertices and for these graphs
our Lemma can be proved by simple inspection.
* Our original proof was more complicated. This simple proof we owe to V. T. S6s.
197 EXTREMAL GRAPH THEORY

A LIMIT THEOREM IN GRAPH THEORY
55
No w we can prove Theorem 2. 1) First assume u > 3. It is known [5] that for every
/• and «>n0(r) every G(n; m(n, «)) contains a Ku{r, ..., r) and an extra edge joining
two vertices of the first r-tuple, and by our Lemma it is easy to see that for r^k
our G(k; /) is a subgraph of this graph.
2) If w = 3, m(G)< 11=6 and G contains no triangle then x(G)^ 3 and x(G) = 3
if and only if G is a pentagon and it is known [4] that in this case
(10) f(n;G)
+ 1 = m(n, 3).
3) Lastly, the case when w = 3 and G contains a triangle was.discussed in [4].
The equality in (9) if u > 3 holds if and only if G contains a K„. This can be
obtained by a simple discussion, which we leave to the reader.
Finally we investigate h(n; G; k) for some special graphs G. Let G be the graph
which consists of I disjoint edges and assume /c>2/. We outline the proof of the
following
Theorem 3. Let n»i0(k,l). Then
h(n;G,;k)= Kj - m(n,k-2l + 2) + 1
and the only graph G{n; h(n; Gt; k) for which every subgraph spanned by k of its
_ k~21+l
vertices contains a G, is Kk__2i+\ (mi, ---^ mk~2i+\) where 2" ni, = n and the m;
i- 1
are all as nearly equal, as possible.
First of all it is easy to see that the subgraph spanned by any k vertices of
-Kfc-2?+ i {m\> ■•■> mk~2i+ i) contains a G}. We leave the simple verification to the
reader. This shows
h(n;G,;k) "'
. m(n,k-2l+2)+l.
To complete the proof of Theorem 3 we now have to show the opposite inequality
in other words we have to show that if G In; II —m{n, k — 21 + 2) I is any graph
then there are k vertices*], ..., xk so that the subgraph of G in; I I — m(n, k — 2/ + 2)1
spanned by these k vertices does not contain a G, and further that the only
G \n; l-J — m(n, k — 2/+2)+ 11 which does not have this property is
Kk-2t+ i("?i> ---^11-21+ i)- These statements will follow immediately from the following
Lemma 2. There is a constant cr >0, independent of n, so, that every
G(n;m(n, r+ 1)), or a G(n;m(n, r + 1) — l), which is not a Kr{mx, ..., mr) (where
Imi = n and w, are all as nearly equal as possible) contains a Kr and crn other
vertices, each of which is joined to every vertex of an Kr.
198 GRAPH THEORY

56
P. ERDOS AND M. SIMONOVITS
Remarks. Turan's theorem implies that every G(n: m(n,r +1)) contains
a Kr+1 and it is known [2], [4] that every such graph contains a Kr+2from which
one edge is perhaps missing i. e. it contains a Kr and two vertices each of which is
joined to every vertex of our Kr. Our Lemma is sharpening of this result.
We supress the proof of our Lemma since it is very similar to the case when
r = 2 which is known [6].
Let now n be sufficiently large and G(n; e) be any graph, for which
v(*t
](>«!,
mk.
■i))
■m(n,k-2l + 2)+l
_j) (where Zm; = n; and mi are all nearly
and which is not a Kk^2i+i(mu ■■■> mk
equal, as possible).
By our lemma, the complementary graph of our G(n, e) contains a Kk__2iw
and 21— 1 vertices, each of which is joined to every vertex of our ATfc_2,+ ], i. e.
there are k vertices, which span in our G(«;<?) a subgraph, which consist of k — 2/+ 1
isolated, vertices and a graph of 21— 1 vertices and hence it can not contain /
independent edges. This completes the proof of Theorem 3.
It is easy to see that if k = 21 the extreme graphs are no longer Turan's graphs,
it is easy to see that in this case
(11)
h(n;Gy,2l) =
-(/-1)//.
To see this observe that if one vertex of G, is not joined to 21— 1 vertices these
2/ vertices can not contain independent edges. This proves
h(n;Gt;2l)
(!-\)n.
On the other hand, the following example shows, that
h(n;G,;2l)
-(1-1)1,.
Let the vertices of Gf be the n-th roots of unity, two such vertices are joined if their
, , 271
distance — on the circle \z\ = 1 —■ is greater, then (I— 1) - .
In this case, if the vertices of our graph are P]5 ..., P„ and At, ..., Ak are k
vertices of them enumerated, as they are on the circle, then A{ and Ai+l,(i= 1,2, ..., /)
will be connected, so there will be / independent edges in the subgraph, spanned
by X,, ..., Ak.
We do not investigate the question of the unicity of the extremal graphs.
Denote by Gj3) the graph consisting of / independent triangles. We outline
the proof of the following
Theorem 4. Let n>«0(/). Then
h(n;G\3); 3/ + 2) = I "J -m(n, 3)+ 1
and the only extreme graph is K2 \\w\ ,\—s— )•
n+ 1
199 EXTREMAL GRAPH THEORY

A LIMIT THEOREM IN GRAPH THEORY
57
On the other hand, if />1
(12) T -Qn3'2</i(n;GP>;3/+l)< !J -ni+" (e,>0).
The structure of the extreme graph (or graphs) is unknown. It is easy to see
that h{n; G,(3);4)= land Kn__i is the only extreme graph.
First of all it is easy to see that every subgraph spanned by 3/ + 2 vertices of
K2 ([|] , f±!j) contains a G<3>.
Let G be any graph for which
n (G) = n, v(G) =
and if v(G)= — then G is not K2 I — I II. If n>n0(l) then it is knowa
[5] that G contains a subgraph of 3/ + 2 vertices xt, x2, x3; yt, ...,737-1 where alL
the edges
(x„ x2); (xh yj) 1S/S3 ls/=»3/-l
are in G. Clearly these 3/ + 2 vertices span a subgraph of G which does not contain
G(j3>. This completes the proof of the first half of Theorem 4.
To prove the second half we observe that it is known that every G{n; [C;«3/2])
contains a K2(2, 31— 1) hence the subgraph of G{n; [C;n3/2]) spanned by the vertices
of our K2(2, 3/- 1) clearly contains no Gj3) this proves the left side inequality of (12).
Now we outline the proof of the right hand side of (12). First of all it is known
[7] that there exists a graph G(n; n1+£') which contains no circuit having ^3/+1
edges. Thus our proof will be complete if we can show that if /> 1 and G(3/ + 1 ;p}
is any graph of 3/+ 1 vertices which contains no circuit then G(3/+ l;p) contains
/ independent triangles. This can be shown easily by induction with respect to I
and can be left to the reader. Thus the proof of Theorem 4 is complete.
(Received October 19, 1964.)
REFERENCES
[1] Brooks, R. L.: On colouring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941)
194—197.
[2] Dirac, G.: Extensions of Turan's theorem on graphs, Acta Hung. Acad. Sci. 14 (1963) 417—422,.
see also [4].
[3] Erdo's, P. and Stone, A. H.: On the structure of linear graphs, Bull. Amer. Math. Soc. 52
(1946) 1087—1091.
[4] See e. g. Erdo's, P.: Extremal problems in graph theory, Proc. Symposium, Theory of Graphs
and its applications Smolenice (1963), 29—36.
[5] Erdo's, P.: On the structure of linear graphs, Israel J. of Math. 1 (1963), 156—160. In this paper
the theorem is proved in detail only for r = 2.
[6] Erdo's, P.: On the theorem of Rademacher—Turan, Illinois J. of. Math. 6 (1962) No. 1. March.
[7] Erdo's, P.: Graph theory and probability, Canad. J. Math. 11 (1959) 34—38.
[8] TurAn, P.: Mat. es Fiz. Lapok 48 (1941) 436—152 (in Hungarian), see also „On the theory
of graphs" Coll. Math. 3 (1954) 19—30.
200 GRAPH THEORY

Studia Scientlarum Mathematicarum Hungarica 1 (1966) 215—235.
ON A PROBLEM OF GRAPH THEORY
by
P. ERDOS1, A. RfiNYI1 and V. T. SOS2
§ 0. Introduction
Let Gn be a non-directed graph having n vertices, without parallel edges and
slings. Let the vertices of G„ be denoted by PY, ..., P„. Let v(Pi) denote the valency
of the point P, and put
(0. 1) V(G„) = max v(Pd-
lSiSn
Let E(G„) denote the number of edges of G„. Let Hd(w, k) denote the set of all
graphs G„ for which V(Gn)=k and the diameter D(Gn) of which is ^d,
(k =1,2, ..., w-1; d=2, 3, ...,/7-1).
In the present paper we shall investigate the quantity
(0.2) F„(n,k)= min E(G„).
G„£Hd(n,k)
Thus we want to determine the minimal number N such that there exists a graph
having n vertices, N edges and diameter ^d and the maximum of the valencies
of the vertices of the graph is equal to k.
To help the understanding of the problem let us consider
the following interpretation. Let be given in a country n airports;
suppose we want to plan a network of direct flights between these
airports so that the maximal number of airports to which a
given airport can be connected by a direct flight should be equal
to k (i.e. the maximum of the capacities of the airports is
prescribed), further it should be possible to fly from every airport
to any other by changing the plane at most d — 1 times; what is
the minimal number of flights by which such a plan can be
realized? For instance, if n = l, k = 3, d=2 we have F2{1, 3) = 9
and the extremal graph is shown by Fig. 1.
The problem of determining Fd(n, k) has been proposed and discussed recently
by two of the authors (see [1]). In § 1 we give a short summary of the results of the
paper [1], while in §2 and 3 we give some new results which go beyond those of
[1]. Incidentally we solve a long-standing problem about the maximal number
of edges of a graph not containing a cycle of length 4.
In § 4 we mention some unsolved problems.
Let us mention that our problem can be formulated also in terms of 0 — l
matrices as follows: Let A/ = (e;j) be a symmetrical n by n zero-one matrix such
1 Mathematical Institute of the Hungarian Academy of Sciences.
2 Eotvos L. University, Budapest.
201 EXTREMAL GRAPH THEORY

216 P. ERDOS, A. RtNYI AND V. T. SOS
that eu= 1,
determine
Clearly
(0.3)
max
l ~' ^«
2 <=.-,
j= 1
. = A'+1 and all elements of Md are i?
n n
Ma(n, k) = min ^ 2 <V
(= 1 J ~ 1
Md(n,k) = 2Fd(n,k) + n.
1. We want to
This formulation shows the connection of our problem with non-linear programming.
We give for the case d = 2 a third formulation of our problem which displays
its connection with the theory of block designs.
Let be given a sequence Au A2, ..., A„ of subsets of the elements 1,2, ..., n
such that if/(E/4j then i£Aj. Let us suppose that denoting by \A\ the cardinal number
of the set A, we have max \Aj\ = k. Let us suppose that for any i (1 s/sn) and
lSjmn
any/V / such thaty$ Al there is a set Ah which contains both i andj (this is equivalent
by our supposition of symmetry to the statement that the sets At and A- are not
disjoint). The problem is to determine
71
(0.4) min 2\^i\ = 2F2(n,k).
§ 1. Some Basic Inequalities, and sonfe Asymptotic Results
It is easy to see that if there exists a graph C„ with V(Gn) = k and diameter
==c/, then
(k-\)d- 1
(1. 1) n s l+k-
k-2
(1.1) can be proved as follows: if V(G„) = k the number of points which can be
reached from a given point, say, Px by an edge is si; the number of points which
can be reached from P^ by a path of length 2 is Sk(k— 1) and finally the number
of points which can be reached by a path of length d is ^k(k— \)d~i. Thus if the
graph has diameter S(i we have
(n- l)?sk(\ + (k- l) + (k- \)2 + ...+(k- I)"-').
This proves (1. 1). If both n and k are odd, then G„ must contain at least one
point of valency S/c— 1 (because the number of points of odd valency cannot be
odd); thus in this case we get
(k — l)d — 1
(1.2) ^1 + (^1)-^ .
Note that for the graph shown by Fig. 1, equality stands in (1.2). For the graph
shown on Fig. 2 (the so-called Petersen-graph) equality stands in (1. 1) with n= 10,
k = 3, d=2.
202 GRAPH THEORY

ON A PROBLEM Ol GRAPH THEORY
>]7
As regards Fd{n, k) we obtain easily the lower
bound
(1.3)
Fd(thk)^
n(n--\)(k-
•2)
"l, '
(1. 3) can be proved as follows: every edge is itself a
path of length 1; it can be contained in at most
2(k— 1) paths of length 2, but in this way each
path of length 2 is counted twice, thus the
number of paths of length 2 cannot exceed E(Gn)(k — 1).
In general each edge can be contained in at most
3(&— 1)2 paths of length 3, but in this way each
path of length 3 is counted three times, thus the
number of paths of length 3 cannot exceed
E(Gn)(k — l)2, etc. As in case Gn has diameter
s=f/ the number of paths of length s d has to be at least
(")■
Fig. 2
we obtain
(1.4)
E{G„)(\+(k-\)+...+{k-\y^)
which implies (1. 3). Note that one has equality in (1. 4) for the Petersen graph shown
on Fig. 2., further for n = 5, k = 2, d=2 because a cycle of length 5 has 5 vertices,
5-4
each of which has valency 2, it has diameter 2 and the number of its edges is 5 = -— .
It is clear from the above proof that one can have equality in (1. 4) only for a
regular graph of order k, i. e. if E(G„) = -— and if any two points are joined by
one and only one path of length 3 d.
The first condition implies that if equality stands in (1.4) then there is equality
in (1. 1) too. For the case d = 2 this means that a necessary condition of equality
in (1. 4) is n = k2 + 1. It has been shown by A. J. Hoffman and R. R. Singleton [4]
that a regular graph of order k, having k2 + 1 points and diameter 2 exists only
for /: = 2, 3, 7 and perhaps for A: = 57. Thus for d = 2 except for these values of
k one has strict inequality in (1. 3). However it has been shown in [1] that there
exists an infinite sequence of pairs (kj, n;) such that kj-
iij ~* oo and
(1.5)
Hm F2{nJ^J)kJ
This is a consequence of the following
Theorem 1. If P is any prime power, there exists a graph G„ of order n = P2 + P + 1
for which V(Gn) = P+ \, which has diameter 2 and for which E(G„)^ ^(n3,2+»).
The graph Gn has also the property that it does not contain any cycle of length 4.
To make this paper self-contained we reproduce the proof of Theorem 1 aiven
in [1].
203 EXTREMAL GRAPH THEORY

218
P. FRDOS, A. RF.NVI AND V. T. SOS
Proof of Theorem 1. Let GF(P) be the Galois field with P elements. Let
us represent the points of the finite plane geometry PG(P, 2) by triples (a, b, c)
where a, b, c are elements of GF(P), not all three equal to 0, and (la, lb, 1c) with
If^GF(P), /.^0 represents the same point as (a, b, c). The number of different
points of PG(P, 2) is P2 + P+ 1. A straight line in PG(P, 2) is the set of all points
(_y, y, z) which satisfy the equation ax + by + cz = 0; we denote this line by [a, b, c\.
The point (a, b, c) and the line [a, b, c] are clearly conjugate elements with respect
to the conic x2 +y2 +z2 = 0. As well known there are P+\ points on each line,
any two different lines have exactly one point in common and through any two
given points there is exactly one straight line. Now we define the mapping T which
maps the point A = (a, b, c) into the line a = [a, b, c] and conversely. We write
TA=oc, Ta = A. This mapping has evidently the properties: if the point B lies on
the line a = TA then the point A lies on the line /?= TB; if C is the point of
intersection of the lines TA and TB then TC is identical with the line passing through
the points A and B; A = {a, b, c) is on TA if and only if a2 +b2 + c2 =0, i.e. if A
lies on the conic x2 +y2 +z2 =0. Now let us define a graph Gn (n = P2 + P + 1)
as follows: the vertices of G„ are the points of PG(P, 2); the vertices A=(a, b, c)
and A' = (a', b', c') are joined in G„ by an edge if and only if A' is lying on TA (and
thus A is lying on TA'). Clearly a vertex A in G„ has the valency P or P+ I according
to whether A is on the conic x2+y2+z2~0 or not.* Thus
(1.6) ~(«3;2-n)s -2P(P2 + P+1)SE(G„)
and
E{Gn)S j(P+l)(P2 + JP+l)S ^(n32 + «).
Finally the diameter of G„ is equal to 2. As a matter of fact any two points
A and B can be joined by the path ACB where C is the point of intersection of the
lines TA and TB. Besides this A and B can be joined by a single edge if A lies on TB.
But the point C such that the edges AC and BC both belong to Gn is in any case
unique; thus C„ does not contain any cycle of length 4.
Thus our Theorem is proved.
We deduce from Theorem 1 the following corollaries.
Corollary 1. Put nk = k2 — k+\; then
,. . n F2(nk,k)k 1
(1.7) hmmf-2-^fr = T.
* If P is prime, there are P->-1 points on the conic and thus
P(P+l)2 1
£(G„) = ni!1 if ngiio.
204 GRAPH THEORY

ON A PROBLEM OF GRAPH THEORY
219
F OF
Corollary
1.
By (1
F2
n
3)
(n,k)k
(«-D
1
2
(1.8)
further if A- = P+ 1, «k = P2+P+l, by Theorem 1
(1.9) F2(P2 + P+l,P+l)^$(P+[){P2
thus in this case
(1.10)
■P+l);
F2(nk,k)k
Mi-1
2 P
this proves our assertion.
Theorem 1 enables us also to solve — at least asymptotically — a problem
which was raised by one of us 27 years ago (see [2]).*
Let C„ denote the class of graphs having n vertices and containing no cycle
of order 4. Put
(1.11)
j.i{n) = max E(Gn).
g„ e c„
The problem is to determine the value of n(n). From Theorem 1 we deduce the
following
Corollary 2. We have
(1.12)
lim
1.1(11)
Proof of Corollary 2. It follows clearly from Theorem 1 that if P is a
prime power, then putting n = P2 + P + 1
0.13)
li (n)
1
(n3'2-«).
It is possible that for these n the graph of Theorem 1 is extremal but we
cannot prove this. Clearly n(ri) is an increasing function of n, and thus it follows that
for any n we have
(1.14) ^n)^ ~[P2 + P+iy!2~(P2 + P+\)]
where P is the largest prime power such that P2 + P +\ Sn. Now evidently for
n^n1 one can choose a prime p so that
(1.15)
yVi-
log n
5/>
VVi-1
* After having written this paper we have been informed by W. G. Brown that
independently of us he has proved (1.12), in the same way as we did. His paper will be published in the
Bulletin of the Canadian Mathematical Society.
205 EXTREMAL GRAPH THEORY

220
P. ERDOS, A. RENY1 AND V. T. SOS
which implies for n^u^
Thus we
(1.16)
and thus
(1-17)
have
for
any
n
n l
2
log n ,
5f+p+lS
= "]
,w 1
„3,2 l
I logn
,. . . /.i(n) 1
iimmf w ^
On the other hand it is easy to see (this follows also from the results of I. Reiman
in [3]) that
(1.18)
lim sup
Kn)
As a matter of fact, let G„ be a graph containing no cycle of order 4. Let Ply P2, ..., Pn
be the vertices of C„ and let us denote their valencies by v1, v2, ..., v„. Now clearly one
can select from the set Et of vertices joined by an edge to P
' (2) pdrS'
and no
pair (Pj, Ph) can be contained in both £, and Et with I ^ i because otherwise f,. f ■ P, Ph
would be a cycle contained in Gn. Thus we must have
(1.19)
Now we have
(1.20)
and thus
(1.21)
K2
Zri'^nZr?
j=i I U=1 ) 1=1
2«
As clearly ,2X = 2.E(G„), we have
(1.22)
which implies
(1.23)
Thus
(1.24)
4£2(C„)^2»£(C„)^»3
E(G„) £
■V
4n 4
„_i-y.+i+-i-
4« 41'n
which implies (1. 18). Thus Corollary 2 is proved.
206 GRAPH THEORY

ON A PROBLEM OF GRAPH THLORY
221
Let us note that weaker results have been obtained previously by E. Klein
(see [2]) and I. Reimann [3], who proved lim inf -^-S—7-- . Reimann's extremal
>.-,«. n6'2 2>'2
graph does not contain triangles either; it is possible that among such graphs it is
optimal.
Note that for the pairs (rij,kj) for which according to Corollary 1 one has
(1.25)
lim
F2(nj,kj)kj
one has k^Yrij. Jt was shown in [1] that there exists another sequence of pairs
(kj, rij) such that
(1.26) limf2(">-yj= 1
;_„ >?;(»,■- 1)
A'2
but for this sequence of pairs one has lim —L = + ->=.
Jt remains an open question what is the value of the function g(c) defined by
(1.27) g(c) = liminfi^*^
k2>nc n(n-l)
for 1 <c-^ + oo; we know only that g(c) is nondecreasing, \ S#(c)and lim g(c) s 1.
§2. Some Exact Results for d=2.
In this § we deal with the exact value of F2(n, k) for ifrsn-1. Evidently,
F2(n, n— 1) = n— 1, because the graph Gn in which one vertex is joined by an edge
with all others, has diameter 2, further V(Gn) = n—l and E(Gn)=n—l. Jt has
been shown in [1] that F2(n, n —2) = 2» —4 (a graph Gn with V(Gn) = n — 2 and
E(Gn) = 2n — 4 and having the diameter 2 is shown by Fig. 3; another graph with
the same properties is shown by Fig. 4), further that F2(n,n — 3) = F2{ii,n—4) =
= 2« —5. (The corresponding extremal graphs are shown by Figs. 5 and 6.)
V(Gn)- n-2
E(Gn)=2n-4
V(G„) = n-2
£fG„)=2n-4
Fig- 3
Fig. 4
207 EXTREMAL GRAPH THEORY

222
P. tRDOS, A. RENYI AND V. T. SOS
We shall prove now
Theorem 2. We have for n s 13
(2.1)
F2 (n, k) = In - 4 for -^—^k^n-5.
P,
VCGn) = n-3
E(Gn) =2n-5
Fig. 5
Proof of Theorem 2. The extremal graph Gn with
n+2
V(G„) = k = n-l,
5ss/
and E(G„) = 2n~4 and having diameter 2 is exhibited by Fig. 7.
n+2
V(Gn) = n~l, E{G„) = 2n-A, 5sl
n ^ 13.
VfS,
EfG,
/■'>• <5
/¾. 7
208 GRAPH THEORY

ON A PROBLEM OF GRAPH THEORY 223
Note that all vertices of C„ except Pt, P2 and P3 have the valency 2, further
v(P1) = n — l, v(P2) = n—l, v(P3) = 2l — 2 and by supposition 2/-2^n — 1. Thus
V(G„) = n — l. Clearly G„ has diameter 2 and the number of edges of G„ is
2(71-/) + 2/-2+2(/7-3)
£(G„) = r = 2n - 4.
5==/ =—-— and diameter
We prove that for any C„ with n S 13, V{G„) = n — l
2 one has E(G„)^2n — 4.
1 "
As E(G„)= — 2v(Pd we may suppose that C„ contains at least one point
2;= i
of degree = 3. If G„ would contain no point of degree 5 2, then let us choose a point
of degree 3; let this point be P,. Let the points connected by an edge with Px be
denoted by P2, P3 and PA. As every point can be reached from P1 by a path of
length = 2, we must have u(P2)+u(P3) + u(P4)an—1.
Now if there would be a point among the points P5, ..., P„ which would be
connected with more than one of the points P2, P3, PA we would have v(P2) +
+ v(P3) + v(PA)^n; as all other points have degree s3 it would follow
n
2 v(pd = n+3(n-3) = 4ti-9
;= l
and thus E(G„)>~2n~ 5 i.e. E(G„) ^2n~ 4, which was to be proved. Thus we may
suppose that all points PX5 = i = n) are connected with one and only one of P2, P3
and PA; similarly we can suppose that P2, P3 and PA are not connected with each
other because this would again imply v(P2) + v(P3) + v(PA) ^n and thus E(G„)^
^2n~ 4. If there is at least one among the Pt with 5^/^n which has degree >3,
it again follows that E(G„)^2n~4. If however all have degree 3, let us suppose
that v(P2) = mm(v(P2), v(P3), v(PA)) which implies v{P2)^sLn—~ . Let P5 be
connected with P2. Then v(P5) = 3 and let the three points connected with ^5 be
P2, Pi and Pj\ clearly />5 and j>/>5. But then v(P2) + v(Pi) + v(Pj)^n-l
and thus
6 = v(Pl) + v(PJ)^-2-in-^
that is n 5 10.
As we supposed 71^13, this case is settled.
The case when there is a point Pt of valency 1 is easily settled, because if this
point is Pu and P1 is connected with P2 only, then P2 has to be connected with
the remaining n — 2 points too, and thus would have valency n — 1. Thus the only
case which remains to be settled is when min v{P;)=2. Suppose u(P,)=2 and
ISiSn
let Pt be connected with P2 and Pz. Then all remaining points have to be connected
either with P2 or with Pz or with both.
Let C, denote the class of points Pt with 1&4 connected only with P2, and
Cj the number of elements of Ci; let C2 be the class of points Pt with /s4 connected
only with Pz and c2 the number of elements of C2; finally let C3 be the class of points
connected with both P2 and P3, and c3 the number of elements of C3. Clearly
209 EXTREMAL GRAPH THEORY

224 P. ERDOS, A. RENYI AND V. T. SOS
c^ + c^ + Ct, = «—3. As the valency of P3 cannot exceed n—l and P3 is connected
with every point in G„ except itself and the points in d, we have 0^/-2^3.
Similarly c2 = I — 2 £ 3.
The number of edges in Gn the existence of which is already established is
clearly c1+c2 + 2c3 + 2 = n + c3 — l. Let us call these the edges of the first kind,
and the remaining edges those of the second kind. As the graph has diameter 2,
every point of Cx has to be connected by a path of length s2 with every point of C2.
Such a path can not contain an edge of the first kind. Thus the graph G" consisting
of the edges of the second kind has to be connected. Now three cases are possible.
Either G" contains besides the points of Qi and C2 at least one further point from
the class C3; in this case it contains at least ct +c2 + l points and thus there are
at least c1 + c2 edges of the second kind, and thus the total number of edges is
E(Gn)^n + c3 — 1 +Ci + c2 = 2n — 4. Or P2 and P3 are connected by an edge;
in this case we get again E{Gn)^2n~4. Or P2 and P3 are not connected and G'
consists only of the points of Cj and C2. In this case the connected graph G' is
either a tree or not. If it is not a tree, it contains at least ct +c2 edges and thus we
obtain again £(G„) ^2« — 4. If G' is a tree, it must have at least two end-points. We
may suppose that d contains an endpoint of G". Let x be the total number of end-
points of G' in C,. Then the sum of valencies (in G') of the points of Cj is at least
x + 2(c1 —x). As G„ has diameter 2 and P2 and P3 are not directly connected, any
endpoint of G' in d has to be connected by a path of length 2 to P3, it follows that
for every endpoint P of G' in d the single edge starting from P ends in C2. Let
y denote the number of points in C2 which are connected with an endpoint of G'
in C] • If Q is such a point, clearly Q has to be connected with every other point
of C2, because otherwise there would not exist a path of length 2 from P to
these points. Now clearly no point of C2 can be an endpoint of G', because it must
be connected to at least one point in d and also to Q. Thus the sum of valencies
in G' of the points of C2 ist at least 2(c2 — y) +y{c2 — 1) + x. It follows that the
number of edges of the second kind is at least
2 (x + 2(c1-x) + 2(c2-y) + y{c2-]) + x) = c,+c2 + j 1-^--1 == c,+c2,
because, as we have shown, c2s3.
Thus we have shown that E(Gn)~2n — 4 and the proof of Theorem 2 is complete.
Note that the restriction «^13 in Theorem 2 is necessary, because for «< 13
2n — 2
there is no value of k between - — and n — 5.
2n~2
As regards the value of F2(n,k) for A'<---■--- we can show that for hS15
r r 'in — 3 , 2n — 2
5n — k — o for —-— ^ A < —-—
(2.2) F2{n,k)=i5n-4k-l0 for 5^3 =§ k < 3^3
4n-2*-I3 for Jl+L * k ^ *HZ*.
210 GRAPH THEORY

ON A PROBLEM OF GRAPH THEORY
225
We give in what follows the extremal graphs for these 3 cases. That these are really
extremal can be proved in a way similar to the proof of Theorem 2, therefore we
leave the details to the reader.
3« — 3 , 2n — 2
The extremal graph for —-— £ k < —-—.
The graph has four points of high degree; let us denote them by A, B, C, D
and four groups of points.
There is a group denoted by AB, the points of which are joined to A and to B.
The group contains 2k —n points. In the group BCD (connected with B, C and D)
there are n — k — 1 points. In the group AC (whose points are connected with A and C)
■ k — 3l
there are -—s points; finally in the group AD (the points of which are con-
-A-3
nected with A and D) there are n—k—3— -—'■—— points. Further the graph
contains the edges AB, AC, AD. The points A and B have the degree k. The whole
graph has 3n — k — 6 edges.
The extremal graph for
5n — 3
a k <
3n-3
9 ~ 5 ■
There are 5 points of high order, A, B, C, D, E.
The group AB has 2k — n points,
\n — k -
The group BCD has
The group BCE has n — A- — 1 —
points.
n — k —
points.
The group AC has 2k —n points.
The group ADE has 2n—3A —4 points.
Further the edges AB, AC, AD, AE, DE belong to the graph. The points A, B
and C have the valency n — k; the total number of edges is 5n — 4A — 10.
The extremal graph for
n+ 1
tk k
5n-3
There are 6 points of high order, A, B, C, D, E, F.
The group AB contains 2k —n points.
\n — k — 1
The group BCE contains
The group BDF contains n
The group ADC contains
2
-k-
-k
1
points.
n — k-
The group AEF contains
2
n — k-
points.
n — k
points.
points.
The graph contains further the edges AB, AC, AD, AE, AF. The graph has An — 2k — 13
edges.
211 EXTREMAL GRAPH THEORY

226
P. ERDOS. A. RENYI AND V. T. SOS
For k-
n+ 1
we cannot determine F2(n,k) exactly. However, we can get
a fairly good upper bound by constructing graphs of diameter 2 by the following
principles. We divide all but I I of the points of a graph G„ into r groups of
approximately the same size. We connect the points of each pair of groups with
one of the remaining points, and connect as many of these points with each other
Fig. 8
as needed. For instance if/- = 4, n = 4/ + 6, we put/points in each of 4 groups, connect
each of the 6 pairs of groups with one of the remaining 6 points, and connect each
of these points with that point which is connected with the other two groups. The
graph obtained is shown by Fig. 8. It follows that
(2.3)
F2{4l+6, 2/+l)s 12/+3.
§3. Some Results for ds3.
We prove first
Theorem 3. We have for every n, every k = n—\ and d^3
(3.1) Fd(n,k)^
Proof of Theorem 3. Let us put
(3.2) 6 = 4
1-4
n
F
Clearly we may suppose ¢5-=1, because otherwise (3. 1) is trivially fulfilled. We have
evidently
4
(3.3)
k
d'3 -
212 GRAPH THEORY

ON A PROBLEM OF GRAPH THEORY
227
We may suppose >i>kd~\ because any graph G„ with diameter ^d is connected
and thus has at least n — 1 edges; thus F^n, k)^n— 1 and if n^Ad~1 the inequality
kd
(3. 1) is trivial. Thus we have to prove (3. 1) only for A''^1 <n<—i.e. for (64n)',d -="
64
:/r<ni/d-i_ *
Let G„ be a graph having n vertices, diameter d and such that V(G„)=k.
An
Let us denote by Xu ..., Xs those vertices of G„ the valency of which is
let Yt, ..., y„_s be the remaining vertices of G„. We have clearly
1
(3.4)
Thus if
we have
(3.5)
E{Gn)
2 v(*i) + 2 »(Yj)
6(1-6)
5 S n 1
£(<?„)
T(]-5)-
2(n — 5-)n
kd~ld'
Thus we have to consider only the case
(3.6)
n\\
5(1-5)
We distinguish two cases. Either every A"; (l^i^s) is connected with at least
6l n
-j- of the vertices Y-, or not. In the first case we have
(3.7)
E(G„) a 5
2 /cd
(1-6)
kd-
Thus we may suppose that there is an X{ — say Xt — which is connected with less
than 1 — — -r^rj Yj-s. We shall show that this case is impossible. By supposition
we can reach, starting from Xu every vertex of G by a path of length ^ d . Let us
consider first those paths starting from Xu the next vertex of which is an Y-.
As Yj can be chosen in < 1 —-y tjzt\ ways, and all vertices of G„ have valency
^k, the number of such pathes is at most
(3.8)
kd
"r(l+(^1) + (^--1)2 + ...+(fc-l)--i) S|l-|
n.
We may also suppose that £^-64.
213 EXTREMAL GRAPH THEORY

P. ERDOS, A. RENYI AND V. T. SOS
Let us count now the pathes of length ^d starting from Xu on which the point
next to Xu is an X{. The number of such pathes is clearly at most
(3.9)
4n
An
1+^^0 + (^) + - + ^-1)^2)
5n
T'
It follows from (J. 8) and (3. 9) that the total number of vertices which can be reached
from Xj by a path of length s.d, can not exceed n
— which is Sr n — 2 it n =—,
6) o
and this is true if n — 1 ^sk §64; thus we arrived to a contradiction and this proves
our theorem.
„2
To show that the order of magnitude —^ of the lower estimate of Fz(n, k)
K
is best possible, consider the following graph G„: Take a complete graph Gr having
r vertices, and connect each vertex of Gr with r—1 new points. Thus we obtain
a graph G„ with r(r — 1) + r ~r2 ~n vertices. Clearly one has k = V(G„) =2r — 2,
T r 1
D(Gn) = 3 and £(G„) = --r(i—1). Thus E(Gn)~-?-.
In this example fc = 2(j/n — 1): by slightly modifying this example we obtain that
FAn,k)-j2{c+r)2[-2+l
if A: ~ en where 0 < c < 1.
To show that F3(n, k) is of order of magnitude--^ for k~kin where
K
0</.< 1 we have to apply a more involved construction. Let us consider a graph
G„ which has the vertices P ■■ where l^gs/, 1 §/Sj, lsjgi and the vertices
Qghi where 1 ^g-<hsl and 1 ^i^s; thus « =
of G„ are as follows:
^fa2
— p. Suppose that the edges
a) Pgij and Phij are both connected with Qghi for 1 Sg </iS/, /,./= 1, 2, ..., 5.
t>) 29/,i, is connected with Qghh for 1
;-S, 1^1,;
c) 2</,/,,i ar|d 232/,2i are connected for 1 =^, </i, = / and
/=1,2, ..'., s.
Clearly
l ssg</i^/;
3Jg2</)2^/,
£(<?„)
further u(Pgij)=s—l and
^ +
'" (£,,»;) = s + l-l +
214 GRAPH THEORY

ON A PROBLtM OF GRAPH THEORY
229
and thus V(G„) = s + l +
- 2. Thus we obtain
F,
ls2 +
+ ,+ (W-I)M
(l]
UJ
'v2 +
\'}
U)
f*l+
U)
11
2
2
By other words by choosing for / an arbitrary fixed natural number and for s tending
to + °°, we obtain an infinite sequence of pairs n, k such that
kJ-^L and F3(n,fc)=E | ^/(/-1).
Thus for arbitrary small i >0 there exists an infinity of pairs n, k such that A- ~ /„]/n and
f3(«, A)
4;.4
Let us study now the behaviour of F3(n, A) for large values of A. Clearly
F3(n, k)=n—] if A ^- because the graph Cn shown on Fig. 9 has diameter 3
V(G„) =A and G„ is a tree, thus it has n~ 1 edges; this result is best possible because
a connected graph G„ cannot have less than n— 1 edges.
Fig. 9
We prove now the following
Theorem 4. If 7+s — 1 =As — + s-2 nTiere ^=1,2,3,
^ s+1 5 ' '
then
^(11^)=71 + (2)-1.
Proof of Theorem 4. The 3 ase j= 1 has been settled above. Let us consider
first the case s = 2. Suppose G„ would be a tree of diameter 3 and V(G„) = k^--,
and let Pl be an endpoint of Gn (such a point exists as every tree has at least two
215 EXTREMAL GRAPH THEORY

230
P. ERDOS, A. RENYI AND V. T. s6s
endpoints). Let P2 denote the single point connected with Pi by an edge, and let
Pz, ..., P, be all the other points connected with P2\ as V(G,,) = k we have l^k+ 1.
The remaining n —k~ 1 ^k~ 1 ^/-2 points have to be connected with one of the
points Pz, ..., P, because otherwise it would be impossible to reach them from
Pi by a path of hnsth ^ 3. But they can not be all connected with the same point
Pj (3^7^/) because this point would have valency >fc. Let Pr and Ps be two
points (/<r<5^n) such that Pr is connected with Pi and Ps with Pj (i^i<j^l).
Then the (unique) path from Pr to Ps has length 4; this contradiction shows that
F3(n, k)^n for k^ —-.
On the other hand Fig. 10 shows a graph G„ with V(Gn) = A where -+1^/:^-
which has diameter 3 and contains exactly one cycle (a triangle) and thus E(G„)~n.
This completes the proof of the fact thr t F3 (n, k) = n for — + 1 ^ k ^ —-.
Note that for n=2k+ 1 there is another extremal graph G2k+i of diameter 3,
for which V(G2k+l) = k and E(G2k+])=2k+], shown by Fig. 11.
P,
D(Gn)--3, VCGn)=k, E(GnUn, ^-^iki-j-
Fig. 10
P2-t1 Pkf3
VCG2kf1) = k, ECG2k„)--2k + 1, DCG2kt,1-3
Fig. 11
Now we pass to the case s^3.
Let Gn be a graph with V(G„) = k
^+1 j
H . 1 / H
and £>(G„) = 3. Let Xu ..., X, be the endpoints of G„. As the remaining n — l points
all have valency ^2, and at least one among them has valency k. we have
£(C„)s }(i + k + 2(n-l-l))=n--!r + t-l.
216 GRAPH THEORY

ON A PROBLEM OF GRAPH THEORY 231
Now if E{G„)S)i + \ l~l; we have nothing to prove; if however E(G„)<n +
+ (2)-1 we get
l>~k — s(s— 1)S5— 1
thus /a2. Let Yt, ..., Yv denote those vertices of Gn which are connected with
at least one X} (lsjS/).
Clearly Y, and Yj are connected by an edge (l^i<j^v) because otherwise
there would not exist a path of length 3 connecting the Xh-s. Thus it is sufficient
to consider the case v^ s, because every connected graph G„ containing a complete
5+1-graph has at least n —1+1 edges. Let us suppose therefore that v^s.
We prove first that cgj. Let the endpoint Xt be connected to Yi. Let Zl5 ... Zr
denote all the points connected with Yt which are not endpoints of G„. As every
point of G„ can be reached from X1 by a path of length ~ 3, if Yi is connected
r
with p endpoints then we have _2 v(Zh) — n ~P ~ 1 thus
thus in case E(G„) < n + — 1 we get
/Sn —r —5(5 — 1).
As however each Yj has valency ^k, it can be connected to at most k of the Xrs,
and Y1 only to k — r A"rs; thus
(v - l)k + k — r ^n — r — s(s — 1)
3
and therefore, in view of s^ \ — we obtain v>s~ 1 i.e. 1)¾. Thus we have only
to consider the case v=s. Now if v=s there exist in G„ at least s points which
are not connected to any of the Yj-s because these have valencies ^ k and thus
the total number of points connected with them is ^ 5-(Ar — (js- — 1)) ^ /i — -?. Let W
be such a point.
Now clearly W has to be connected with each Xh by a path of length 3 and
therefore with each y. by a path of length 2. Let U1, ..., Ut be the points connected
with W, then each Y- is connected with some Uz Thus it follows
E(Gn) a }-(2i + s(s-l) + 2s + 2t + 2(n-l-s-t-l)) =
2
?+ ^l+J + f + n-/-5-<-l =n +
Thus Fj(»,t)5»+ -1. On the other hand consider the graph G„ of the
following structure: let us take a complete graph Gs+1 having 5 + 1 points, and connect
217 EXTREMAL GRAPH THEORY

232
P. ERDOS, A. RENYI AND V. T. SOS
each of these points except one with k — s endpoints, and the last with n — s(k — s) —
-(J+l) points. (Clearly 0^n-s(k -s)-(s+ l)s=fc -s).
Thus we obtain a graph G„ with V(G„) = k, D(G„) = 3 and E{G„) = n + I I — 1.
This completes the proof of Theorem 4.
Let us consider now FA(n, k). Clearly
FA(n, k) = n-l if k^n^l.
This can be seen as follows. Fig. 12 exhibits a tree of diameter 4 showing that
FA(k2 + l,k) = k2
Clearly if (k — 1)2 + 1 < n < fc2 + 1, we obtain a graph Gn exhibiting FA(n, k) =
= n —1 by omitting from the graph on Fig. 12 k2 + l—n endpoints. We shall
prove now
Theorem 5.
FA(k2+2, k) ^k2 + l + ~ Yk {k = 2,3,...).
Proof of Theorem 5. Let Gk2+2 be an
extremal graph i.e. one which has k2 + 2 points,
diameter 4, satisfies the condition V(Gk2+2) = k and
has FA(k2 +2, k) edges.
Let Xu ..., Xm be the points of Gk2 + 2 having
valency ^2, and let G* be the subgraph of Gk2 + 2
spanned by these points. We assert that each point
Xj has the valency ^2 in G* too. Suppose that
Xt is an endpoint of G*, and that X2 is the only
point of G* to which Xi is connected. Clearly A\
is connected with at least one endpoint Yx of
Gki + 2 because it has valency ^2 in Gki+2, thus it
is connected with some point of Gki+2 different
from X2 and this point cannot be in G* and
thus is an endpoint of Gk2 + 2. Every point of Gk2 + 2 can be reached by supposition
from Y1 by a path of length ^4. However the number of points which can be
reached from Y1 by such a path is clearly
^2k~l+(k~l)2 = k2
which is a contradiction. Thus in G*„ each point has valency ^2. As the diameter
of G* is ^4, it follows from (1. 1) that G* contains at least one point of valency
Fig. 12
Ym — 1; thus the number of edges of G* exceeds (m ~ 1) + ? (]/w — 1). Each point
in G* can be connected with at most k — 2 endpoints of Gk2+2 thus k2 + 2^m +
k2+2_
+ m(k ~ 2) = m(k— 1) and therefore m
1
E(Gk2 + 2) \
Thus Theorem 5 is proved
k2+l + -
~ k-
4
{Y~m-
zk + 1; thus
1)
l + ^K.
218 GRAPH THEORY

ON A PROBLEM OF GRAPH THEORY
233
Note that the statement of Theorem 5 is trivial for /:5=16, because it states
only what we know already that if D(Gk2 + 2) = 4 then Gk2 + 2 can not be a tree.
To get an upper estimate for F4(Ar2+2, k)~k2 consider the following graph.
Take a graph Gk + S with V(Gk + 5) = k, D(Gk + s) = 2 and E(Gk + 5)=2k + 6; such
a graph exists according to Theorem 2 if fc^8 (see Fig. 7 with /=5). This graph
has k+2 points of valency 2. Connect k out of these points with k — 2 new points
each and one with A: —3 new points. Thus we get a graph G„ with n~k2+2 points,
such that V(Gn)=k, D(Gn) = 4 and E(Gn) = k2+ k + 3. Thus
F4(k2+2, k)^sk2+k + 3.
§ 4. Some further Remarks and Unsolved Problems
First we formulate some general principles of construction which were implicitely
used above.
If Gn is a graph of diameter d, and such that V(G„) = k, then if Gn is not regular,
we may construct from Gn a graph Cv of order N=n + kn~ E(Gn) with V(Gy) = k
and diameter d + 2, by connecting each vertex P: of Gn which has valency v(P{)<k
with A: —v(Pj) new points. Thus
(4.1) Fd + 2(n + kn-2Fd(n,k),k)skn-Fd(n,k).
For instance we have shown that F2(n, n — 5) = 2n — 4. It follows immediately
from (4. 1) that
F4(«2-8n + 8, n~5)^n2~ln + A.
Notice that for each value of d, the extremal graphs Gn with V(G„) = k, D(G„) = d
and having a minimal number of edges, are trees if k is sufficiently large, k ^ Ud(n) say.
We have implicitely shown that
(4. 2) U2(n) = n-\
(4.3) f3(n)=f
(4.4) UA(n) = in-\.
It can be shown that
(4 5) </,<») = !±J|Ld>
further that for any fixed ^^3 and n-*~
(4.6) U2s(n)^in
and
(4.7) U3s+1(n)>
The extremal tree of diameter 2s has a center, while the extremal tree of diameter
2s + 1 has a central edge.
219 EXTREMAL GRAPH THEORY

234
P. ERDOS, A. RENYI AND V. T. SOS
1 V
Notice that if k decreases by one below the critical value Ud(n), i.e. to Ud{n) —
there is a considerable increase in the value of Fd(n, k) if d is even, but not if d
odd. As a matter of fact
F2 (n, U2 (n) - 1) - F2 (», U2 (n)) = (2n - 4) - (n - 1) = » - 3
F3(2A + 1, A)-F3(2/c+l, A+ 1) = (2A + 1)- 2A = 1
and
F3(2A + 2, k) - F3(2k+2, k+l)= (2k + 3)- (2A +1)=2
further as proved by Theorem 5
F4(A2+2,A)-F4(/c2+l,A)
The situation is similar for (Z>4.
We call attention to the following problems, left open in this paper:
Problem 1. Is the graph of Theorem 1 extremal in the sense that among all
graphs with n vertices and not containing any cycle of length 4 does it have the
maximal number of edges? (We have proved only that it is asymptotically extremal.)
We can prove the following result, which is connected with Problem 1.
Theorem 6. If Gn is a graph in which any two points are connected by a path
of length 2 and which does not contain any cycle of length 4, then/; = 2/: + 1 and
G„ consists of k triangles which have one common
vertex (see Fig. 13).
Proof of Theorem 6. Let G„ be a graph
with the required properties. Let P^ be a point of
Gn having maximal valency. If P1 is connected with
all the remaining points of Gn then evidently these
have to be connected by pairs, and G„ is of the
type described in Theorem 6. Thus we may suppose
that Gn contains at least one point P2 which is
not connected with P1. It is easy to see that in this
case V(P2)=V(Pl).
As a matter of fact there is a point P3 in Gn
F'S- 13 which is connected with both P1 and P2. As there
must be a path of length 2 between P1 and P3
there is a point F4 which is connected with both P1 and P3. As there has to be a path
of length 2 between P2 and P3, there is a point F5 connected with both P2 and P3,
which is clearly different from P1, P2, P3 and F4. Let Qx, Q2, ..., Qk-2 be the
remaining points (besides P3 and F4) which are connected with Pt. Clearly P2
and Ps are not among the Q,; we have A'54 because v(P3)^4 and by supposition
P1 has the maximal valency.
Now from each of the points Q, there is a path of length 2 to P2; thus for each
Q, (i= 1, 2, ..., k — 2) there exists a point R, which is connected with both Q, and P2.
Clearly R-^Rj if jV; because otherwise G„ would contain the cycle P\Q\R\Qj.
Further R, is different from P3 because if R, would be identical with P3 G„ would
contain the cycle PiQiP3P^ Finally R, is different from P5 because otherwise G„
220 GRAPH THEORY

ON A PROBLEM OF GRAPH THEORY
235
would contain the cycle PiQ-,PsP3- Thus v(P2)^k and as k=V(G„) we obtain
v(P2) = k = v(P1). Thus any point of G„ which is not connected with Pi has the
valency k = v(P1). Repeating the same argument with P2 instead of Pi it follows that
v(Qi)—k (i= 1,2,...,^-2). As P3 is not connected with Q1 (because otherwise G„
would contain the cycle P1 Qt P3 P4) repeating the same argument for Q instead of
P1 it follows that v(P3) = k. Thus the graph G„ is regular.
Now if V(P:) = k (7= 1,2,..., n) and G„ does not contain a cycle of length 4
and between any two points there is a path of length 2, then clearly if S,- denotes
the set of points connected with Pt then the sets S,- and Sj have exactly one point
in common, and for any two points P, and Pj (7V 0 there is exactly one point Ph
such that Sh contains both P; and Pj. Thus if we define the sets of points S,- as lines
we obtain a finite plane geometry, with k = P+l points on a line, and thus having
n~P2 + P+] points. But then in this geometry there would exist a one-to-one
mapping beween points and lines such that no line contains the point corresponding
to it, and such a mapping is known [5] to be impossible. This proves Theorem 6.
Problem 2. To determine the exact value of F2(n, k) for k<—, or at least
the asymptotic value of F2(n, [nc]) with 0<c< \.
Problem 3. Is the lower estimate in Theorem 3 asymptotically best possible,
_i
i.e. do there exist for each d^ 3 a sequence of graphs G„(n-*°°) withK(G„) = fc~ cnd~L
71 11
where oO is a constant, D(Gn) = d and E(Gn) ~ rjzri ~ d_ , ?
/c c
Problem 4. Determine asymptotically FA(k2 + 2, k) — k2.
Problems similar to those considered in this paper can be asKed for directed
graphs. We hope to return to these problems in an other paper.
(Received February 1, 1966.)
REFERENCES
[1] Erdos, P.—Renyi, A.: On a problem in the theory of graphs (in Hungarian, with English and
Russian summaries), Publ. Math. Inst. Hung. Acad. Sci. 7/A (1962) 623—641.
[2] Erdos, P.: On sequences of integers no one of which divides the product of two others and
on some related problems, Mitteilungen des Forschungsinstitutes filr Math, und Mecha-
nt\ Tomsk, 2 (1938) 74—82.
[3] Reiman, I.: Ober ein Problem von K. Zarankiewicz, Acta Math. Acad. Sci. Hung. 9 (1958)
269—278.
[4] Hoffman, A. J.—Singleton, R. R.: On Moore graphs with diameter 2 and 3, IBM Journal
of Research and Development 4 (1960) 497—504.
{5] Baer, R.: Polarities in finite projective planes, Bulletin of the American Math. Soc. 52 (1946)
77—93.
221 EXTREMAL GRAPH THEORY

SOME RECENT RESULTS
ON EXTREMAL PROBLEMS IN GRAPH THEORY
(Results)
P. ERDOS *
Three years ago I gave a talk on extremal problems in graph theory at
Smolenice [2]. I will refer to this paper as I. I will only discuss results which
have been found since the publication of I. n(1&) will denote the number of
vertices, V(@) the number of edges of <&. 1S{n ; /) will denote a graph of n vertices
and / edges. The vertices of <& will be denoted by letters x, y, ... ^0(xu ..., xk)
will denote the subgraph of <& spanned by the vertices xu ..., xk. v(x), the
valence of x, denotes the number of edges incident to x. xC&) will denote the
chromatic number of &, Kr the complete graph of r vertices IS j r ;
Kr(pu ..., pr) denotes the complete r-chromatic graph with pt vertices of the
i-th colour in which every two vertices of different colour are adjacent. C„
is a circuit having n edges.
Denote by f(n ; ^x, ..., ^k) the smallest integer so that every
»(n ;/(«;»!, ...,^)
contains at least one of the graphs ^,-, /=1, •-., fcasa subgraph. For the older
literature on this subject I refer to I. Here I only state that in 1940 Turan [9]
determined f(n ; Kr) for every r ^ 3, he proved
(0 /(/1^,)=(1+0(1))^(1-^).
In fact Turan gave an explicit formula for f (n ; Kr) and determined the
structure of the unique <0{n;f(n;Kr) — l) which does not contain a Kr,
but (I) suffices for our purpose.
In a forthcoming paper [3] Simonovits and I proved the following result :
Put
r = min x(@t) ■
Then
(2) Km f(n ; 9U ..., $k)/n2= ~(l - —L-) .
(*) Magyar Tudomanyos Akad6mia Matematikai Kutat6 Int6zete, Budapest, Hungary.
G))'
222 GRAPH THEORY

118
P. ERDOS
In other words the asymptotic relation (I) holds in the general case too.
(2) Follows easily from a theorem of Stone and myself [4] which states that
if n > n0(plt ...,pr, e) then every
'(-t('-tV.))
contains Kr(p1, ..., pr).
Recently I succeeded in proving the following stronger theorem : There is
an A = A(e, r) so that for n > n0(e, r) every
'(•'t('-tV'))
contains a
(3) Kr(pu ..., pr) with pr > — — .
A Pi ■■ ■> Pr - 1
In particular (3) implies that every
*(-T('-^7-))
contains a
(4) Kr([c.(log «)1/r-'], ..., [c.(log n)1/r-J]) .
In [4] we only proved (4) with (logr_! n)1'8 instead of ce(Iog n)1/r-1 (logrn
is the r-fold iterated logarithm). It seems probable that both (3) and (4) are
best possible, but this has been proved only for r = 2. In this paper we will not
prove (3) and (4).
I have obtained the following sharpening of the theorem of Simonovits and
myself :
Consider any one of the extremal graphs ^(n ;f(n ; &\, ..., ^k) — 1) which
does not contain any of the graphs <&u ..., t§k. Then there is a graph
r_1 n
Kr-i(Pi, •••> Pr-l), I! Pi = n , Pi = (1 + 0(1)) -, i= 1,...,/- - 1,
i=i r — i
so that our ^(n ;f(n ; <&u ..., ^k) — 1) is obtained from this
Kr-i(j>i, ...,Pr-i)
by adding and subtracting o(n2) edges.
In fact I can prove the following stronger.
Theorem. Let r ^ 3, and ^(n ; I),
223 EXTREMAL GRAPH THEORY

SOME RECENT RESULTS ON EXTREMAL PROBLEMES
119
be a graph which does not contain a Kr(t,..., t)for some fixed t. Then there is a
r_1 n
^-i(Pi.-.P,-i)>Zft = ^ Pi =0 +0(1))- f» i=l,..., r-1
i=i ' l
which can be obtained from our 9(n ; /) by adding and subtracting o(n2) edges.
Let x(^) = r. Then for sufficiently large t, & is contained in a Kr{t, ..., t).
Thus our theorem is stronger then the previous assertion.
We will later prove the theorem for r = 3 and only outline the proof for
r > 3.
Now we state without proof some further recent results : Let /(^) = 2. The
graph (9 ; k) is defined as follows : Let y1}..., yk be k new vertices; (^ ; k) is
obtained from 9 by adding the k new vertices yu ..., yk and by joining each of
the yt, i'=I, ..., k to all the vertices of <&. I proved
(5) f{n;(9;k))<!j + ckf(n;9).
Kovari and the Turans proved that [8]
(6) /(«;X2(r,r))<Cl«2-1/r.
In I. I stated that every 1§[n ; — + cn2~1/r\ contains a K3(r, r, r). In view of
(6), (5) immediately implies this result.
Simonovits and I conjectured that for every <& :
(7) f(n ; 9) = y (l - y^j + en' + o(n% x(9) = r
for some 0 < a = a(^) < 2. (7) if true will not be easy to prove. If /(^) = 2
(7) would imply
(8) lim /(/z; 9) In' = c (c > 0)
n= oo
for some 0 < a < 2. We are very far from being able to prove (8). If <S = C4
then recently Brown and independently V. T. S6s, Renyi and I [5] proved that
Iim/(n;C4)/«3/2 = i
n— oo
(In I. I conjected that the above limit is 1/2 y/2).
Brown [1] recently proved that
f(n;K2(3,3))> c2n5'3.
Brown's proof seems to break down for r > 3 and does not prove the
existence of
Iim/(n;X2(3,3))/n5/3 .
224 GRAPH THEORY

120
P. ERDOS
The determination of a(^) seems to be a very difficult question. Perhaps the
following result holds : Let the vertices of 9 be x1; ..., x„. Put
v(9) == min u(x,), v*C&) = ma\v(9(xi, ..., xk))
1 $ i $ n
where x1;..., xk runs through all the 2" subsets of xx, ..., x„. Then
(9) f(n;9) < cn2'1'"'^ .
(9) is known if 9 is K2(r, r). I can also prove (9) if 9 is the graph determined
by the vertices and edges of a cube. In this case probably
a(9) = 2 - -1— = -.
v\9) 3
a(9) = 2 —certainly does not always hold. In fact we have v*(C6) = 2 and
v*(9)
f(n ; C6) < erf13. A very special case of (9) was conjectured in [6]. It seems
lat a(^) can take on only the values 1 + - , k = 2, 3,... and 2 — ~ ,k = 1,2,...
k k
Now we prove our theorem. Assume first r = 3. We have to show that if
there is a
that a(^) can take on only the values 1 + - , k = 2, 3,... and 2 — ~ ,k = 1,2,
k k
we prove our theorem. Assume
n2
/ = -(1 + 0(1)) and 9(n ; /) does not contain a K3(t, t, t) for some fixed Jthen
n
of valence <•=;(! — c). But then we see by a simple computation that
n n
K2(Pl,P2)> Pi + P2 = n , Pi = (1 + 0(1))^, P2 = (1 +0(1)) 2
and 9(n ; /) differs from K2(j>i,P2) by o(n2) edges.
First of all we can assume that all but o(n) vertices of our ^(n ; [) have
valence > - (1 + o(l)). For if this would not be true let xu ..., xk, k = [en]
(e > 0 is small positive number independant of ri), be the vertices of 1§(n ; I)
of valence
(c > c(e))
n2 kn2 In — k~\2
(10) v(9(xk+1,...,xn))>^(l +0(l))-^-(l ~c)>{ 4K> (1+n),
n = n(e, c) > 0 .
(10) implies by the Erdos-Stone theorem that 9(xk+1, ..., xn) and therefore
our 9(n ; I) contains a K3(t, t, t) which contradicts our assumption.
Let now
(11) xu ..., x,„, m = (1 + o(l))n
225 EXTREMAL GRAPH THEORY

SOME RECENT RESULTS ON EXTREMAL PROBLEMES
121
Tl
be the vertices of ^(n ; I) which have valence > ~-(l + o(l)). By (II) it follows
that the valence of all the vertices of ^(x1; .... x,„) (in ^(xu ..., x,„)) is
y(l +0(1)) = ^(1 +0(1))
also, by (11)
2 2
ryt fj
(11) V(9(xu ..., xj) = — (1 + 0(1)) =-(1+ o(D).
Thus to prove our theorem it will suffice to show that there is a
K2(Pi,P2), Pi+Vi=™, Pi = (1 + o(l))—, p2 = (1 + o(l))—,
which differs from ^(x1; ..., x,„) by o(m2) edges. Thus it is clear that without
loss of generality we can assume that all the vertices of our 'Sin ; /) have
valence ^=(1 + o(l)), and henceforth we will alwaysmake this assumption (*).
An edge of 'Sin ; I) is called bad if it is contained in only o(n) triangles of our
'Sin ; I). Assume first that 'Sin ; /) has at least en1 bad edges. By the Erdos-Stone
theorem there are vertices xu ..., x, ;y1, ..., y, so that all the edges (x;, yj),
1 < /, j < t are bad. Now since each of the vertices x,- and y- have valence
(1 + o(l))^ and each of the edges (x;, yj) are contained in o(n) triangles, a
simple argument shows that the remaining n — 2 t vertices of our 'Sin ; I) can
be divided into two classes (neglecting o(n) vertices)
Tl Tl
zu...,zui; Wi, ..., w„2, «! = (1 + o(l)) j, u2 = (1 +0(1)) 2
so that all the xr are joined to all the z's and all the yj are joined to all the w's.
If both graphs ^(^, ..., zui) and ^iw1, ..., w„2) have o(n2) edges then a simple
computation shows that the K2iult u2) having the vertices zlt..., zUl ; wt, ..., wU2
differs from our 'Sin ; I) by o(n2) edges which proves our theorem (the remaining
n — u1 — u2 = o(n) vertices can clearly be ignored). If say "Sizu ..., zuj) does
not have o(n2) edges then by the Erdos-Stone theorem it contains a K2it, t)
having the vertices z1;..., zt; zt + 1,..., z2t. But then the graph "Sixu ..., x„ z„ ...,
z„ z, + 1,..., z2t) clearly contains a K3it, t, t) which contradicts our assumption.
Henceforth we can thus assume that there are o(«2) bad edges. Thus there
n2
are -r (1 + o(l)) edges each of which are contained in rn triangles. We now
n2
(*) We now no longer will have to use I = -r (1 + o(l)), I > en2 would suffice, but this is no
real gain since our assumption y(*() = (1+ o(l)) r , /= 1 n already implies I = -r (1 + o(l)).
226 GRAPH THEORY

122
P. ERDOS
deduce from this assumption that ^(n ; /) contains a K3(t, t, t) (in fact we will
only use that ^(n ; /) has en2 edges each of which are contained in rn triangles).
Let ev, ..., es, s > en2 be the edges of 1§(n ; /) each of which are contained in at
least rn triangles. Let x*0, ..., x,',' be the vertices which form a triangle with
eh rt > rn. Form all possible Muples from the x\l), 1 < j < r; for all i,
I < ;' =¾ s. In this way we get
Muples. Since the total number of Muples formed from n elements is | I
there is a Muple say zv, ..., zt which corresponds to at least £(½) n2 edges
et — in other words each of these e{ form a triangle with all the z,-, j = 1,..., t.
By the theorem of Erdos-Stone these edges determine a K2(t, t) with the
vertices xu ..., x, ; yh ..., yt. Thus finally ^{xu ..., x„ yv, ..., y„ zu ..., zt) contains
a K3(t, t, t) as stated. But by our assumption our (&{n ; /) does not contain
a K3(t, t, t). This contradiction completes the proof of our theorem.
By slightly greater care we could prove the following stronger statement :
n2
Let ^(n ;/),/=—(! + o(I)) be a graph which does not contain a K3(t, t, t)
for t = o(logn). Then there is a
r_1 n
K,.-.!^!, •••, Pr-i), E Pi = n> Pi = (1 + 00))- 7> (=1,...,/--1
;= i r — i
which differs from our 1§(n ; /) by o(n2) edges.
The proof for r > 3 does not substantially differ from r = 3. As in the case
r = 3 we first show that without loss of generality we can assume that every
vertex of ^(n;l) has valence > n I I + o(l)J. An (r — l)-tuple
x£l,..., x,.^ formed from the vertices xx, ..., x„ of ^(«; /) is called bad if there are
only o(ri) of the Xj so that each edge of the complete r-tuple { x£l, ..., x,r_l, Xj }
belongs to our ^(n ; I). Now as in the case r — 3 we first assume that there are
more than el J bad (r — l)-tuples. We complete the proof as in the case
r = 3 only instead the Erdos-Stone theorem we here have to use the following
result [7] : Let eu ..., ek, n > n0(e, k, r) k > ei I be a family of (r - 1)-
tuples formed from the vertices xt, ..., x„. Then there are (r — 1) t vertices
x\}1, 1 «S i < t; 1 «Sj «S r - 1 so that all the t1"1 (r - l)-tuples { x^, x[22),...,
xt-i"* } are e's-
In the second case there are only ol I bad (r - l)-tuples. It is not
difficult to deduce then that our ^(n ; /) contains more than eri"1 complete
227 EXTREMAL GRAPH THEORY

SOME RECENT RESULTS ON EXTREMAL PROBLEMES
123
(r — l)-tuples so that to each of them there are at least rn vertices each of
which are joined to all the vertices of the (r — l)-tuple. Using this result and
[7] we can complete the proof as in the case r = 3.
By the same method we can prove the following sharper result :
every 9\n ; ^{l - --^ - ej J
which does not contain a Kr(t, •••, 0. contains for n > n0(r, t, e) an (r - 1)-
chromatic subgraph having more than ^-(1 r — cr e I edges. I just
learn that Mr Simonovits proved this independently by a different method.
REFERENCES
[1] The Brown's paper will appear in the Bulletin of the Canadian Math. Soc.
[2] Erdos, P., Extremal problems in graph theory, Theory of graphs and its applications
Proc. Symp. Smolenice, Plublishing House of the Czechoslovak Academy of Science,
Prague, 1964.
[3] Our paper will appear in Studia Scientiarum Mathematicarum Hungarica.
[4] Erdos, P. and Stone, A., On the structure of linear graphs, Bull. American
Mathematical Society, 52, 1946, 1087-1091.
[5] Brown's paper will appear in the Bull. Canadian Math. Soc, our paper will appear in
Studia Scientiarum Mathematicarum Hungarica.
[6] Erdos, P., on some extremal problems in graph theory, IsraelJ.ofMath.,3,1965,113-116.
[7] Erdos, P., On extremal problems of graph and generalised graphs, Israel J. of Math., 2,
1964, 184-190.
[8] Kovari, T., S6s, V. T. and Turan. P., On a problem of K. Zarankiewicz, Coll. Math.,
3, 1954, 50-57.
[9] Turan, P., Eine Extremalaufgabe aus der Graphentheorie (written in Hungarian),
Mat. is Fiz. Lapok, 48, 1941, 436-452. See also. Turan, P., On the theory of
graphs, Coll. Math., 3, 1954, 19-30.
228 GRAPH THEORY

Extremal Problems in Graph Theory*
Paul Erdos
Without any doubt, Paul Erdos is one of the most famous mathematicians in
the world. Bis areas of interest include number theory, set theory, probability,
analysis, and graph theory. He has probably written more papers for
mathematical journals than any other living mathematician. He is very well known for his
fondness of travel and has surely lectured in more universities than anyone else.
Erdos usually writes a joint paper with one or more of the mathematicians at
each university he visits. So it was not entirely unnatural that a fantastic (false)
rumor was spread about him to the effect that he even wrote a joint paper with
a railroad conductor while traveling from one university to another. After his
home base at the Mathematical Institute of the Hungarian Academy of Science
in Budapest, his favorite locations are the Technion in Haifa, Israel, and the
University College, London.
Everyone who has met him knows the classical Erdos terminology. An
epsilon is a child. A married couple consists of one boss and one slave, the wife
and husband respectively. When a couple is married, the boss is said to have
captured the slave. After a divorce, the slave is liberated. If the man should
remarry, he is said to be recaptured, and so on.
As indicated in the lecture itself, the subject of extremal problems in graph
theory was initiated by another Hungarian mathematician, Paul Turan.
Presently, it is the most popular area of graph theory in Hungary with papers
on the subject written by Erdos jointly with each of the mathematicians,
Andrasfai, Bollobds, Gallai, Hajnal, and Posa (who was an epsilon of only
thirteen when his prodigious talent was discovered by the lecturer).
F.H.
The starting point for many extremal problems in graph theory is the
work of Turan [13] and [14] who initiated this topic in 1940 while in a labor
camp. He showed that every graph with n points and I + [n2/4] lines contains
a triangle. Throughout this lecture, G(n; m) will denote an arbitrary graph
*Another article with the same title appeared in Theory of graphs and its applications,
edited by M. Fiedler. Prague, 1964, pp. 29-36. All results given without references are
unpublished.
229 EXTREMAL GRAPH THEORY

EXTREMAL PROBLEMS IN GRAPH THEORY
55
with n points and m lines. Also, K(m, n) will denote the complete bicolored
graph with m points of one color and n of the other. Similarly, K(nv,n2, ■ ■ -,«t)
will denote the complete fc-colored graph with «, points of the ith color.
Turin's result asserts that every graph G(n; 1 + [«2/4]) contains the
complete graph K3 . This is the best possible result because there exists a
graph, namely K([(n + l)/2], [n/2]), with [n2/4] lines but no triangles. These
facts are illustrated in Fig. 8.1 where n = 5; graph GY has seven lines and a
triangle, whereas G2 has six lines and no triangle. More generally, for each
integer p <n, Turan determined the least integer m(n, p) such that any graph
G(n; m(n,p)) contains the complete graph Kp. He also showed that the only
graph G(n; m(n, p) — 1) which does not contain Kp is K(ni, n2, ■ ■ ■, ^-^),
where the «,- are as nearly equal as possible. Dirac [5] showed that in addition
to containing Kp, any graph G(n; m(n, p)) contains Kp+i - x, the graph
obtained from Kp+i by deleting one line.
The graphs G(n; 1 + [«2/4]) have other interesting properties. When
n > 4, each such graph contains the first graph shown in Fig. 8.2, and when
n is sufficiently large, it contains all of the graphs with one cycle as shown in
Fig. 8.1. Furthermore, it contains any of the four possible graphs G(5; 6) with
five points and six lines, all shown in Fig. 8.3. In addition, for n large enough,
any graph G(n; 1 + [n2/4]) contains both of the graphs with five points and
seven lines shown in Fig. 8.4. These two graphs are special cases of the following
result. For any integer r, there is an integer n0(r), such that when n > n0(r),
each graph G(n; 1 + [«2/4]) contains a subgraph of the form
K([(r+ l)/2],[r/2]) + x,
obtained by adding a line to the complete bipartite graph. Similarly, for n
large enough, every graph G(n; 1 + [«2/4]) contains the cycle C2k + 1 .
A variation of this problem is the determination of the value off(n, k, r) ,
the smallest integer m such that every graph C(«; m) contains some graph with
k points and r lines. As indicated above,/(«, 3, 3),/(«, 4, 4),/(«, 4, 5),f(n, 5, 5),
and f()i, 5, 6) all have the same value, 1 + [«2/4], for n sufficiently large. In
general, not much is known about f(n, k, r), but we do have the following
results.
If r <k/2 , then f(n, k, r) = r.
If/c/2 < r <k, then/(-w, k, r) =f(n, 2/- + 2 - k, 2/- + 1 - k).
Lastly,f(n,k,k -1) = 1 + [(kk~-~ ■
Questions remain concerning f(n, k, k). There are only two graphs C(4; 4), the
first graph of Fig. 8.2 and C4. The extremal graphs for the former have been
found, but for the latter, the problem remains unsolved. In 1938 Erdos proved
the existence of a constant c such that every graph G(n; [en312]) contains a
230 GRAPH THEORY

56 EXTREMAL PROBLEMS IN GRAPH THEORY
cycle of length 4 for n large enough. Reiman [12] showed that for arbitrarily
small e and sufficiently large n,
(1 -*)2 /2</(«,4,4)<(l +^)-2-,
and it has just been shown by Brown, Renyi, Sos, and Erdos that
hm^V^1-
«3/2 j
Fig. 8.1.
1 n ru A
AvAhu
Fig. 8.2.
Fig. 8.4.
231 EXTREMAL GRAPH THEORY

EXTREMAL PROBLEMS IN GRAPH THEORY
57
Kovari, Sos, and Turan [10] have shown that for some constant c, every
graph G(n; [cn2~1/k]) contains K(p, p), which is a special case of a problem of
Zarankiewicz [15]. Except when p = 2, however, it is not even known if this
order of magnitude is best possible. Another open problem is the
determination of how many lines a graph must have to ensure that the graph K^ + K(p, p)
is a subgraph. Again, Dirac and Erdos have settled this independently when
/7 = 2. When/7 = 3, it is conjectured that the extremal graph is the join of two
Cycles ('[(n+1)/2] + ^[n/2] •
In 1941 Rademacher (see Erdos [6]) showed that every graph G(2n; n2 + 1)
contains n triangles. Generalizing this, Erdos [7] proved the existence of a
constant c such that when k < en, every graph G(2n; n1 + k) contains kn
triangles. This is false when k = n since the graph Kn_^ + C„ + 1 contains
n2 + n lines but only n2 — 1 triangles. In proving this, several interesting
lemmas were required. For example, there exists a constant c < J/j, such that
every graph G(2n;n2 + 1) contains a line belonging to at least [en] triangles.
It has also been shown that every graph G(3n; 3n2 + 1) contains n2 cycles of
length 4 and that the result is best possible.
Ore [11] proved that every graph G(n;2 + n(n + 1)/2) contains a
hamiltonian cycle, clearly another best possible result. Dirac [3] has shown
that any graph in which all points have degree at least n/2 is also hamiltonian.
We have proved that every graph G(n;r) is hamiltonian if every point has
degree at least k and r > I J + t2 for every t satisfying k < t < n/2.
This result is also best possible. The proof uses the following theorem of Posa
which generalizes Dirac's result: A graph with n points is hamiltonian if
whenever /c < n/2, there are at most k — 1 points with degree less than k + 1.
It has been conjectured that any graph with rm points, all of degree at
least m(r — 1), contains mKr. Its validity when r = 2 follows from Dirac's
result [3] on hamiltonian cycles; Corradi and Hajnal [2] proved it when r = 3.
Erdos and Gallai [8] have shown that every graph G(n;r) contains m
independent lines if
f(2m - 1\ , „ , ,o (m - 2\]
r > max , I , n(m - I) - (m - ly + i I .
In this problem the extremal graphs are K2m-1 and Km-x + Kn_m+1 .
It was shown by Dirac [4] (see also Erdos and Posa [9]) that when n > 4,
every graph G(n;2n — 2) contains a subgraph homeomorphic to K± , still
another best possible result. It has also been conjectured that when n > 5,
every graph G(n; In — 5) contains a subgraph homeomorphic to Ks.
Bollobas and Erdos [1] proved that every graph G(n; [(in — 1)/2])
contains a cycle and another point adjacent to two points of the cycle, and
thus every such graph contains two points which are joined by three line-
disjoint paths. These results are best possible. It was conjectured that every
graph CI 1 + n(m — 1) ; 1 +«(->)) contains two points which are joined by
232 GRAPH THEORY

58
EXTREMAL PROBLEMS IN GRAPH THEORY
m disjoint paths. The graph K1 + nKm shows that, if true, this is best possible.
Bollobas proved this for m = 4; in fact, he showed that every graph
G(n; In — 1) contains two points which are joined by four line-disjoint paths
and this is best possible. Perhaps every graph G(n; In — 2) contains a cycle and
another point adjacent to three points of the cycle. If true, this would strengthen
Dirac's result mentioned above because in the subgraph homeomorphic to A"4,
the three paths incident with one point would consist of a single line. It is
obvious that every graph G(n; n) contains a cycle. A complicated result due to
Erdos and Posa [9] is that every graph G(n;(2m — \)n — 2m2 + m + 1)
contains m disjoint cycles if n > 24m. The result still holds when a line is
removed if doing so does not yield the graph K2,n-i + Kn _ 2m +1. The condition
that n > 24m can be weakened somewhat. The contribution made by Posa,
who was thirteen years old at the time, was the following ingenious argument,
which can be generalized, to show that every graph G(n; 3« — 5) contains two
disjoint cycles if n > 6.
That this result holds when n = 6 may be verified by considering the
various possibilities. Assume that it is valid when » < / — 1 and consider any
graph G(t; 3r — 5). There is some point, say u'0, in which the degree d(\v0) < 5.
Suppose that vv0 is adjacent only to the five points wu w2, w3, u'4, w5. If the
subgraph induced by these six points contains at least 13 lines, the result
holds, since it has already been established when /; = 6. If not, then there is a
point vi'j which is not adjacent to at least two of the other points, say \r2 and vr3.
Add the new lines vv',H'2 and m']H'3, and remove u'0 and the five lines incident
with it from the original graph. There remains a graph G(t — 1; 3r — 8) which
contains two disjoint cycles by the induction hypothesis. At least one of these
two cycles does not contain either of the lines it'i vv2 or m,1vv3. In any case, it is
easily seen, by including vr0 in a cycle if necessary, that the original graph must
contain two disjoint cycles if d(w0) = 5.
Next, consider the case where w0 is adjacent only to the four points
vt-j, w2, n'3, n'4. It may be assumed that the subgraph generated by
W = {vi'0, vi'j, w2, w3, h'4} is #5 since otherwise an argument similar to that
used above yields the required result. Let H = G(t; 3r — 5) — W, the graph
obtained from the original graph G(t; 2>t — 5) by removing the points of W.
If any point in H is adjacent to two or more points in W, the result is clearly
true. So it may be supposed that no point in H is adjacent to more than one
point in W. Remove vi0, h^, and w2 from the original graph. The remaining
graph has n - 3 points and at least (in — 5) — (« — 5) - 9 = 2n - 9 lines.
Since 2« — 9 > n — 3 if n > 6 , this remaining graph has at least one cycle.
This cycle and w0vV]W2n'0 form two disjoint cycles in the original graph.
When the degree d(w0) < 3, the result is shown by applying the induction
hypothesis to the graph G(t; 3/-5)- \v0 . This suffices to complete the
proof by induction.
We close with a proof of the following result by Posa. Every graph
G(n; n + 4) contains two line-disjoint cycles. The result is proved for multi-
233 EXTREMAL GRAPH THEORY

EXTREMAL PROBLEMS IN GRAPH THEORY 59
graphs. This clearly holds when n = 1. Assume that it holds when n = t — 1
and consider any graph G(t; t + 4). If the graph contains a cycle C of length
3 or 4 the result certainly holds since removing the lines of C yields a graph
with a cycle line-disjoint from C. Hence it may be assumed that every cycle
has length greater than 4. If there is a point of degree 1, the result follows by
removing this point and applying the induction hypothesis to the remaining
graph. If some point v0 is adjacent to only two points vt and v2, then the
induction hypothesis can be applied to the graph obtained by adding the new
line v1u2 and removing v0. If one of the two line-disjoint cycles in this graph
contains the line viv2, then this line is replaced by the lines uiv0 and v0v2 in
the original graph. Hence, it may be assumed that the degree of every point in
the graph G(t; t + 4) is at least 3, which implies that 3//2 < t + 4, that is,
t < 8. But it is not difficult to see that there are no multigraphs with fewer
than nine points in which every point has degree at least 3 and every cycle has
length greater than 4. Therefore, the result holds when n = t and hence in
general by induction.
References
[1] B. Bollobas and P. Erdos, On extremal problems ingraph theory (in Hungarian).
Mat. Lapok. 13(1962) 143-152.
[2] K. Corradi and A. Hajnal, On the maximal number of independent circuits in a
graph. Acta Math. Acad. Sci. Hung. 14(1963) 423-439.
[3] G. A. Dirac, Some theorems on abstract graphs. Proc. London Math. Soc. (3)
2(1952)69-81.
[4] , In abstrakten Graphen vorhandene vollstandige 4-Graphen und ihre
Unterteilungen. Math. Nachr. 22(1960) 61-85.
[5] , Extensions of Turan's theorem on graphs. Acta Math. Acad. Sci. Hung.
14(1963) 417-422.
[6] P. Erdos, Some theorems on graphs. Riveon Lematematika 9(1955) 13-17.
[7] , On a theorem of Rademacher-Turan. III. J. Math. 6(1962) 122-127.
[8] and T.Gallai, On maximal paths and circuits of graphs. Acta Math. Acad.
Sci. Hung. 10(1959) 337-356.
[9] and L. Posa, On the maximal number of disjoint circuits of a graph.
Publ. Math. Debrecen 9(1962) 3-12.
[10] T. Kovari, V. T. Sos, and P. Turan, On a problem of K. Zarankiewicz. Colloq.
Math. 3(1954)50-57.
[11] O. Ore, Arc coverings of graphs. Ann. Mat. Pura Appl. 55(1961) 315-322.
[12] I. Reiman, Ober ein Problem von K. Zarankiewicz. Acta Math. Acad. Sci.
Hung. 9(1958) 269-279.
[13] P. Turan, Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok.
48(1941)436-452.
[14] , On the theory of graphs. Colloq. Math. 3(1954) 19-30.
[15] K. Zarankiewicz, Problem 101. Colloq. Math. 2(1951) 301.
234 GRAPH THEORY

ON SOME NEW INEQUALITIES CONCERNING EXTREMAL PROPERTIES
OF GRAPHS
by
l\ EHDOS
Mathematical Institute of the Hungarian Academy of Science*
Budapest, Hungary
Denote by G(n; I) a graph of n vertices and I edges. x(G) will denote the
chromatic number of G. K^p^, ..., pr) denotes the complete r-ehromatie
graph with p, vertices of the i-th colour where any two vertices of different
colour are joined. Kx(p) is a graph consisting of p isolated vertices.
(G:Kr(pv .. -,pr)) is obtained from (7 by adjoining a,Kr(pv.. -,pr), and by
joining every new vertex to all the vertices of 67. Clearly *((67 : Kr(pir . . ., pr))=
= > (67) 4- r.f(n; 67) is the smallest integer so that every G^n; f(n; 67)) eon-
tains 67 as a subgraph. The graphs G'(n) = G'(n; f(n; 67)-1) which do not
contain 67 as a subgraph arc called the extremal graphs belonging to 67.
The vertices of 67 will be denoted by x, xu . . ., y, . . ., the edges will be
denoted by (x, y). The valence of a vertex x of G is the number of edges
incident to x. jc(G) denotes the number of vertices, v(G) the number of
edges of 67. If 67' is a graph and xv . . ., xk are some of the vertices of 67' then
6r'(Xj, . . ., xH) is the subgraph of 67' spanned by £-,, . . ., xk. c, ca, . . . denote
absolute constants not necessarily the same if they occur in different
formulas.
In a previous paper [1] I stated without proof that
!D
f(n ; Kr(t, ..., t))<
1
r - 1
+ ail~
In the present paper I will prove that (1) is a special case of a mure general
theorem. A recent result of SimonovIts and myself states [2] (x(G) = r)
(2)
f(n; 67) =
r - 1
4- o(n2) .
In this paper I will prove
Theorem 1. Let >.(G) — 2. Then for n > n0(t)
1
f(n;(G:Kr_,(t,...,t)))
+
independent of t
+ (1+0(1))(,.-1)/(
G\ + cn
235 EXTREMAL GRAPH THEORY

v. erdOr
first we deduce (1) from Theorem 1. A well known result of Kovari
and the Turans [5] states that
:3)
f(n;K2(t,t))<cri2-V>.
Clearly Kr(t, ...,t) = (K2(t, t) : Kr_2(t, . . .,t)). Thus from Theorem 1 (G =
K2(t, t)) we immediately obtain (1). (1) is probably best possible for every
r and t but I can prove this only for t <^ 3.
Theorem 1 immediately implies that for n > n0(l)
(4) f(n;Kr(t,t,l, ...,1)) |l
1
< ci(r — 1) n2~]jt + ¢2 n ■
l)
r-1)
where both cx and c2 are independent of I. In faet perhaps for n > n0 (lit I,
(5) \f(n;Kr(t,t,llt ....h)) - f(n;Kr(t,t,l2, ... ,l2))\ <cn,
but I am very far from being able to prove (5).
It seems likely that in contrast to (4) and (5)
<c';ri2-Vt
7)2 I 1 \
-W-llt <\f(n; Kr(t, l,..., 1)) - — 1
where C/ —*■ 00 and c[ —*■ °o as I —*■ 00. The upper bound follows easily from
Theorem 1 and the known result
(6)
K2(t,l) <c"n2']H .
((6) follows e.g. by the method of [5]), but I can not prove the lower bound.
By more complicated methods I can prove the following strengthening of
Theorem 1.
Theorem 2. Let x(G) = r and put
,2
f(n ; G)
n*
r-1)
+ h(n ; G)1
Let d = d(G) be sufficiently small. Then for n > nQ(G, d)
f(n -,(0:^ ([dn])) < — (l~-\ + c, h(n ;G)+c2n.
Theorem 2 in particular implies ( (G) = 2)
n
f(n ; (G : Kr_,(t, ...,t, [dn]))) < -^-( 1 - --!—
2 1 r — 1
+ (1 +0(1)) (r-1)/
We do not prove Theorem 2 in this paper.
1 By [2] h(n; G) = o(n2).
+
; G + en
236 GRAPH THEORY

NEW INEQUALITIES CONetOH.VIXG HXTliEMAL PROPERTIES 79
In a recent paper [3] I proved the following sharpening of (2):
n
2
Theorem A. Let 1 = (1 +0(1))— 1 and assume that G(n; I)
r-1
does not contain a Kr(t, . . ., t) as a subgraph. Then there is a
r-l n
C<) Kr-i(Pi, ■ ■ ■ ,Pr-i),J? Pi = n, 2),- = (1 +o(\)) , i = 1, . . .,r — 1
i=i r — 1
which differs from our G(n; I) by o(n2) edges.
The principal tool in the proof of Theorem 1 will be
Theorem 3. Let G'(n) be any extremal graph belonging to G (x(G) = r).
Then the vertices of our G'(n) can be partitioned into r — I classes
each containing (1 + o(l)) of the xt so that for every e > 0 all but <\
r — 1
of the Xj are joined to all but en of the x's which do not belong to the same class
as Xj.
Observe that Theorem 3 does not contain Theorem A, though the
conclusion of Theorem 3 is stronger its assumption is also more stringent.
To prove Theorem 3 we need a lemma which is of independent interest.
Lemma. Let G'(n) be one of the extremal graphs belonging to G. Then every
vertex of G'(n) has valence greater than (1 + o(l)) n 1
{ r — 1
Assume that the lemma is not true and let y be a vertex oiG'(n) whose
1 — 1. It easily follows from Theorem A
r — 1 /
that for every k, if n > n0(k), G'(n) has k vertices xlt . . ., xl( each of which
is joined to ylt ..., ys, s = (l + o(l,))n 1
valence is less than (I — e)n
. The existence of these
r — 1,
vertices is clear since by Theorem A all but o(n) vertices of the first colour
in K(px, ..., Pr-i) are joined in our G'(n), to all but o(n) other vertices of
different colours. Delete now all the edges incident to y and replace them by
the edges (y, y()t i = 1, . . ., s. The new graph has more than Gx(n; f(n; G))
edges and clearly can not contain G as a subgraph since if it would
contain G and if k would be greater than n(G) then the subgraph
G'(xlt . . ., £ft, yx, . . ., ?/s) of G'(n) would also contain G as a subgraph,
which contradicts our assumption. This contradiction proves our lemma,
Not to complete the proof of Theorem 3 assume for the sake of simplicity
that r = 3 the case r > 3 can be settled similarly. Let
71
K2(Pi, P-z), 2>x + P2 = n, pt = (1+ o(l)) — , i = l,2
he the graph (7) and let xlt . . ,, xpv ylt . . ., ypi be the vertices of colour one
and two, respectively. By Theorem A all but o^2) of the edges (xh y^) occur
in our G'(n). By our Lemma we can further assume that the valence (in
237 EXTREMAL GRAPH THEORY

8(»
P. ERDOlS
G'(ti)) of all the x( and ?/; is |> (l + °(l)) - - and that each x is joined with at
least as many y's than x's and each y is joined with at least as many x's
than y's (for if say xx is joined to more x's than ?/'s wc put it amongst the
y's). Thus, each vertex is joined with at least (1 -4- o(\)) — vertices of the op-
4
posite colour.
Assume now that Theorem 3 is not true. Then we can assume that for
a fixed f > 0 and for every k if n > n0(k) there are vertices xv . . ., xk
n '
k > kn(f) each of which are joined to fewer than (1 — f)— y's. But then
hv our lemma each xh i = I, . . ., /t: is joined to at least — n x's, I now show
2
that this leads to a contradiction, since then our G'(n) will contain G as a
subgraph, in fact for large enough k > kn(e, I) it contains a Ks(t, t. t) which
of course contains our G if t ^> n(G).
Applying twice the lemma on p. 185 of [4] it easily follows that if k > k0(e, t)
there are t x's say xx,
x's and > rj n y's say xu ,
and more than rj n, rj = rj(e, &, /) other
^, ..., 2/,, .s- > rj n so that every x,-,
1, , . ., t is joined to every xun i = 1, . . ., <s' and to every y]t j = 1,
, ,. ,s-. By Theorem A all but o(s2) of the edges (xm, yj) occur in G'(n),
hence by the theorem of Kovaki and the Turans [5] there arc vertices say
xlh. . . .,-xui; yt, . . ., yt so that all the edges (xUi, yj), I <C i, j <^t occur in
G'(n) but then clearly G'(xly . . ., xt, xUl, . . ., xUl, yi, . . ., yt) contains a
K..(/, t, t). This contradiction completes the proof of Theorem 3.
Theorem \ follows easily from Theorem 3. 1 ^ct G>_2 be an extremal graph
of a vertices with respect to (G : Kr_.z(t. . . ,, /)). To prove Theorem \ wc
onlv have to show
(8) v(G'r_.,) <
(l+o(l))(r-l)/
; G\ +cn.
We now use Theorem 3. Let xv . . ., xt, I < cB be the exceptional vertices
of 6>_2 whose existence is permitted by Theorem 3. The other n — I
vertices of G>_2 can by Theorem 3 be partitioned into r — I classes each of
which has p,- = (1-)- o(l)) vertices and each of these vertices is joined
r — 1
to all but en vertices which belong to different classes. The graphs spanned
by the Pi vertices of the i-th class can not contain G as a subgraph, for if
this statement would be false let yv . . ., ym m = n(G) be the vertices of the
i'-th class which span a graph containing G as a subgraph. By what has been
just said the yu i = 1, . . ., m are joined to all but en vertices of the other
classes, and since each of these vertices are again joined to all but en
vertices of the other classes we obtain by a simple but not quite short
argument that for n > na (r, t, I) our G>_2 contains a (67 : Kr_2 (t, . .., t)) which
contradicts our assumption.
238 GRAPH THEORY

NEW INEQUALITIES CONCEKNING EXTREMAL PROPERTIES
!1
Thus, the number of edges which join two vertices belonging to the same
class is less than
G l
— 1
(9) ^-/b/;«)< (i +o(i)) (»■-i)/
In (9) wc used that if u1 = (l + o(l)W2 then
(10) f(Ul; O) = (1+ o(l)) f(u2;G),
the proof of (10) is easy and can be left to the reader.
The number of edges which join vertices belonging to different classes is
clearly not greater than
(11)
2? Pi Pj <>
1 < i <j <^n
n1-
r — 1^
2 (r- 1)2
The number of edges incident to the I < cE exceptional vertices is clearly
less than cen, hence (9) and (11) imply (8), which proves Theorem 1.
REFERENCES
|1] Khdos, P.: Extremal problems in graph theory, Theory of graphs and its
Applications-, Proceedings of the symposium held at Smolenice in June 1963
29 — 36.
[2J JObdOs, P. and Simonovits, M.: A limit theorem in graph theory, Sludia Math.
Sci. Hui,rjar. 1 (1966) 51—57.
[3] Ebdos, P.: Some recent, results on extremal problems in graph theory, Actes des
journees d'eludes sur la theorie des graphes, I. C. C. Dunod, 1967. 117 —130.
|4] Erdos, P.: On extremal problems of graphs and generalized graphs, Israel J.
Math. 2 (1964) 183 — 190.
[5] Kovabi, T., S6s, V. T. and Turan, P.: On a problem of K, Zarankiewicz, Goll.
Math. 3 (1954) 50 — 57.
239 EXTREMAL GRAPH THEORY

AN EXTREMAL PROBLEM IN GRAPH THEORY
P. ERDOS and L. MOSER
To Bernhard Hermann Neumann on his 60th birthday
(Received 6 January 1969)
Communicated by G. B. Preston
G(n; I) will denote a graph of n vertices and I edges. Let f0(n, ft) be
the smallest integer such that there is a G(n; /0(w, ft)) in which for every set
of ft vertices there is a vertex joined to each of these. Thus for example
/0(3, 2) = 3 since in a triangle each pair of vertices is joined to a third.
It can readily be checked that /0(4, 2) = 5 (the extremal graph consists of
a complete 4-gon with one edge removed). In general we will prove: Let
n > ft, and
(1) f(n,k) = (k-l)n-
/k\ Vn—k
+ i;
then f0(n, ft) = f(n, ft).
It will be convenient to say that the vertices xlt . . ., xk of G are visible
from xk+1, if all the edges (xit xk+1), i — 1, • • •, ft occur in G. A graph is said
to have property Pk if every set of ft of its vertices is visible from another
vertex. Gn will denote a graph of n vertices (the number of edges being
unspecified) and G(m) denotes a graph having m edges. Let GJt0) = (Gn;
/(w; ft)) be defined as follows: the vertices of G{°} are x1,---,xn. The
vertices xu i = 1, • • •, &—1 are joined to every other vertex and our Gjt0)
has [w — ft-4-2/2] further edges which are as disjoint as possible. In other
words if n—ft+1 is even G^0) has the further edges (xk+2j, xk+2j+1), j = 0, • • -,
O —ft —1/2], if n—ft+1 is odd the edges are (xk,xk+1), [xk,xk+i), (xk+i+1,
xk+i+2)> /=1>---. [w—ft—2/2]. It is easy to see that G^] has property Pk.
Now we prove
Theorem 1. A graph G(n;f(n, ft)) has property Pk if and only if it is
our graph G^K
Theorem 1 is vacuous for n :£ ft and it is trivial for n = ft+ 1, thus we
can assume n 5: ft+2. Clearly Theorem 1 implies (1). To see this it suffices
to observe that if a G(n; f(n, ft) —1) would have property Pk we could add
to it a new edge so that the resulting G(n; f(n, ft)) would not be a G^K
42
240 GRAPH THEORY

[2]
An extremal problem in graph theory
43
Since G'°] has property Pk we only have to prove that a G(n; f(n, k))
has property Pk then it must be our G<,0). Before we give the somewhat
complicated proof we outline a simple proof of (1) for k = 2.
Lemma. Let Gn have property Pk then every pair of its vertices is visible
from at least k—1 vertices.
Assume that the Lemma is false. Then say x1 and x2 are visible
from only ylt . . .,yu I f^ k — 2. But then the set of 1+2 :£, k vertices
xi> x2< Vi< ■ ■ •. Ih would not be visible from any vertex of G„, which
contradicts our assumption.
Let now xt, i -- I, . . ., n be the vertices of Gn and assume that t, is
the valency of x{ (i.e. x{ is joined to v{ vertices of G). Our Lemma implies
since the number of pairs of vertices visible from xt is \"S\.
From (2) it is easy to deduce (1) for k = 2. To see this observe that the
number of edges of a graph is \ 2XiV
BY (2) 2Li (¾) - (2) and thus by a simPle argument \ ^Li vt will be
at least as large as in the case that one v{ say vl is as large as possible i.e.
vx = n—1, and v2, . . ,, vn are as small as is consistent with (2). Now it is
easy to see that P2 implies vt >. 2 for all i. Hence
(3) ££^£(n-l + 2(n-l))=f(n-l)
which agrees with (1) for k = 2 if n is odd. If n is even a similar but
somewhat more complicated argument proves (1).
It does not seem easy to deduce (1) from (2) for k > 2. One could
easily obtain
/(n,A)= {k-\)n+0{\)
but a more precise estimation seems difficult. Hence to prove (1) and
Theorem 1 we shall use a different method.
We say that G{m) has property 0t if it contains a set S of 1 vertices
xlt . . ., xt each of which is joined to some vertex of G(m) not in S. G is the
complementary graph of G i.e. two vertices are joined in 0 if and only if
they are not joined in G.
Put n = k+t—1. Then
Now a simple argument shows that the fact that G(n; f(n, k)) does not have
241 EXTREMAL GRAPH THEORY

44
P. Erdos and L. Moser
[3]
property Pk is equivalent to G(n; f(n,k)) ~ G\yA — [(t-\-\)l2]\ having
property 9t_j. Thus Theorem 1 is equivalent to the following
Theorem 2. Every Gul) — [(2+1)/2]) has property 6t_1 except if it
is a (5<°>.
Clearly our (%°> is a G{t,, (£)-[(*+1)/2]) where the missing [(2+1)/2]
edges are as disjoint as possible.
Theorem 2 is vacuous for t < 2 and trivial for 2 <: 3. Henceforth
assumed 2^ 4.
To prove Theorem 2 let GUfy — [(2-1)/2]) = G be any graphs which
does not have property 9t_j. We will show that it must be a 0^\ First of
all we can assume that all vertices of our G have valency ^ t—2. For if not
then say xt is joined to ylt . . ., yt_t which shows that G has property 0t_t
which contradicts our assumption.
Assume next that G has a vertex x of valency t — 2 (this will be the
critical case). Denote by ylt . . ., yt_2 the vertices joined to x and let zlt . . .
be the other vertices of G. Clearly no two z's can be joined. For if (z1( z2)
would be an edge of G then zlt ylt . . ., yt^2 are t—\ vertices each of them
are joined to a vertex not in the set, or G has property 0t_1. Also no y
can be joined to two z's. For if y1 is joined to Zj and z2 then the t—\ vertices
2i> 22< 2/2. • • •, Ut-2 would show that G has property 0t_1.
Next we show that at least t— 3 j/'s are joined to some z (as we know
each y can be joined to at most one z). Assume that u y's are joined to
some z(u < t— 3). Clearly (v(G) denotes the number of edges of G)
(4)
HG) - I -
"2+1-
C;')
-N or m— iV =
■1,
where N is the number of the edges of the complete graph spanned by
V\> ■ ■ ■• 2/(-2 which do not occur in G. Now clearly
"M+r
(5)
N >
since a y joined to a z cannot be joined to all the other y's (since otherwise
Its valency would be 2—1), hence a missing edge (i.e. an edge not in G)
is incident to every y which is joined to a z and this proves (5). From (4)
and (5) we have
(6)
(6) clearly implies u ^ 2—3 as stated.
Hence either u = 2 — 3 or u ~ 2 — 2. (4) and u fg 2—2 implies that we
must have equality in (5) i.e. iV = [(m+1)/2].
242 GRAPH THEORY

[4]
An extremal problem in graph theory
45
First we prove Theorem 2 if u = 2 — 3. (6) implies that if u = 2— 3, 2 is
odd and since N = [(m+1)/2] + [m/2] = [(2-2)/2] and every i/ which is
joined toaz must be adjacent to a missing edge we obtain that the [w/2]
missing edges must be isolated. In other words we can assume that our G
contains all the edges of the complete graph spanned, by x, ylt . . ., yt_2
with the exception of the edges (y2i, yii+1), i =1,..., [(2-2)/2]. Further
every yi,i = 2,...,t—2 is joined to exactly one z. If all these z's coincide
then G is spanned by x, ylt . . ., yt^2, z ar)d is clearly our (¾°) and Theorem 2
is proved in this case.
To complete our proof of the case u = 2 — 3 assume that yt is joined
to zt and y} to zs, (zi ^6 z}), 2 <, i < j ^ 2 — 2. But then the 2—1 vertices
x, zit zjt {«/(} 1 5S I 1=k t — 2, I ^ i, I ^ / show that our G has property 9t_j
(a- and z,- are joined to yt, zs is joined to y, and every other yxl ^= i, I =£ j
is joined to yt or ys [since the missing edges were isolated]). This
contradiction completes the proof of Theorem 2 if u ~ 2 — 3.
Assume next u = 2— 2. Then each ?/ is incident to at least one missing
edge and since the number of missing edges is [(m+1)/2] = [(2-1)/2] we
obtain that for even 2 there are (2-2)/2 isolated missing edges. Just as in
the case u = 2 — 3 we see that all the 2—2 y's must be joined to the same z.
But then we again obtain our Gj,0'. This disposes of the case u = 2 — 2,
2 even.
Assume next u — 2 — 2, 2 odd. These are [(2—1)/2] missing edges and
since each y is incident to one of them we can assume without loss of
generality that the missing edges are (y1,y2)l (ylt y3), (y2i, y^i+i), [ = 2. •••.
[(2-2)/2]. If all the y's are joined to the same z we again get our G^K Thus
we can assume that not all the y's are joined to the same z. Now to complete
our proof we have to distinguisn two cases. Assume first that there is a z
say zi which is joined to only one y say yi. This case can immediately be
disposed of since the set of 2—1 vertices x, zit {y^}, 1 :¾ I tst t— 2, I =£ i
shows that our G has property 0t_1 (x and zi are joined to y{ and all other
y's are by our assumption joined to a z different from z{). This contradiction
proves Theorem 2 in this case.
Assume finally that every z is joined to more than one y and there
are at least two z's. Let, say, zt be joined to yt and ys and z2 to yr. Observe
now that either every y is joined in G to one of the two vertices yt and yr
or every y is joined to one of the two vertices y^ and yr (this follows from
the fact that the missing edges are either isolated or have at most one vertex
of valency two). Assume thus that every y is joined either to yt or to yr.
But then the set of 2—1 vertices x, zx, z2, {y^\, 1 ^ I ^ 2 — 2,1 ^ i,l ^ r
show that our G has property 04_j (x and zt are joined to yit y2 to yr and
every yx, I ^ i, I ^ r is joined either to yt or ?/r). This contradiction
completes the proof of Theorem 2 if G has a vertex of valency 2¾ 2 — 2.
243 EXTREMAL GRAPH THEORY

46
P. Erdos and L. Moser
[5]
Assume now that all vertices of G = GIL) —[(£+1)/2]) have valency
< t—2. We will show by induction with respect to t that then our G must
have property 0t_1 and this will complete the proof of Theorems 2 and 1
and also of (1).
Assume that the maximum valency of a vertex of our G is r < t— 2.
Let x be joined to yx, . . ., yr. Denote as before by zlt . . . the other vertices
of G and let u be the largest number of z's joined to a y. Assume that y1 is
joined to zx, . . ., zu. We evidently have
(V
u ^L min (i — r—1, r—1)-
To prove (7) observe that u ^ r would imply v(yx) > r and u S; t—r
would imply that G satisfies 0t_j (consider the vertices y2, . . ., yr, zlt...,zu).
Denote by ui the number of z's joined to yt (% = u) and by wi the
number of y's joined to yt. By (7) v(yt) = l + M,-+ze\ Ss r—1. Thus by (7)
the number E of edges incident to the vertices x,y1, . . ., yr equals
(8) E = r+ 2 ("i+k>>i)^r(u+l) +
r(r-
-1) r{r+u+l)
(8) follows from the fact that E is evidently maximal if all the ut are u= r—l
(i.e. they are all as large as possible) and if wi = r—u — 1 = 0. From (7)
we have (G3 is the graph spanned by the z's)
(9)
"Pi)
Q - 0
*+i-
Assume first r 5S ¢/2. Then we obtain from (9)
(10)
HGi) >
(7)
-*-r+r
Hence by our induction assumption G1 has property 9t_r_1 i.e. it
contains a set of vertices zx, . . ., zt_r_^ each of which is joined to some z},
j > t~r—1. But then the t—1 vertices zlt . . ., zt_r_1, ylt . . ., yr show that
G has property Bt_1, which proves Theorem 2 if r ^ t/2.
Assume next t\1 < r ^ t—3. Fiom (7) we have ut 5S <—r—1 and by (8)
E is maximal if all the ui are £—r-
by (8)
(11) £^r+r(;!-r-
From (11) we have
1+1
1 and w,
-1)+ -(2f-0
1
2r—;. But then
2"
•« > Q
rt
(7)
't—r + l-
244 GRAPH THEORY

[6]
An extremal problem in graph theory
47
Thus the proof can be completed as in the previous case, and the proof of
Theorem 2 is complete.
Denote by f0(n, k, r) the smallest integer for which there is a
Gin; f0(n, k, r)) in which every set of k vertices are visible from at least r
vertices. We say that a graph has property Pk r if every set of k of its
vertices is visible from at least r vertices. Just as in our Lemma we can show
that if Gn has property Pk then every pair of its vertices is visible froii at
least k-\-r— 2 vertices (our old property Pk is Pk-1).
Thus we obtain as in (2) that if Gn has property Pkr then if k > 1
From (2') we can deduce that if n < n0(k, r) then
(12) /0(n, k, r) = /0(n, k + r-1) = f{n, k+r-1).
(12) certainly does not hold for every n, k and r. It is easy to see that
/,,(10, 2, 6) = 40 but /(10, 7) = 41. Our Theorem 1 states that (12) always
holds for r = 1 and perhaps it always holds for r = 2 also if k > 1. For
& = 1 every Gn each vertex of which has valency S; r clearly has property
Plr, thus f0{n, \,r) = [(ra+l)/2], in other words if k = 1, r > 1 then (12)
is not true. We hope to return to these questions on another occasion.
Finally we can ask the following question: Denote by F(n, k) the
smallest integer for which there exists a directed graph G[n; F(n, k)) so that to
every k vertices x1,...,xk of our G theie is a vertex y of G so that all the
edges (y, xt) i ~ 1, . . ., k occur in G and are directed away from y. It is
easy to see that for n S; 3, F(n, 1) = n (for n ^ 2 there clearly is no
solution). It is not hard to show that for n S; 7, F(n, 2) = 3w and for n < 7
there is no solution. For k ^ 3, we only have crude inequalities for F(n, k).
We say that Gn has property Sk (after Schiitte who posed the problem) if
for every set of k nodes (xlt . . ., xk) there is at least one node y in Gn so
that all the edges (y, x{), i = 1, . . ., k occur in G and are directed away
from y. Denote by f(k) the smallest value of n for which an Sfc-graph of n
vertices exists. We have
(13) (£-1)2^+3 ^ f(k) < ck22k.
(13) is due to P. Erdos, E. Szekeres and G. Szekeres (Math. Gazette 47 p. 220
and 49 p. 290). We can show that for n > n0(k)
(14) /(A-l) " n g F(n, k) ^ f(k) ■ n.
University of New South Wales and
University of Hawaii
245 EXTREMAL GRAPH THEORY

COLLOQUIA MATHEMATICA SOCIETATIS jANOS BOLYAI
4. COMBINATORIAL THEORY AND ITS APPLICATIONS. BALATONFtTRED (HUNGARY), 1969.
Some extremal problems in graph theory
by
P. Erdos and M. Simonovits
Budapest, Hungary
We consider only graphs without loops and multiple edges. Gn
denotes a graph of n vertices, \/(G) , e(G) and ^CG) denote the number of
vertices, edges, and the chromatic number of the graph G respectively. The
star of a vertex u. will be denoted by st v. (that is the set of vertices joined to
x.), the valency of x. will be denoted by cC*.). Ktm,n) denotes the complete
bichromatic graph with m and n vertices in its classes. {KCm,n)-r} is the
graph obtained from K( m,n) omitting r ( r < min ( m,rO) independent edges.
Thus { K(4,4)- 4} = G is the graph formed by the vertices and edges of a cube.
Let us denote by •fCn;L1,...>L>,) the maximum number of edges
a graph G can have if it does not contain any L; as a subgraph.
If it does not cause any confusion, f C n •, L^,..., Lx) will be
abbreviated by fCnl.
According to [1]
(1) f(n; K(«,m)) = OmCn *) U<m)
- 377 -
246 GRAPH THEORY

This result is sharp if I = 2,3 [l] [2]
(2) fCn, K(2,2» = (Uon))^-
iln5/s < f(n; K(3,0)l c'3e n 3
(3) c
Hereinafter c, c^, C[: , ... will denote positive absolute constants,
and if c^ is used in different formulas, it can have different values.)
TURAN asked for the determination or estimation of f ( n •, C) .
A very special case of a result of ERDOS gives [3]
fCn;C) <cn5/3
Further ERDOS showed that [4]
C1n3/2<f(n;{C}-U})<c2n3/2
where {C]-{x} is the graph, obtained by omitting a vertex from the cube. We
shall prove a stronger assertion, namely
(4) Cjn3/M(n-, { C- 1 ) ) < c^2
where {C-1; is the graph, obtained by omitting an edge from the cube. ERDOS
5/,
conjectured that n ,3 is also the lower bound for C, but this conjecture is
false. In fact,
(5) fCn;C) « 0(n8/s)
ERDOS conjectured [5] that for any graph L , if xCL) = 2 ,
then there exists an <* =<x(L) such that
(6) Urn f Cn; L)/ n*
- 378
247 EXTREMAL GRAPH THEORY

exists, and perhaps
(7) o( = 1 + 1 or * = 2 _ -L
k K
would also hold in all cases (where k is an integer). This conjecture is
disproved by the following results of this paper:
Let D(k,-C) be the graph, obtained by joining two given vertices x
and y by k independent paths of length i and let E(t,k,t) be the graph
obtained by joining each vertex of the first colour of a K(t,t) to each one of
the first colour in D(k,£) and joining the vertices of the second colour of
K(t,t) to the vertices of the second colour in DCk,-t). (D(k,-t) can be coloured
by 2 colours in exactly one way, thus E(t,k,l) is well determined). Then
2k + 2t 2
2 2-
(8) C|Citn 3k^ *2tck+,)-1 <f(n;ECt,k,3)") < C'kjtn
2-1*3
where the exponent in the lower estimation tends to the exponent in the upper
one, therefore (7) does not always hold.
Besides, we shall also obtain that
(1 ±+—\
(9) f(n> ECt.a.t)") = Oln ^*-0 + t )
Further, we obtain that
(10) f(n;{K(r,r)-3D = 0 [n 2r"} J
which is a generalization of (5). (10) is trivial from {KO,r)-3} = E(r-3,2,3)
and from (9),
All these results are obtained from two main theorems of the paper.
In order to formulate Theorem 1, we need
DEFINITION 1. Gn is d-regular, if
- 379 -
248 GRAPH THEORY

d min crtx.) > mcu o"( x.) (d'O
xeG" xeG"
^r+1
THEOREM 1. If eCGn) > nU<* and d =10.2062 then G(rn
contains a d-regular subgraph G such that
e(.o ; 2 — m
«1"«
and man ^01 unless n is too small.
COROLLARY. If f^( n) denotes the maximum number of edges a d-
gular graph can have if it does not contain any L|_,
re;
and if
fdCn) = 0(n1+")
then
f(rv, Lr.-,LA) = 0( nU")
THEOREM 2. Let LCt) be a graph obtained from L where XfL> = 2.
by joining each vertex of its i-th class to each vertex of the i-th class of
KCt.t) U = 1,2).
If
(10) f (n-, L) = OCn1-*) CoteCO.O)
and
then
(12) f(n;LCtfl = 0(n2"P)
The latter theorem is a recursive one. It can be used for many
estimations if x^L) = 2. .
380 -
249 EXTREMAL GRAPH THEORY

PROOFS.
PROOF of THEOREM 1. Let A be a large number and let us divide
the vertices of Gn into 2A classes of equal size. (Hereinafter no difference will
be made between x and [x] . This is allowed here.) The i-th class will be
denoted by C[ and we may suppose that xeC, , y ^ C, implies that <5(x) 2 ff(y)
There are two cases.
a) If C1 represents less than -i- n1+<* edges, we consider G™1 =
= Gn- C, . eCGm<) 2 -y n1+a . If G™1 contains a vertex x, of valence
s An", we omit it, Gm2 = G^ - {*,} .
If G l contains a vertex jc2 of valence * — nw we omit it, and
mr
so on. At last we cannot omit any it.- from G J and since we omitted less
than ^-n1+<* edges, eCGmO >-|- n<+<* . Further
2e(G) £ vCG)- max trU)
xeG
implies that
(13) vCGmJ) = m.-> A-n
J 5 A
Here we used that
(14) max SCO "£ A n*
X6G J
but (14) follows from the fact that G1 contains a vertex x0 of valency
and if y 4 G,,
<y(y)r£G-(x0) = An0'
381
250 GRAPH THEORY

1 Ot
Therefore, the maximal valence in G is less than An and the
minimal one is greater than — n . Thus G is 10 A -regular and
eCG J) 2 4mi > mi2 -5T"n
5 J J 5A
In this case the theorem is proved.
b) If G represents more than — n edges, we consider the
graphs G' Cj = 2,...,2A) spanned by C-uC( . Since
.e(G:) >-ln1+"
•I 2
there exists a j0 such that
e((V2-2A'Tn =TA"'n
m1
Let us put G; = G and apply to it this splitting method again.
0 m
Either we obtain a 10 A -regular graph by a) or a G 2 by p) from it. In the
m
latter case we apply the method to G z , and so on.
m:
We prove here that the iteration gives a G ■' at last, which
posseses the properties, described in Theorem 1. Clearly
eCG ) 2 n
(4A)k
and
mk« Jjj-.n
2
Since v (G) > e(G) for every G
, , A ,-k 1+oi 2 A_zk
(4-A) . n -S n • A
382 -
251 EXTREMAL GRAPH THEORY

Thus
(A)" * nH-*
and consequently
k £ (1-o() logn/log -7-
On the other hand
Log m k « Log —r- = Log n - k Log A .
Ak
Therefore
Logm, > 0-CI-oO ^ )
(15) Log mk > ^.1-(1-0() ^-J Log n .
Lo9T
Therefore, if c<'< <*. and A is large enough,
(16) mk > 1-/
This shows that the procedure stops at last, and the obtained graph
G will have at least n* vertices, if A is large enough. Clearly G is
10A -regular and
e(G> ; > -L m
5
«"2+1
E.g. if A = 2 , then a short computation shows that
1-«
(17)
Here d=l0'2 , and this completes our proof.
„« +1
PROOF of the COROLLARY. There exists ac(>1 such that
(18) f ,( n) ^n1"1""
- 383
252 GRAPH THEORY

Let now G" be arbitrary, but having at least 3c,n1+<,( edges.
According to Theorem 1, it contains a d-regular Gm such that e(Gm)>c1m1+0(
and m>n . According to (18) Gm contains at least one L^, therefore Gn
also contains an L^ . Thus
fCn) = 0(n<+ot) .
PROOF of THEOREM 2. We use induction on t . Let L, = L(t)
and L-2 = L^-l) . Then L2 = L(t+l). Therefore the induction is trivial, if
we know the theorem for t = 1 .
A triple (x,y,z.) will be called a cap, if x,y are joined to z .
First we estimate the number of caps in G by eCGn) then the number of
K(2,2) and finally the number of K(2,2) containing a fixed edge. This will
give the recursion.
Let G be a graph of E edges, E > 100O n and let it be
d -regular, where d is a large integer fixed later.The valency of X' is vCxi),
therefore the number of caps is
r trCxps
2. n 2^- ^Xi; - 2 ^ n ^> - IT
for sufficiently large n. Here we applied the Cauchy inequality.
Let us denote by v(Jo,y) the number of caps (v.,y,z) and by Mk
the number of K(2,2) in G". Clearly
384
253 EXTREMAL GRAPH THEORY

'Zv(x,y)4(i-Nt]>ii
since Mc^ 106 • nZ . Thus
Therefore, there exists an edge (x,y") such that (*.,y) is contained
in at least
3 n*
K(2,2) -s. Since the graph is d-regular, cc*), Q-(y) 5. and the even
graph, obtained by considering only the edges joining st jc to st y , has at least
± li
3 n +
edges. If Gn does not contain LH), then this graph does not contain L, thus
^ e» <rf 4dE .L^cf frdE ^-^
3 n* U n 'L> - °l n j
Let
(20)
—2
and a = (0-2^ +1 . Then
c(4d) - E • n > 3 ■ E . n .
- 385 -
N, =
, j. («<■■»').
(19)
"* (J)
= 1
254 GRAPH THEORY

Thus we obtain
(21) c(n > E
where C, is a constant, depending on C, d and ot . (20) and (21) imply that
E s c2n r
i.e.
fdCnj LCO") = 0(n2~h
Applying Theorem 1 we obtain
f ( n; LCO) = OCn2^)
which was to be proved
APPLICATIONS.
1) Let us denote by D(k,t) the graph obtained by joining two given
vertices x. and y by k independent paths of length I.
An unpublished result of Erd6s states that
1 + T
f(rr,D(!U)) = 0( n l)
Let us denote by E(t,k,i) the graph, obtained by joining t
vertices of the first class of a K(t,t) to the vertices of D(k,i) having the same
colour, the other vertices of K(t,t) to the other vertices of D,(k,£) . (I.e. ,
if L=D(k,-t) then E(t,k,l) = L(t)). If L is connected and bichromatic,
L(t) is uniquely determined, thus E(t,k,-t) is also uniquely determined.
According to Theorem 2
f Cm E(t,2,D) = OCn *C*-0 + l )
- 386 -
255 EXTREMAL GRAPH THEORY

On the other hand, the method used in [7] gives that
(23) f(n; L) > n e'1
where
v = v(L), e = eCL)
Therefore
2-2 n
f(*.: E(t,2,£)) > C, . .n t2 + 2U+2t-1
1 1
If t—»oo, t is fixed, the exponents tend to 1- -r and 2 -
t t +-1
respectively. In this sense the estimations are not too bad.
2) It can be proved that
fCn; D(k,3)) =0(nV3)
Therefore
2- *
fCn; E(t,k,3j) = OCn n + 3) .
(23) gives
2k + 2t
2
(24) f(n; E(t,k,3^ > ct k 3 n 3k+tz + 2t (k + 1)-1 ,
2
If t is fixed, k—»oo, the exponent in (24) tends to 2- . .This
shows that the estimation is not too bad and at the same time it disproves the
conjecture of Erd6s, mentioned in the introduction. If k= 2 we obtain
estimations for f(n;C), where C is the cube, moreover, for -f-(n;{K(4,4)-3})
(see in the introduction) but (24) is not better than the trivial n 'z obtained
from (2). (Clearly, if L, c L2 , f ( n; L1 ) « f (n; Lz) and here C o K(2,2) .)
- 387 -
256 GRAPH THEORY

3) Though it is not a new result, it is interesting to see that (1) is
an immediate consequence of Theorem 2. Indeed, if L = K(1,1 ) and t = r-\,
then LCt) = K(r,r) thus
= r-1
cxr CX.]
defines an «r such that
f(rr, KCr.r)) = O ( n2_0<r)
Since 0(.,=1, = r and this gives (1). Since (1) is sharp if
r = \, 2, 3 , thus Theorem 2 is the best possible in a certain sense.
4) Let T be any tree, then f(n ,T) = 0(n). Therefore
fCn; T(t)) = 0(n t + 1 ) .
If e.g. T is a path of length 5, then f(n;T(1J) = 0(rT2)
But TC1) = {C-1} and this proves (4).
Here we stop our investigations, though Theorem 2 has many
further applications.
(The reader can easily check that all the results, stated in the
introduction, were really proved.)
OPEN PROBLEMS
By the method of random graphs we can show that for every d and
■3 1
t there is Gn, e(Gn) = [ n ] , which does not have a d-regular subgraph
G such that e(G ) > eV~n" m .
Many open problems remain, we just state two of them: Is it true
:ry £ and ex if n > n
contains a d-regular subgraph
that for every £ and ex if n>n0(t,oO and d>d-(t,cx) every G e(G ) > n +<*
- 388
257 EXTREMAL GRAPH THEORY

G, m > n , e(G ) > tm L.
Is it true that every Gn, e(G ) = [nlogn] contains a d-regular
subgraph G , e(Gm) > £ m log m where m tends to infinity together with n2.
It would be interesting to determine the correct order of magnitude
Of f(n,C).
389
258 GRAPH THEORY

REFERENCES
[l] W. G. BROWN, On graphs that do not contain a Thomsen graph, Canad.
Math. Bull. 9 (1966), 281-285.
[2] P. ERDOS, A. RENYI and V.T. SOS, On a problem of graph theory,
Studia Sci. Math. Hung. 1 (1966), 215-235.
[3] P. ERDOS, On an extremal problem in graph theory, Coll. Math. 13
(1965), 251-254.
[4] P. ERDOS, On some extremal problems in graph theory, Israel J. of
Math. 3 (1965), 113-116.
L5J P. ERDOS, Some recent results on extremal problems in graph theory,
Theory of graphs, International Symposium. Rome 1966, Dunod
1967.
[6] T. KOVARI, V.T. SOS and P. TURAN, On a problem of K. Zarankiewicz,
Coll. Math. 3 (1954), 50-57.
[7 J ERDOS, P.: Graph theory and probability, II. Canadian J. of Math. , 13
(1961) 346-352.
390
259 EXTREMAL GRAPH THEORY

COLLOQUIA MATHEMATICA SOCIETATIS jANOS BOLYAI
4. COMBINATORIAL THEORY AND ITS APPLICATIONS, BALATONFORED (HUNGARY), 1969.
Some remarks on Ramsey's and Turan's
theorem
by
P. Erdos and Vera T. Sos
Budapest, Hungary
l.In this paper we are going to discuss some special cases of a
general problem which might be considered as being on the one hand a
generalisation of the problem raised and solved by the well-known theorem of
Turan, on the other hand as the well known problem of the Ramsey-numbers.
Before going to explain this in details, we give the notations we
shall use:
G(n) is a graph with n vertices
G( n ;e) is a graph with n vertices and e edges
eCG) denotes the number of edges of G
G is the complementary graph of G
K(\>) is the complete graph with v vertices
HCn; k,t) is the class of G(n) graphs, where G(n) contains no
k(k) and G(n) contains no KU)
H(rv, k) is the class of G(rO graphs, where G(n) contains no K(k)
- 395 -
260 GRAPH THEORY

fCn-, k.O d=ef
max e(Q) if H (n; k,t ) ± 0
G € HCn; k,«)
0 if HCn-, k.-C) = ¢)
f(n;k) ^f may e(G)
G 6 H C n •, k )
G(x...,..., *k1 denotes the subgraph of G spanned by the vertices
x1 ,..., xk .
The well-known, special form of Ramsey's theorem [5] asserts
that for any k,-t there exists a N(k,t) such that if n > N(k,t) then
H(n-, k,t) = 0 .
The well-known theorem of Turan [6] gives the exact value of
f (n-, k) namely that
f(n;k) = 4r ^^ C n2 - r2) + (!j where n = rmod(k-1) 0 5r«:k--l.
1 2 k-1 L
The only "extreme graph" in H(n;k) with e = f(n-,k) is the
complete k-1 chromatic graph in each class having \——1 resp. ["—^—1+-)
vertices. It is worthy of note that for this graph G(n) contains a rather
"large" complete graph (with [ n 1 vertices).
Now the general problem is to determine f ( n -, k X ~) ■
In the special - extremal-case when -(,= n + 1 (i.e. if there is no
condition on the complementary graph), f(n;k,l) = fCn;k) is determined
by Turan' s theorem.
In the other special case, when k and I are fixed and n is large
enough, f(n;k,0 =0 by Ramsey' s theorem. The exact determination of
f(n,k,l) is probably hopeless, since this would imply the determination of
the Ramsey-numbers. But one might expect - having in mind the remark in
connection with Turan' s theorem, - that f ( n-, k.O is essentially smaller,
than {( n; k) when I is supposed to be much smaller than [ n ] .
396 -
261 EXTREMAL GRAPH THEORY

It is easy to show that for every c < 1
(,1) fCn-.k.c-j^j-) < gCO±.
\ k-2 n2
k -1
with g(c)< 1 , but we cannot determine the exact value of g(c). We do not
prove (1) in this paper, but hope to return to it, and to other related questions,
at another occasion.
2.In this paper we first investigate the case when k is fixed and
I = o(n).
TriVally f(nj 3,1)½ ^- since if G contains no triangle and has
a vertex of valency \i, the \i vertices joined to this vertex must be independent.
Therefore { (n ; 3,t) = oCn1) if -t = o(n).
For the general case we prove
THEOREM 1. If I = o(n) then
(4) f (n-, 2r+1,0 = i. (1 --L") n2 C 1 + oCO) .
REMARK:
We cannot settle the case k = 4. Perhaps
(3) f(ns4,0 = o(nz)
if £ = o(n). We only get crude upper bounds for f(n-, 4,-t)
If (3) holds, we can deduce for each fixed r and •{. = o(n)
(4) f(n;2r+2,i) = 1 ( i - -1 ) nZ ( 1 4- o ( O) .
2 r
Now we prove Theorem 1. First we prove it if r = 2, i.e. we prove
that if I = o(n) then
"in some cases -f(n;3,t) = ^- . See [l] , [2].
jce l
397
262 GRAPH THEORY

(5) f (n; 5,1) = (1+ 0(1))^- .
First we show that for sufficiently large n
(6) f(n; 5, cnTlog2n ) > -~ .
It is well known [3j that there is a G(m) which contains no triangle
and for which G(m) contains no K([ cm'2 Log2m ]") .
LetG^t-^-]) and G2([ ni. ]) be two such graphs which do not
have a common vertex. Join every vertex of G1 to all the vertices of G2 .
The resulting graph clearly proves (6).
To complete the proof of (5) we have to show that if n > nc(£.)
and G ( n ; [-^-(1 ^ £-)]) does not contain a K(5) then G contains a K(lctn])
where c, depends only on e .
First we show the following
LEMMA. Let 0<a<j and G(n -, [c*nzO + ol) be any graph. Then
there is a subgraph G(m), m>c£ ^n each vertex of which has in G(m)
valency greater than 2otm (1 *■ -|-) .
Let us assume that our Lemma is false. Then we can write the
vertices in a sequence jc1 ,.., xn so that for every k < U-c)n the valency
of x, in G(x-k,..., x-n) is less than 2ot(n-k)(l + 4-) . But then
eCG(n)) - [«n2(H--f)] < 2ot(U-|-) Z.Cn-U) + ( C" ) <
which is an evident contradiction if c < Vex-£- .
Now we use the Lemma with ex * -7- . Let G(m">, m > en
4
263 EXTREMAL GRAPH THEORY

be a subgraph of our G (n ; [-^- (1 + e) J each vertex of which has valency
2 4
Let G(jt,, x2, x3) be a triangle of our G(m) (clearly every edge
of GCm) is contained in a triangle). Let y ,..., ym3 be the other vertices of
our GCm). Each vertex of G(m) has valency at least -^-(l + -|-), hence more
than^-m edges of type (j^,y-) A * I f. 3 , li.jsm-3 are in our G(m).
Thus more than ^- y^' s are joined to the same two x^' s say x
and x2. If these y-'s are independent we have found an K(Lcn]) in GCm).
If yr and y are joined, then GCx1, x2, yr, y ) is a K(4) in our
0(01). Henceforth we can thus assume that G(m) contains a KC4).
Let GCi,, z2,z3,zv) be a K(4) of our GCm) and co1 , .., c^m_^.
are the other vertices of it. At least 2 m (1 + ^0 + 0 C1) edges of the form
(z^.w.O belong to GCm) C1£ i £ 4, 1 ij-s m-4). Thus by a simple computation
there are at least -j-^r vertices co; which are joined to the same three z,' s.
These tj.-'s must be independent since otherwise GCm) contains a K(5) and
this completes the proof of (5).
Now we prove (2) for general r. First we show
(7) fCn; 2r+1,e) > 1-0 - -1) n2
The proof follows the proof of (6).
Let G- ; 1 £ i. ^ r be graphs of [—J vertices (with disjoint set of
vertices) which contains no triangle, and where G^ contains no
K([cn/2lo92n]) .
Join every vertex of G^ to every vertex of G; for every 1 £ I < j £ r .
The resulting graph proves (7).
To complete the proof of (2), assume that it holds for 2r-i and we
prove it for 2r+1 . Thus we have to prove that every
- 399
264 GRAPH THEORY

G(n.,[|0-± + £)n2])
either contains a K(2r+0 or G contains a K([cn}) where c depends only on
t and r. The proof will be very similar to that of (5). First of all, from our
Lemma we obtain that we can assume that our G(n) contains a subgraph 6(m)
with m > c. _ n each vertex of which has the valency > m (i - — +■ -|-)
Clearly for this GCm)
Hence by our induction hypothesis we can assume that our GCm)
contains a K(2r-0 whose vertices are x,,.-, i2 _i ■
Denote by y. ,..., y _2 +i t'ie other vertices of G(m). At least
(2r-l)0 --i- -l--|-) m + 0(1 ) > (2r-3)m+ ^-
edges of type (x^.y:), 1 - i- 2r-1, "lij i m-2r +1 belong to G(m).
Thus as in the proof of (5) we obtain that there are at least c, m
C C1 - C,Cr)") vertices of G(m) which are joined to the same 2r-2 jt-Js,
since all these vertices cannot be independent, two of them must be joined,
thus our GCm) contains a K(2r).
Let now 2.^,.,.,-z.^ be the vertices of this K(2r) and let
031,.-., um42 be the other vertices of G(m). At least
2r (1 - — - y) m + 0(1) = (2r-2)m + trm +0(1)
of the edges (z^coO, 1 s L i 2r, 1sj im-2r belongs to our G(rr0. Hence by
the same argument as used in the proof of (5) at least ct r m vertices w; are
joined to the same 2r-1 z.j_' s. If two of these z{ s are joined, GCm) contains
a K(2r+1), if no two of them are joined, GCm) contains a K([c£ r m])
and since m >c1n the proof of Theorem 1 is complete.
3.We remark that (6) is nearly best possible. In fact we prove
- 400 -
265 EXTREMAL GRAPH THEORY

(8) f(n; 5,[cn1/2]-) < ±-(1 + £.) n2
for every c and t if n>n(t,c).
Let G(n ; [4-^ + ^-) n2]) be any graph for which G does not
o
contain a K([cn'2]) . We will show that it must contain a K(5). First of all,
observe that by our Lemma it must contain a subgraph GCm), m>cfcn each
vertex of which has valency > -7-(1 + 4-) m and therefore
(9) eCGCm)) > ±(u-f-)m2.
Secondly observe that
V2 >
(10) f ( n;4,cn/z) = 0(n2) .
Namely if (10) would be false, there would exist a G(n ; [6 n2])
which contains no k(40 and G contains no K(Ccn 2]). G clearly contains a
vertex of valency [26nl i.e. G has a vertex x which is joined to y ,.,., y ,
S ^ [26n] .
By a result of Graver and Jackel [4] G(y ,---,y ) must either
contain a triangle or G(y,,...,y ) contains a K(Cc,n'2]). Both assumptions
clearly lead to a contradiction. Thus (10) is proved.
(9) and (10) clearly imply that G(m) contains a K(4) with vertices
( x.,, x2, x3,x4). Since each of the x-'s (1<i,£4) have valency >— (1-k-4-)m,
there clearly are ctm> c1£^n vertices y ,... ,y» Ct>c1£-m) which
are joined to the same two x^'s say to x. and x2. GCy.,..., y.) cannot
contain a kCLcfn]) thus by [4] G(y,,..,y») contains a triangle, say
G(y1,y2,y3) but then GCx^Xj, y1,y2,y3) is a K(5) of our G(n) , which
completes the proof of (8).
401
266 GRAPH THEORY

Perhaps
fCn; 5,[cnVn) = o(n2)
is true, but we could not prove it.
4.As to the case k -lr, we prove that assuming fCn-,4,-Q = o[nz)
for I = o(n) we have for every fixed r
(11) -f Cn ; 2r-k-2,t) = 1-(1--1) n2( 1 voCO)
For the sake of simplicity we only prove (11) for r = 2 .
The proof of the general case is the same, only slightly more
complicated.
2 n2
f(n;6,0 > — is trivial, (it follows from f(n;5,d)> -j-) ■
Thus to prove (11) for r =2 we only have to show that for every t>0 there is a
C >0 so that for every G(n ; [-^- (1 + E.V]) which contains no K(6) G contains
aK([c£n]) (we of course assume -f(n;4,t)= o(n2)).
From Lemma it follows that our GO) has a subgraph G(m) with
m > c,n so that every vertex of G(m) has in G(m) valency greater than
— (1 +—)m. Let x be any vertex of G(m), denote by Sex) the set of vertices
of G(m) joined to s...
We evidently have
(12) I S(x) n SCy) I > -^- •
Put
M = max |5COn S(y)|
where the maximum is taken over every two vertices x and y of GCm) which
are joined. By (12) we have M > —^- .
- 402 -
267 EXTREMAL GRAPH THEORY

Assume that for jc1 and x2 we have | S(jc.) n S(x.,)| = M
and let y.,.-,yM be the vertices of G(m) joined to both x.. and*.2. Our
assumption f(M-,4,-d) = o(M ) clearly implies
(13) eCGCz,,...,*M3) = o(M2) .
To see (13), observe that GC z1,..., zM") cannot contain a KC4)
thus if (13) would not hold, then G( z 1,..., zM) would contain a K(Lc£ml),
which is impossible.
From (13) it immediately follows that for all but o(m) = ofM)
vertices the valency (in GC z1,..., zM)) is oCM). Hence there is a subgraph
GCz,,,..., zN ) of G(z1,--,zM) with N = 0 + o(-0)M each vertex of which
(in GCz1,.., zN )) has valency o(N). Since N > ^- we can assume that the
vertices z,.,.-.,zN are not all independent, without loss of generality we can
assume that zfl and z2 are joined.
Now we prove
I SCz,) n S(z2)I > M
and this contradiction will prove our assertion.
Let y ,..., ys be the vertices of our G(m) different from
z^,...,zN. Clearly both z1 and z2 are joined to at least — (1+-y)m +o(m)
of the y;,' s. Thus we evidently have
|SCz1)nSCzz)|>rn(++-|-)-S + oCm) = M+-|-m-t-oCm).
This contradiction completes the proof of (11).
Incidentally it is easy to see that if f (n ; 4,1) 4 o(rf) then
i
Kn;6,i) > —-(1 + t) for infinitely many n and -fc = o(rO .
To see this let G1 and G2 both have n vertices, every vertex of G.
is joined to every vertex of G2, G1 contains no triangles, G2 no k(4), G2 has
more than en edges and both G1 and G>2 do not contain a k(t).
- 403 -
268 GRAPH THEORY

REFERENCES
[l] B. ANDRASFAI, Graphentheoretische Extremalprobleme, Acta Math.
Acad. Sci. Hungar. 15 (1964) 413-438.
[2] P. ERDOS, On the construction of certain graphs, J. of Combinatorial
Theory 1 (1966) 149-153.
[3] P. ERDOS, Graph theory and probability, II. Canad. J. of Math. 13 (1961)
346-352.
[4] J.E. GRAVER and J. JACKEL, Some graph theoretic results associated
with Ramsey' s theorem, J. Combinatorial Theory 4 (1968) 125-175.
[5] F.F. RAMSEY, On a problem of formal logic, Collected papers, 82-111,
see also P. Erd6s and G. Szekeres, A combinatorial problem in
geometry, Compositio Math. 2 (1935) 463-470.
[6] P. TURAN, Eine Extremalaufgabe aus der Graphentheorie (in Hungarian),
Mat. es Fiz. Lapok 48 (1941), 436-452.
- 404
269 EXTREMAL GRAPH THEORY

Chapter 6
Circuits
All of these papers could have been placed in Chapter 3, Extremal Graph
Theory. Graphs with a specified number of vertices and edges are discussed.
Questions considered include; What is the maximal length path (or circuit)
that must be found in such a graph? What is the maximal number of vertex
disjoint circuits that must be found in such a graph?
Papers [367], [440] are extended by J. W. Moon, On edge disjoint
cycles in a graph, Canad. Math. Bull., 7(1964)519-523.
In theorem 2 of [406], Bondy (University of W'aterloo) has shown
ci = 2 t0 be best possible. The conjecture on page 157, line 11, has been
proven by Bondy and Simonovits (University of Budapest).
Papers in Chapter 6
[301] (with T. Gallai) On the maximal paths and circuits of graphs
[367] (with L. Posa) On the maximal number of disjoint circuits of a graph
[392] (with G. Dirac) On the maximal number of independent circuits in a
graph
[406] On the structure of linear graphs
[440] (with L. Posa) On independent circuits contained in a graph
271 CIRCUITS

ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS
By
P. ERDOS (Budapest), corresponding member of the Academy,
and T. GALLAI (Budapest)
Introduction
In 1940 TurAn raised the following question: if the number of nodes,
n, of a graph1 is prescribed and if / is an integer ^ n, what is the number
of edges which the graph has to contain in order to ensure that it
necessarily contains a complete /-graph? TurAn gave a precise answer to this
question by determining the smallest number depending on n and /, with the
property that a graph with n nodes and with more edges than this number
necessarily contains a complete /-graph ([9], [10]). More generally, the
question can be posed, as was done by TurAn: given a graph with a
prescribed number of nodes, what is the minimum number of edges which
ensures that the graph necessarily contains a "sufficiently large" subgraph
of a certain prescribed type? An alternative formulation of this question is as
follows: the number of nodes being fixed, we seek the maximum value of //,
u being such that there exists a graph with p edges which does not contain
a subgraph of the type in question with more than a certain given number
of nodes. In our paper we are concerned with this problem for the case in
which the types of graphs considered are paths, circuits and independent
edges. (These terms are defined in § 1.)
Our results are not exhaustive, because, in general, we only give an
estimate of the extremal values, only in isolated cases — for certain special
values of the number of the nodes — do we succeed in determining the
extreme values and the "extreme" graphs completely. Here are some of our
results capable of simple formulation:
Every graph with n nodes and more than (n—1)//2 edges (/g2)
contains a circuit with more than I edges. The value (n —1)/2 is exact if and
only if n = q{l—1)+1, then there exists a graph having n nodes and
1 The graphs considered in this paper are all finite, every edge has two distinct end-
nodes, and any two nodes are joined by at most one edge.
1. c. letters always denote non-negative integers, n always denotes an integer ^2 1.
A complete l-graph is a graph with I nodes, every pair of distinct nodes joined by an edge.
A graph is said to contain its subgraphs. (See § 1 of this paper and [6], pp. 1—3.)
273 CIRCUITS

338
P. ERDOS AND T. QALLA1
(n —1)/,2 edges which contains no circuit with more than / edges. (Theorem
(2. 7).)
For all n ^ (k-\- l):i, 2, k^ 1, every graph with n nodes and more than
nk—k(kJr 1) 2 edges contains a path or a circuit with more than 2k edges.
The value nk—k(k+ 1) 2 is exact. (Theorem (3.6).)
For our proofs we need a group of theorems different from the above,
which are of interest in their own right. In these it is not the number of
edges, but the fact that every node has a sufficiently high degree (the degree
of a node is the number of edges incident with the node), which implies
the existence of a "sufficiently large" subgraph of a prescribed type. This
class of problems was first considered by Zarankiewicz [11] and Dirac [3].
From among these older results we require two theorems due to Dirac ([3],
Theorems 3 and 4), for which we give new simple proofs in § 1. (A simple
proof of Theorem 3 can be found in [8].) We also prove some new theorems
of the type just now discussed.
In § 1 we present the necessary preliminary notions and some lemmas,
and we prove the theorems pertaining to the Zarankiewicz—Dirac field of
problems. In § 2 we carry out the estimations connected with problems of
the TurAn type. In § 3 we determine two extremal values exactly for a
sufficiently large number of nodes. In § 4 we determine the maximum
number of edges in a graph of n nodes and at most k independent edges. We
distinguish our more important results from the less interesting assertions leading
to them by the appellation "Theorem".
§1
(1. 1) Let M = {Pu ..., P,,} be a finite non-empty set and let the set
of all unordered pairs of distinct elements P,Pj = P,P, (i=f=j) of M be
denoted by N. (If n =^=1, then A^ is the empty set.) The elements of M are
called nodes, the elements of A' are called edges, and the edge P,P; is said
to be incident with the nodes Pt and P,. Let Nt be an arbitrary subset of
N, M and A/, are said to define a graph F=(M, N{). The elements of M
and A\, respectively, are the nodes and edges of F. If P,P/f/V,, then P,
and P; are said to be joined (in F), or we say that the edge P;Pj exists (inF).
The graph F-=(M,N) is called complete, more exactly a complete
n-graph. The graph F=--(M,N—M) is the complement of the graph
F^iM.NJ.
Let A" denote the set of pairs of elements of the finite set M' and let
N[^N'. The graph F' = {M', N[) is called a subgraph of the graph
r=(M, A\) if fcjw and A/,'c/v,. We also say that F contains F' and
274 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS
339
that F' is in F. If F' is a subgraph of F, then the graph [F'] = (M', N'nNi)
is called the subgraph (of F) spanned by F' or by M' in F. If P is a node
of the graph F, then F—P denotes the graph obtained by deleting P and
all edges incident with P from F.
(1.2) If a (1 — 1) correspondence can be established between the nodes
of the graphs A and Fi so that nodes joined in one graph correspond to
nodes which are joined in the other and, conversely, then the two graphs
are regarded as identical, and this is expressed in symbols by Fl = F2-
The number of nodes and edges in the graph /' is denoted by ;r(r)
and v(F), respectively.
The number of edges incident with the node P in the graph F is called
the degree of P in F. If there is no room for misunderstanding, then we
speak of the degree of P for short, and denote it by o(P). If p(P) = 0, then
we call P an isolated node of F, if q(P) = 1, we call P a terminal node of F.
A graph L is called a loop, more accurately a P-loop, if a series'2
P,,..., P«, P,+i («£l) can be constructed from the nodes of L so that every
node of L appears in the series, P= Pi, the nodes Pi,...,P„ are all
distinct, Pll+l=f=P„, and if n>\, then P„+i=f=P„-i, and the set of edges of L
consists of PiPi+i (i= 1,..., rc). It is easy to see that we can form in at
most two ways from the nodes of a P-loop a sequence of the required
properties and that P„+i is uniquely determined. P is the initial node of the
loop and P„fI the final node. The loop is also said to start from P and to
lead to P,H\. We call the P-loop directed if one of the above-mentioned
sequences is made to correspond to it.
If P,+ i is different from Pu...,Pn, then the loop is called a path,
more accurately a P,Pi+,-path, if Pn+i=--Pi, then it is called a circuit.
Paths and circuits will be designated by the common term arc. If P,ii = P/
(lS/^n-2), then the nodes P}, Pj+1,..., Pn+1 and the edges PTpTi
(/=/, ...,n) together form a circuit which is called the circuit of the loop L.
The number of edges of L is called the length of L. Paths, circuits and arcs
of length / are called l-paths, l-gons and l-arcs, respectively. The equation
L = (Pu...,P„,P,,« = Pj) (l5E/=Sn-2)
states that the graph L is a Pi-loop wich is composed of the nodes P,,..., P„
and the edges PP+i (/=--1,..., rc). The equation
W = (Pl,...,Pn+l)
~ If j and g are natural numbers and j<g or j>g, then P., ..., P^ denoles the set
of nodes P, where / runs through the natural numbers from j to g. If j--g, Ihen P,, ...,P,
means Pj by itself.
275 CIRCUITS

340
P. ERD(3s AND T. QALLA1
states that W is a PiP,i+i-path composed of the nodes Pi,..., Pn+i and the
edges PiPi+1 (/= 1,..., n).
A set of edges of r is called independent if no two of them have a
node in common. We shall say that the maximum number of independent
edges is k, if there exists a set of k such edges and there does not exist
a set of A:+1 such edges.
(1.3) The graph F is connected if it consists of a single node or if
corresponding to any two distinct nodes P and Q F contains a PQ-path.
The "maximal" connected subgraphs of F are called its components. The
subgraph F' of the graph /"" is maximal with respect to some property if F
contains no subgraph with this property of which F' is a proper subgraph.
If F' is a component of F, then F—F' denotes the graph obtained from F
by deleting /"".
The nodes P,,...,P,- (/=^1) are said to separate the two (distinct)
nodes A and B in the connected graph r if Pi=f= A, Pt=f=B (/=1,...,/)
and every /46-path in F contains at least one of the nodes Pi,...,Py.
The nodes Pi, ..., Pj divide the connected graph F if F contains two nodes
which they separate.
The graph F is n-fold connected (n & 2) if it is connected and if no
set of fewer than n nodes divides it. A complete /-graph is said to be rc-fold
connected for all n.
The maximal twofold connected subgraphs of the connected graph F
are called the members of F. Every edge of F is an edge of some member
and every member, except for the graph consisting of a single node, contains
more than one node. If F is connected but not twofold connected, then it
has more than one member, and it may be verified ([6], pp. 224—231) that
two of its members have at most one node in common and that such a node
divides F.s Furthermore, it may easily be verified that F has at least two
members containing only one cut-node each. Such members are called
terminal members of F. If F' is a terminal member of r, then F—F' denotes
the graph obtained from F by deleting all edges of F' and all nodes of
F' except its cut-node.
(1.4) If in the graph F every node except the single node A has
degree ^k (k^2) and rr(F)^k2k, then, if ?(/l)^2, F is twofold connected
and if g(A) = \, then F—A is twofold connected and F is connected.
Proof. Suppose first that ()(/4)¾2. Then it follows from our
assumptions that every component of F and every terminal member of F (if any)
contains at least 3 nodes, and at least two of these have degree ^k. If F
:) Such a ziode is called a cut-node of the graph.
276 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS
341
is not twofold connected, then it has two components or two terminal
members. At least one of these components or teiminal members has not more
than k nodes. The nodes in it with at most one exception have degree <k,
which is a contradiction.
If ¢)(/1)=1, then let the edge incident with A be denoted by AN.
If ^3, then the degree of A in F— A is clearly at least 2. If k-=2, then
only .t(F) = 4 is possible, in which case the degree of A' in /"—A is 2.
In both cases the theorem with q(A)^2 can be applied to F—A. F is
obviously connected.
(1.5) A circuit of the graph F which contains all nodes of /' is called
a Hamiltonian line of F, H-line for short. A path of F which contains all
nodes of /' is called an open H-line of F. Two distinct nodes P and Q of
F are said to be H-independent in F if F contains no open //-line starting
in P and ending in Q.
Our later reasoning is based on the following lemma:
Lemma (1. 6) If the circuit C = (P,..., P», P„+,= Pi) is an H-line of
the graph F and if Pi and Pj (i,j=/=n+ 1) are H-independent nodes of F, then
9(Pi+i) + 9(Pj+i)^T(C) = n;
p(P) denotes the degree of the node P in F.
Proof. It may be assumed that /=-1 and \<j^n. Because two
neighbouring nodes of C are not //-independent, 3^j^gn — 1. Accordingly,
(1^4 and the nodes Pi,P>,P/, Pj.H are distinct.
If F contains the edge P>P,j (3^g^j), then it does not contain the
edge PjfiPy-i . For if this edge belonged to F, then the path
W = (P„ Pn, . . . , Pj,U Py-lt..., Pit P,j, ..., Pj)
■ would be an open //-line of F starting in Pr and ending in Pj.
It follows that PiPj+i does not exist in F, since P>Ps does.
If the edge ~p7p, (/ + 2^/^ n) exists in F, then the edge P^P^
cannot exist in F. For if this edge existed, then the path
W = (Pj, ..., Po, P, />■+,,/>,+,,..., P,+ I)
would be an open //-line of r starting in P,- and ending in Pn+I = Pi.
Accordingly, with every node, other than Pi, joined to P> in F there
can be associated a node not joined to P,+1 in such a way that these
associated nodes are all distinct. It follows that the number of nodes not joined
to PJ+l is at least e(P-i)—l, so that p(Pi+I) =£(/? —1)-(e(P2) —1) = n — «(P>).
277 CIRCUITS

342
P. ERD6S AND T. OALLAl
Note. It follows from Lemma (1.6) and its proof that, under the
hypotheses of the lemma, o(Pi-i) + p(Pj-i)§/! and the edges P,+1P;+i and P,^Phi
do not exist in F (Pti=P„).
Let the loop Z. = (P,, ...,Pll( P,1+i = Pw) (l^m^rc — 2) be a subgraph
of the graph F. A node P,- belonging to the circuit C = (P,„, ..., P„, PnH = P„)
of the loop L and different from the terminal node Pm of L is called H-node
of the loop L (with respect to /"") if F contains a P7P,H-path containing all
the nodes of C and no other nodes, or, otherwise expressed, if the graph [C]
spanned by C in F contains an open //-line starting in Pj and ending in P„.
(1.7) Let L = {Pl P„,P«+i = P,„) (l^m^n — 2) be a loop of
the graph F and let the circuit of L be denoted by C. If every H-node
of L(with respect to F) has degree ^k (k^2) in [C] and if :x(C)^2k—\,
then every node of C different from P„ is an H-node of L.
Proof. PmH and P„ are clearly //-nodes. Our theorem is established
if we prove the following assertion: If P/rI (m<J<n) is an //-node, then so
is Pj. To see that this is so, let it be assumed that P,tI (m<j<n) is an
//-node and P7 is not. Then P„ and P,- are //-independent in [C], and, since
C is an //-line of the graph [C], it follows from Lemma (1.6) that «'(P„iu) +
+ p'(P,ti)=-"t(C), where (/(P) denotes the degree of the node P, in [C].
But it was assumed that p'(PiH)si and (?'(Pm+i)=^k. We have a
contradiction !
Let A denote a node of degree ^ 1 of the graph F and let the degree
of all nodes of F other than A be u2. If U/-=(P,..., P„) (P, = A) is
any path of F which starts from A, then, the degree of P„ beings2, F
contains an edge P„P,,,i incident with P„ and different from P„.P„-i. This
edge and W together form an /4-loop which is longer than W. Thus to
every path W starting in A there exists an /4-loop longer than W, hence the
longest .4-loops of F are not path (i. e. they contain circuits).
These longest A-loops of F which possess the longest circuits will be
called maximal A-loops.
(1.8) Let A be a node of the graph F with degree &1 and suppose
that the degree of every node of F other than A is & k where k=i2.
Further, let L be a maximal A-loop and let the terminal node of L be
denoted by B and its circuit by C. Then if :r(C)^2k — \, it follows that [C\ is
either a component or a terminal member of F and in the latter case the cut-
node of [C] is B. In both cases every node of [C] distinct from B is
connected to B by an open H-line of [C] and :t(C)&A:+ 1.
Proof. It follows from the assumptions made that P contains a
maximal /4-loop. Let L = (Plt..., P,,P„+i = P„) (\§m^n — 2,P1 = A,Pm = B)
278 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS
343
be the maximal /4-loop concerned. If P,- (m<j^n) is any //-node of L,
then Pj is not joined by an edge in F to any node which is not in C. For
if PjP is an edge of r and if W denotes an open //-line of [C] leading
from Pj to P,„, then if we add the edge P3P and the path (Pu...,Pm) to
W the result is an /4-loop which is as long as L and which is a path if
P=j=Pi (z= 1,..., n) and which has a longer circuit than C if P=P,j
(1 ^g<m). A contradiction with the maximal nature of Z, is therefore avoided
only if P = P,j (ffl^s^n). It follows that the degree of every //-node of
L in [C] is 1 k, therefore if n(C)^2k—1, then according to (1.7) every
node of C other than B is an //-node of L. There follows firstly the
existence of the open //-lines asserted in the theorem and secondly that every
node of C other than B is joined exclusively to nodes of C. The latter fact
implies that [C] is a component or a terminal member with cut-node B.
([C] is obviously twofold connected because of C.) :t(C)1/:+1, because
&(Pn) = k and Pa is joined exclusively to nodes of C.
If the graph F of Theorem (1.8) has at most 2k — 1 nodes, tlien it
follows from Theorem (1. 4) that [C]==T or [C]=- F—A according as o(/l)s2
or q(A) = 1 and that, if :r(F)^2k and e(A)=i2, then ;i{C) =2k.
The following two theorems can be deduced:
Theorem (1.9) If the node A of the graph F is not isolated and the
degree of every node of F distinct from A is ^k (ig 2) and if n(F) ^_2k — \,
then A is connected by open H-lines to every node of F.
Theorem (1. 10) (Dirac) If every node of the graph F has degree ~^k
(k^2), and if :i(V)ti2k, then F has an H-line.
(1.9) obviously implies the following theorem:
Theorem (1. 11) If every node of the graph r has degree ^k (^2),
and if n{F)^2k—1, then any two distinct nodes are connected by an open
H-line.
If the graph F of Theorem (1.8) is twofold connected and if ;t(P) =g
s=^2k, then yc(C)^2k. From this it follows that the node A of Theorem
(1.8) is the initial node of a path of length 12/:-1.
The following two theorems can be deduced:
Theorem (1.12) If F is a twofold connected graph and every node
with the exception of one single node A has degree ^k (A:= 1) and if in
addition :c(F)^2k, then F contains a path with at least 2k—1 edges which
starts from A.
(This theorem is trivial for /:=1.)
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344
P. ERDOS AND T. QALLA1
Theorem (1. 13) (Dirac) If the degree of every node of the twofold
connected graph F is ^k (k^2) and if ^i{F)^_2k, then F contains a
circuit with at least 2k edges.
Remark. It follows from the above considerations that the assertions in
Theorems (1. 10) and (1. 13) remain true if the degree of every node except
one node A is at least k (k^2) and q(A)^2. If two nodes have degree
<k, then these theorems are not generally true.
Theorem (1. 14) If F is a connected graph and the degree of every
one of its nodes is ^k (k^ 1) and if :x(F)^2k-\- 1, then F contains a path
with 2k or more edges.*
Proof. If k— 1, the theorem is trivial. In what follows it will be assumed
that k ^2.
If F is twofold connected, then, by (1. 13), 7" contains a circuit
C=-(Pl,...,,Pnl,Pm+l = Px) {m^2k). If m>2k, W=-(Pi,.... Pm) is a path
of tlie required kind. If m = 2A:, then, by our assumptions, F contains a
node P which is not in C and which is joined to a node of C, say Pt.
Then W = (P, Pi, ..., Pm) is a path of the required kind.
If F is not twofold connected, then let Fi and F> denote two terminal
members of F, and /4, and A> their cut-nodes. Fi and F> are twofold
connected and apart from At and A-> their nodes have degree s/c in F\ and F>,
respectively. This is possible only if ;t(Fi) i= k-\- 1 and n;(F*)^k-\-1. From
this and from (1. 9) and (1. 12) it follows that I\ contains a path of length
is A: which starts in Ai and F> contains a path of length §fc which starts
in A-i. If A\=- A->, then these two paths together constitute a path with at
least 2k edges, and if A\=f=A±, then these two paths together with an
^1/4^-path of F—Fi — F-j, constitute a path with more than 2k edges.
Remark. Theorem (1. 14) can be proved easily whithout using the
preceding theorems.
(1. 15) The "accuracy" of Theorems (1. 13) and (1. 14) is demonstrated
by the following graph F:
F consists of the nodes Pi, ..., Pk, Qi, ••-, Q„-k (2^k^n—k) and
of all edges P«Q, (/ = 1,..., k; y= 1, ..., n — k). F is an even graph ([6],
p. 170). The degree of every node of F is & k and it may easily be
verified that 7" is fc-fold connected, further that the longest circuits and paths
in 7" have 2k edges.
It is seen from the example of this graph that in the theorems in ques-
4 This result was obtained independently by G. A. Dirac.
280 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS Oh GRAPHS
345
tion it is not possible to assert the existence of longer paths and circuits
than those proved to exist, even if the connectivity is assumed to be higher.
(1. 16) Using an altered form of Theorem (1.8) and Menqer's well-
known "rc-path theorem" the following theorems can be established:
If the degree of every node of the twofold connected graph F is ^k
(/eg 2) and if n(F)^2k, then through each node of F there passes a circuit
having at least 2 k edges.
If the degree of every node of the twofold connected graph F with the
exception of the two nodes A and B is ^k (A: is 2), then every node of F
lies on an A B-path having at least k edges.
These theorems are not proved in this paper.
§2
(2. 1) Let the classes of all graphs containing exactly n nodes and
containing, respectively, no paths, circuits, arcs with more than / edges (/^1)
be denoted by F{n,l), G(n,l), H(n,l). The graphs in each class which
contain the most edges are called the extreme graphs of the class concerned,
and the number of edges in these graphs will be denoted by f(n, I), g(n, I)
and h(n, I), respectively. So if the graph /' is a member of, respectively,
F(n, I), G(n, I), H(n, I), then the following inequalities hold :
(*) r(Os/(n,/), >'(F)^g(n,l), r{F)^h{n,l),
and if F is an extreme graph of the class concerned, then equality holds
under (*).
We wish to estimate or determine f, g and h and to find the extreme
graphs.
Clearly, if n^kl (/^=2), the only extreme graphs of F(n, I—1), G(n,l),
H(n, I) are the complete n-graphs.
(2. 2) Our method of estimating the values in question from below is
to construct graphs belonging to the classes F,G,H and containing as
many edges as possible.
In this paper /If (lsKn) denotes the graph which consists of the
nodes Pi,..., Pk, Qu ..., Q„-fe together with all the edges PiPj (/",/=--•= 1,..., A:;
i =/= j) and all the edges AQ, (/== 1,..., k;j= 1,..., n — k).
If \^k<n — 1, then Ff+i denotes the graph obtained from the graph
F^ by the addition of the edge Q1Q2. Accordingly, the graph F'n is defined
only for /s2 and n> [(/+1)/2]. In what follows the graphs F;, will be
called stars and the graphs rt will be called J-stars for all values of n.
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P. ERDOS AND T. QALLAI
Witli the notation
7-(n, 2k) = ( * ) + (''-*)*--=»*-(* J'j'
<f(n,2k+l)=-<r(n,2k)+l
we have r(r'N) = <f(n, I) for 17=.2 and n > [(/+1)/2].
It may easily be verified that the graph r'„ has the following
properties: it is [/2]-fold connected, it contains no path and no circuit with more
than / edges, if n^/&3, then it contains an /-gon, and if n>l, then it
contains an/-path. Hence /',', is a member of each of the classes F(n, I),
G(n,l) and H(n, I), and so — having regard to the remarks concerning the
case n =s/ in (2. 1) —
(2.3) f(n,l)-<f(n,l), g(»,/):^(//,/), h(n,l)><f(n,l) (/¾ 1).
(2.4) The graph /+,■ (/i?" 1) is defined as follows: Let n = qj-\-r
where /•</. /+,- has exactly n nodes, and if /-- 0, it consists of q
components, while if r>0, it consists of ¢+1 components, if r = 0, all its
components are complete /'-graphs, and if r>0, then q of its components are
complete /'-graphs and the remaining component is a complete /--graph. (I\t
consists of a single node.)
G*(n,j) (/'s 1) denotes the following class of graphs: Let n^q(j—1) + /-
where l^kr^j—1. G*(n,j) is the class of connected graphs containing
exactly n nodes which have q members if n>\ and /"=—1, and q + 1
members if n>\ and /->1, every member is a complete/'-graph if n > 1 and
/■= 1, and q members are complete /'-graphs and one member is a complete
r-graph if n>\ and r>\. G*(1,/) =---= {rlA\ for all /=^1.
r*j is to denote that element of G*(n,J) which contains at most one
cut-node.
In the notation
V(»> i>'') = - 2 "■/~~ Tr(/ + * — 0
it is found that
r(/+ju)= i/;(n,/,/■) where n-= ¢(/+1) + /- (/•</+ 1)
and that
if r^G*{n,j), then r{.r)=y(nj,r) where n = ¢(/-1) + /- (1 ^/-^/-1).
The following statements may easily be verified:
/;.,„ 6 /=■(«, /), G*(n,l)c:G(n,l), l\,$H(n, I) (/1=1).
282 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS OF OKAPHS
347
Hence
i f(n, I) r- V(n, I, r) (n = ?(/ + 1) + r, r<l+ 1),
(2.5) ; g(n,[)^y(n,l,r) (n =</(/-1) +/-; 1=5/-2/-1), (/^1).
( h(n,l)^>p(n, l—\,r) (n --<?/+ /•; /•</)
The statements below follow from a simple calculation:
(1) If l = 2k (/:^1), then rp(n,2k)^ ip(n,2k,r) and equality holds only
if /-== A: or /-== k + 1. So (2.5) gives a better estimate of / and ^ unless
/•=k or /- = /:+1, in which cases (2.3) and (2.5) are equally good.
(2) If l = 2k (fcsl) and n>k(k+l), then (f(n,2k)>«j{n,2k—\,r).
Here (2. 3) gives a better estimate of /2.
(3) If/ = 2/:+1 (A ^ 1) and n >A +3, then y(rt,2A+l)< 1//(/1, 2/:+1,/7.
For this case (2. 5) gives a better estimate of / and g\
(4) If /=2/:+1, then, according as (a) /-=-/: or /- = /-+1, or
(b) /- = /:-1 or /- = /: + 2, or (c) r<k—\ or r>k + 2, tp(n,2k+\)>,=,
or < i//(n, 2/:, /-) and, accordingly, (2. 3) gives a better or equally good or
worse estimate of h than (2. 5).
In order to estimate f,g and h from above we need the theorems of § 1.
Theorem (2. 6)
/(",/) = y«/ (/^1),
equality holds only if n = q(l+\), in which case F,,.i+i is the only extreme
graph of the class F(n, I)
Proof. If n^ /+1, then a graph F having exactly n nodes cannot
contain a path with more than / edges, so
KO ^-^=^¾
and equality holds here only if F is a complete (/+ l)-graph. So the theorem
is true if n ^/+ 1.
Now let n'>/+1 and suppose that the theorem is true for all n such
that n<n'. We prove that in that case the theorem is also true for n'.
Let F € F{n', I).
(1) If r is not connected, then let its components be FU...,FP (/7&2),
;T(Fi)=n; (/=1,...,/7). Then nH +rc7, = n', /?,</?' and Fi£F(n;,l)
(/=1,...,/7). So by hypothesis
r(0 = z+r.) + • • • + r(FP) :§ /i, y + • ■ ■ + /i„ | = n' \,
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348
P. ERDOS AND T. OALLA!
and equality holds only if /'(/',-) — n,--^- (/=1,...,/7), i.e. if f\,...,[], are
all complete (/ + l)-graphs. In this case our theorem is therefore true for //'.
(2) Suppose that /' is connected. We show that there exists a node P'
whose degree in F, 'j(P'), is at most 12. For if no such node P' exists,
then, whether l = 2k or / = 2/:+1, g(P)^k+l for every node P. If / = 2/:,
then ;r(F)>2k+\, and if :T(F)>2k + 2, then, by (1.14), /' contains a
path having at least 2/: + 2 edges, while if :t(F) 2A'-| 2, then, by Theorem
(1.10), 7" contains a path having 2/:+1 edges. If /==^--2/:+1, then .t(7')>
>2A:+2 and so, by (1.14), /' contains a path having 2/: + 2 edges. In
every case we have a contradiction.
Suppose therefore that p(F) ^//2 and let /"=--/'— P\ Then F'£F{ri—\,l).
F' cannot contain a complete (/ + l)-graph because if it did, then F would
contain an (/+l)-path. So by our induction hypothesis >'(F')< (n'—1)//2,
and therefore
r(/')==P(n+''(n<^+(n'-l)4-/''y-
Theorkm (2. 7)
g(n,l)^~(n-\)l (/ir2),
equality holds only if n--~q(l—1)+1, in which case the extreme graphs of
the class G(n,l) are the elements of the class G*(n,l)
Proof. The theorem is trivially true for n=\. If 1 <//=£/ and
F{G(n,l), then
n(n~l) ^ (n-l)l
V 7 -' 2~ - 2 '
and equality here can hold only if /' is a complete /-graph. The theorem is
therefore true for //s;/.
Suppose that n'>l and suppose that the theorem is true for all n if
n<n'. We show that it is then true also for n'. Let F^G(n',l).
(1) If 7" is not connected, let its components be Flt...,FP (/7^2) and
let ;t(77) = //,-. Then //i + h 'h = «', ";< n' and 7', £ G (//,-, I) (i —- 1,..., p).
By our hypothesis therefore
7*(O-r(r0+••• + '•(/•,)=£(«,-1)//2+.-- +
+ (/7,,-1)/2 = (//-/7)/ 2< (//'— 1)//2.
(2) If P is connected but not twofold connected, then let 7r denote a
terminal member of F and let F> = F—Fl. ;t(/'i)=.-//i, ;t(7'2) = //2. Then
284 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS
349
n'=ni + n2 — l,fh<n',ti2<n', fi f G((ii, /), A € G(^,/), and so, according
to our hypothesis,
1^^)=--1-(/-,)+1^(/0)^(^-1)/,2 + (/12-1)//2 = (^-1)//2,
and equality can hold only if l\£G*(ni,l) and /ofG*(i!2,/). But then
r$G'(n'l).
(3) Let r be twofold connected. We show that P then has a node P'
of degree-£//2. For if no such node exists, then p(P)g^+l for every
node P both if l = 2k and if /=-2/:+1. If /=---2k, then •r(/,)>2/r, in
which case if .-t(,T) + 2/: + 2, then by (1.13) /" contains an m-gon with
m 3=2/: + 2 while if :T(F) = 2k+\, then by (1.10) P has an //-line and
therefore contains a (2/:+l)-gon. If /-2/:+1, then :t(/-) > 2/:+1 and so
F contains an m-gon with m + 2/: + 2 by (1. 13). We have obtained a
contradiction in every case.
So assume that (>(P')=//2 and let r = r—P'. Then /" 6 G(rc'—1,/).
/"" is connected and p(P') is 2. /v is not an element of G*(n'—1,/). For if
this were the case, then P' would lie on a circuit with more than / edges.
From our induction hypothesis it follows that
r(D = p(/y)+»'(r')<//2 + (n'— 2) //2 = («'— 1)/2.
This proves the theorem.
For the investigation of h{n,l) it is useful to consider the cases /=2/:
and / = 2/:+1 separately.
Theorem (2. 8)
/?(/?, 2k) si (n —l)fr (£=1),
if n = \, then equality holds for all k and l\i is the extreme graph of
H(\,2k), if n>\ and /: = 1, then equality holds for all n and the star
with n nodes P'l is the only extreme graph of the class H(n, 2), finally if
n >1 and k>\, then equality holds only if n = 2/:, and the complete (2/:)-
graph is the only extreme graph of //(2/:, 2k).
Proof. Because //(«,/)£= G(n, I), we have that h(n,2k)^g(n,2k).
By (2.7), g(n,2k)^(n — \)k, so
h(n,2k)^=(n — \)k.
Equality can hold only if n~-^q(2k—1)+1 and if //(n, 2/:) contains an
element of G*(n, 2k). But an element of G*(n,2k) belongs to H{n,2k) only
if it contains no path with more than 2k edges. This holds only for the
graphs described in Theorem (2.8), whether /:=1 or k>\.
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P. ERDOS AND T. OALLAI
If / = 2/:+1, then, since H(n, 1) -■=---F(n, \), we need only consider
the case kj^ 1.
Theorem (2.9)
h(n,2k-ir\)^nk (t>l),
and here equality does net hold ///:=1 and n = 1 or n ---=- 2, and it does
hold if k-— 1 and n >2, in which case the extreme graphs are those of which
all the components are J-stars; if k>\, then equality holds only if
n = q(2k-ir\) (q>0), and the only extreme graph is /'„.-.-,.-1-
The proof which is similar to the proof of (2.6) will be left to the reader.
§3
In this paragraph we determine the graphs with the most edges among
the connected graphs of the class F(n, 2k) and the extreme graphs of the
class H(n, 2k) for sufficiently large values of n.
We denote the classes of the connected graphs in F(n, I), G(n, I), H(n, I),
respectively, by F(n, 1), G(n, I), H(n, I), and the number of edges in the
extreme graphs of these classes (the graphs with the most edges) by
f(n> 0> g(}1, 0> h{n, I), respectively. From the fact that /',! is connected it
follows that each of these maximal numbers of edges is ~- (f(n,i).
(3. 1) If lT is an extreme graph of any one of the classes F(n,2k),
G(n,2k), H{n,2k) (k ^2) and if //>/-'—/■+], then F contains a 2k-gon.
Proof. If /' contains no 2/r-gon, then F£G(n,2k—1), and so by
(2.7) t'(F)'i(n —1)/-— . On the other hand, it follows from the extremal
property of /' and from the remark on the graphs /",' made above, that
v(F) s r(/f )==/iAr—k(k + 1) 2. Accordingly,
(n-1)(/--^-) z-nk — k(k+ 1)2,
from which it follows that n ty k-— /:+1. This contradicts our hypothesis.
From Lemma (1.6) we deduce the following lemma:
Lemma (3.2) Let the graph F have an H-line and let ;/■(/") ==/7/^4. If
F contains p mutually H-independent nodes S ,5,,, then /?=gm 2 and
(1) ''(/)^|2| 2- + 1 2 |.
286 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS 351
and if p = m/2, then equality in (1) holds only if /'=/'„/' and the nodes
Si,...,Sit span a complete p-graph in I'.
Proof. The theorem is trivially true for /» = 4. In what follows it will be
assumed that m>4. Let C=(Pi,..., P,„, P„,j-i = Pi) bean //-line of V and let
the nodes S- — Pa; (i---=1, ..., p) be mutually //-independent. It may be
assumed that \ = oi<---<o,J^m. Further let P,- = P0-,-i, 7 = Pa.M, q(7) =q,
(/ = 1, ...,p; Po = P,„) and let the elements of the sets {Pi,..., P,,}, {Si,..., S,,]
and {71,...,7,,} be named P-, S-, 7-nodes, respectively.
Because the S-nodes are //-independent, two S-nodes cannot be
neighbours on C, and therefore this is true of the P- and 7-nodes also, so that
an S-node cannot coincide with a 7-node or with an P-node. It follows
that p g m/2.
By (1.6)
(2) o/ + 9 j S m (/, / = I,..., p; ij=J).
Adding these inequalities together
(3) pi-| \-o,^pm2.
Let /" denote the complement of the graph I" and let the degree of
7 in /' be denoted by T>, (/---1,...,/7). Then
(4) P(. + p,. = fli-1 (/=- 1,. .., p).
By the remark after Lemma (1.6) neither two P-nodes nor two 7-nodes
can be joined in l\ Consequently, all the edges T;Tj (i,j=\, ...,p;i=f=j)
are in /', and so the number of edges of /' at least one end-node of which
is a 7-node is «i+--H -p> .
Accordingly,
(5) r(m-»H---+^-( 2 )'
and so, having regard to (4) and (3),
mp | p \ pm | p + I
2 ! 2 2
Hence
(6) ''O^U.H'^^UI 2- + 1 2 I'
Equality holds here only if it holds in (2) and in (5) throughout. If
p = ml2, then this is the case only if p, = ■■■ =o(, = m/2 and every edge
287 CIRCUITS

352
P. ERDOS AND T. OAI.LAI
of r is incident with some 7-node, that is to say any two nodes which are
not T-nodes are joined by an edge in l\ In this case, however, every
P-node is also a 7-node; and these are all joined to every 5-node, further
any two 5-nodes are joined to each other. This implies that /=- Pi1,', and
that the S-nodes span a complete /?-graph in /'.
(3.3) If the connected graph F contains a 2k-gon (k:^2) but does
not contain any path having more than 2k edges and if n =:t(F)^
:=3/: + 2, then r(F)^==y(n,2k), and equality holds only for r=--Ff.
Proof. Let C = (Pi, ■■-, Pa, /Vi = Pi) be a 2/c-gon of 7". The nodes
of C will be called P-nodes and the remaining nodes of P will be
called Q-nodes. Let the Q-nodes be denoted by Qi,...,Q,, where g = rc—
— 2k3ik+2.
Not two Q-nodes are joined in /". For if the edge QiQi exists, for
example, then, since F is connected, there is a path W in F which starts
in Qi or Q> and ends in a P-node, say Pr, and does not contain any
P-node other than Pi, and contains only one of the nodes Qi,Q^ — say Qi.
Then the edge QiQ^ and the paths W and (Pi, ..., P>,;) together constitute
a path with at least 2/:+1 edges. This contradicts our hypotheses.
From this and from the fact that /" is connected it follows that every
Q-node is joined to some P-node.
If P and Pj are distinct P-nodes and if there are two distinct nodes
Q,, and Q,, such that the edges PiQ;l and PjQ,, exist, then P, and Pj are
//-independent in [C]. For such an open //-line leading from P to P, in
[C] would, together with the edges PiQ,, and P,Q/,, constitute a (2/:+l)-path.
We divide the P-nodes into three classes. A P-node will be called an
a-node if it is joined to at least two Q-nodes, it will be called a p-node if
it is joined to exactly one Q-node and it will be called a -/-node if it is not
joined to any Q-node. The number of a-, t3- and /-nodes will be denoted
by pa,Pa and /?,,, respectively. pa+pi + py-- = 2k.
According to the above, any pair of ^-nodes are //-independent in [C],
and so are any ^-ntide and any /j-node. Since two neighbouring nodes of
C are not //-independent in [C], the neighbours on C of an «-node can
only be -/-nodes. It follows from this that pa^py, and so p„^k and
(1) p?^2k—2pa.
C is an //-line of the graph [C], so Lemma (3. 2) applies to the
rt-nodes. Accordingly,
(2) ^)-(^)-^ + (^+1).
288 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS
353
The number of edges which join a P-node to a Q-node is
(3) "rv^qpa+p?,
and so it follows from (1), (2) and (3) that
>'(r)=,i[C])+r1.v^[2£) + 2k + (q-k-2)pa + ( P"+l |.
Because q^k + 2, the expression on the right attains its maximum
value in the range Og/7,,^/: only if pa = k, and a simple calculation shows
that this maximum is equal to nk— ~l \ =^q(n,2k). So r(F)^<p(n,2k)
and equality holds only if pa = k and equality holds in (1), (2) and (3).
But if pa=^k, then Pi = 0 and equality holds in (1), and further, by (3. 2),
equality then holds in (2) only if [C] =/!>£ and the ^-nodes span a
complete /c-graph in [CJ. Finally, because P/s = 0, equality holds in (3) when
every Q-node is joined to every «-node. From the properties which have
been enumerated it follows that F=Ft.
Theorem (3.4) ff n>k- — k + 6 (k^l), then f(n,2k)--=fp(n,2k), and
the only extreme graph of the class F(n, 2k) is ff.
Proof (I) First assume that /:=1. By (2.6) f(n,2)^n with equality
holding only if n = 3q, and then /\3 is the only extreme graph. From this
and from f(n,2)^f(n,2) it is seen that f(n, 2) ^ n— I ==-cp (n, 2) except if
« = 3. But if V is connected and :i{F) = n, v{F) = n— I, then F is a tree
([6], p. 47), and therefore does not contain a path with more than 2 edges
only if /"=/",;. The theorem is therefore true if /:=1 and n>3.
(2) Assume that k^2. Then, because n>k2 — /: + 6, by (3.1) any
extreme graph F of F(n,2k) contains a 2/c-gon, and since k- — /: + 6=¾
^3/: + 2, it follows from Theorem (3.3) that v(F)^(p(n,2k), equality
holding only if F = rT. Our theorem follows from this and from r!,k 6 F(n, 2k).
(3. 5). ff F is an extreme graph of the class H(n, 2 k), then F has
not more than (/:+1)/2 components.
Proof. Suppose that the components of Tare Fu ..., I], and :7(/^) = 7^
(/==--1,...,/7). Then rii-\-• • • + np =--= n and /*; is an extreme graph of the
class H(nt,k) for / = 1,..., p. Then, according to Theorem (2.8), 7-(/^) =
=5(72,-1)/: (/=1,...,/7), and so v{F) = v{l\)-\ \-v{F,)^(n—p)k.
On the other hand, v(F)'^-i {n,2k) by (2.3), therefore
(n—p)k>_nk—k(k+l) 2,
and hence /?=£(/:+ 1)/2.
289 CIRCUITS

354
P. EKDOS AND T. GALLA1
Theorem (3.6) If n>- (k+\f, then h(n,2k)=-if(n,2k) and if
is the only extreme graph of the class H(n, 2k).
Proof. For k ----- I the assertion of our theorem is contained in (2. 8).
Suppose that /:==:2, n>^(k-\-\)\ and /" is an extreme graph of the class
H{n,2k). According to (3.5), f then has a component /" such that
n'=■ :t(/')>(/:+ 1)-. /" is an extreme graph of the class H(n',2k) and
therefore contains a 2/c-gon by (3.1). (k+ l)-> 3k + 2 because /:==2, so
by (3.3) r(r')i5cf(n',2k), equality holding only if /"=/""'■'. But
/f- 6 H{n', 2k), and so /"=/•?' and /<(/") =-?.(«', 2k). We show that
/"=/'. For suppose that 7"^=/'and let /'"---/'—/". «"=/'(/"') =- n —n'
and f"£H(n",2k). By (2.8) /-(/") :S («"—!)£, and so >'(/')- >'(/") +
+ /'(/"') =5«'/:— (A' 2 ' | +(n"—l)k-'(f(n,2k) — k. But this contradicts (2. 3).
Conjectures. We conjecture from the above that all extreme graphs
of the classes occurring in § 2 and § 3 can be found among the graphs
/t, /"„..' and the members of the class G*(n,l). Among the twofold
connected graphs /'', is probably in every case the only extreme graph if n>/+l.
§4
We are going to prove the following
Theorem (4. 1) Let ;r(/') ; "• Assume further the maximum number
of independent edges is k (k=g I). Then
>>([ )s max|( 2 ), k(n — k)-\-\2J)-
Equality can occur only if T — T1,!', or if one component of f is a
complete (2k-f \)-graph and the other components are isolated nodes.
Proof. We can clearly assume n>2k. Choose k independent edges
and call these a'-edges and the remaining edges fi-edges. The n — 2k nodes
of the graph which are not incident with «'-edges we call unsaturated.
Following Berqe ([2], p 176) we add a node U to T and connect U with
every unsaturated node by an edge. The new graph we call /" and the new
edges and the old (/-edges we call a-edges (U is incident with n — 2k
r^-edges, every other node is incident with exactly one «-edge).
Let L be a directed LMoop. We call L alternating if its edges are
alternatingly ^-edges and ,i-edges in the ordering determined by L. A node
290 GRAPH THEORY

ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS 355
P of /" is called ('.-accessible if there exists an alternating L'-loop of final
node P whose last edge is an fj-edge and it is called J-accessible if there
exists an alternating {/-loop of final node P whose last edge is a /i'-edge.
Further, by definition, U is called /j-accessible. The nodes which are
^-accessible but not ^-accessible are called ,-i-nodes. Denote the number of /j-nodes
by /( + 1-
It is easy to see that U is not ^-accessible, thus U is a ,j-node ([I],
[2], p. 176; [5], p. 140).
Denote the components of the graph obtained from /" by omitting the
,j-nodes and the edges incident to it by /\,...,/',„ (m^l). /, is called
odd or even if ;t(/"",■) is odd or even. If /', is odd, put ;*(/',) = 2a,■ + I,
if /', is even, put ;t(/',) = 2a,. An f.'-edge one node of which is a .i-node
and the other node of which belongs to one of the /',■ we call an entering
edge of 1,.
The following facts are well known ([I], [2], pp. 169—170; [5], pp.
141 —142).
Every cc-edge incident to a J-node is an entering edge of some odd F
and every odd /, lias exactly one entering edge.
From this it follows that every f.'-edge is either an entering edge (of
some odd /",) or is an edge of some /",. Further /", contains exactly
<7, f^-edges.
The o-edges in /", (ls/<m) are clearly f+edges. We obtain then-
number by subtracting from A the number of ^'-edges incident to the
.i-nodes, i.e. their number is A- — u. Thus
If /, is even,
Thus
i_-1
2, a, -= k— u.
then r(r,)^[2"''). If /, is odd, then
,-(/^ f2%+^)-(^1 +2a
i(^|+V2„,.("-2"| + 2*-2.
I 2A-—2/( + 1 |
Equality is only possible if every /", is odd and is a complete graph and
a, = 0 for al! i^2.
The number of edges in /" incident to the .i-nodes is less than or
equal to («—.«)."+ ? ■
291 CIRCUITS

356
P. ERDOS AND T. (3ALLAI." ON MAXIMAL PATHS AND CIRCUITS (IF GRAPHS
Since every edge of /" either belongs to one of the r, or is incident
to one of the i-nodes, we obtain that
r(r^{2k-lfl + l)+(n-fl)ft + ^) ^/00.
Since f(n) is a convex function of .» and 0-^u^Ek, we obtain
;•(/-) 2 max (/,0), /(A)) = max ((2*J '), (n-k)k+[^j ).
Equality is only possible if ,«==0 or ,u=---k, if ,« = 0, one of the r, must
be a complete (2/: + I)-graph and the other /", must be isolated nodes.
If u = /:, all the F, must be isolated nodes and every ,i-node must be
connected with all the nodes of f, i. e. F - ff.
(Received 24 June 1959)
Bibliography
[1] H. B. Belck, Regulare Faktoren von Graphen, Journal/, reine u. angew. Math., 188
(1950), pp. 228—252.
[2] C. Beroe, Theorie des graphes et ses applications (Paris, 1958).
[3] G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. (3), 2 (1952),
pp. 69—81.
[4] P. Erdos and A. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc,
52 (1946), pp. 1087—1091.
[5] T. Gallai, On factorisation of graphs, Acta Malh. Acad. Sci. Hung., 1 (1950), pp. 133—153.
[6] D. Konio, Theorie der endtichen und unendlichen Graphen (Leipzig, 1936).
[7] T. Kovari, Vera T. Sos and P. Turan', On a problem of K. Zarankiewicz, Colloquium
Math., 3 (1954), pp. 50—57.
[8] D. J. Newman, A problem in graph theory, Amer. Math. Monthly, 65 (1958), p. 611.
[9] P. Turan, Egy grafelmeleti szelsoertekfeladatril, Mat. es Fiz. Lapok, 48 (1941), pp.
436-452.
[10J P. Turan, On the theory of graphs, Colloquium Math., 3 (1954), pp. 19—30.
[11] K. Zarankiewicz, Sur les relations symetriques dans l'ensemble fini, Colloquium Math.,
1 (1947), pp. 10-14.
292 GRAPH THEORY

On the maximal number of disjoint circuits of a graph
By P. ERDOS and L. POSA (Budapest)
Throughout this paper Gk will denote a graph with n vertices and k edges
where circuits consisting of two edges and loops (i. e. circuits of one edge) are not
permitted and Gk will denote a graph of n vertices and k edges where loops and
circuits with two edges are permitted. v(G) (respectively v(G)) will denote the
number of edges of G (respectively G).
If xu x2, ■■■, xk are some of the vertices of G, then (G —xt — ... —xk) will
denote the graph which we obtain from G by omitting the vertices x1; ..., xk and
all the edges incident to them. By G{xx, ..., xk) we denote the subgraph of G spanned
by the vertices xt, ..., xk. The valency of a vertex x — v(x) — will denote the number
of edges incident to it. (A loop is counted doubly.) The edge connecting xx and x2
will be denoted by [xt, ,x2], edges will sometimes be denoted by elt e2, .... (xt, x2,...xk)
will denote the circuit having the edges [x1,x2], ..., [xk~i, xk], [xkxx].
A set of edges is called independent if no two of them have a common vertex.
A set of circuits will be called independent if no two of them have a common
vertex. They will be called weakly independent if no two of them have a common
edge.
In a previous paper Erdos and Gallai [1] proved that every
(1)
G\+i where / = max
2k2 M, (£-1),,-(^-1)2+(^1
contains k independent edges.
In the present paper we shall investigate the following question of Turanian
type (see 1): how many edges are needed that a graph should have to contain k
independent or weakly independent circuits? Put
f(n, k) = {2k -\)n-2k2+ k.
Our principal result will be that for n^n0(k), k> 1 every G/"^*) contains k
independent circuits except if it contains 2k — 1 vertices of valency n — 1 (its structure
is then uniquely determined). If k=l trivially every G^P contains a circuit, but
there are of course graphs G(„"l i where no vertex has valency n — 1 and the graph
nevertheless contains no circuit. Thus the restriction k>l is necessary.
Clearly n0(k)^3k (since a circuit contains at least three vertices). For k = 2
and k = 3 n0(k) = 3k, but in general n0(k) =-3k, but we will prove n0(k)^24k.
293 CIRCUITS

4
P. Erdos and L. Posa
Perhaps the following result analogous to (I) holds: Every
(2)
Gi+'i with 1--
■ max
3fc-l
+ n-3k + 2, (2k-l)n-2k2+k
contains k independent circuits.
Denote by g(k) the smallest integer so that every g£+9(*) contains k weakly
independent circuits. We will show that g(2)=4 and that for every k
ctk log k<g(k) -=■ c2k log k
where ct and c2 are suitable absolute constants (the c's throughout this paper will
denote suitable absolute constants). The exact determination of g(k) seems to be
a very difficult problem and we cannot even show that g(k)/logk tends to a
constant. Further we do not know the value of g(3).
It is easy to see that g(2)^4 i. e. we will show that for every n ^6 there exists
a Gi + 3 which does not contain two weakly independent circuits. To see this let
the vertices of our graph be xx, ..., x„ and its edges
[Xi,Xjl, 1
:3<j = 6 and [xk, xk+1], 65iSn-l.
A simple argument shows that this G„ + 3 does not contain two weakly independent
circuits.
After completing our paper we found out that some of our results were known
to G. Dirac but he published nothing on this subject. In particular he proved that
for n^6 every G(3n-s contains two independent circuits and that every G(„"lA
contains two weakly independent circuits. He also proved that for n^6 every G(n)
where the valency of every vertex is 53 and the valency of every vertex with at
most one exception is ^4 contains two independent circuits and conjectured that
for ns3& every G(n) which is 2/c-fold connected (i.e. which remains connected
after the omission of any 2k — I of its vertices) contains k independent circuits.
Theorem 1. Let k>\, n^24k then every Gf^k) either contains k independent
circuits or 2k— I vertices of valency n — \.
Our Theorem clearly implies that for n^24k every G(") , , contains k inde-
pendent circuits (since a simple computation shows that a G'"' k which has 2k — 1
vertices of valency n — \ has all its other vertices of valency 2/c — 1 and its structure
is thus uniquely determined). n^24k could easily be improved a great deal, but
our method does not give any hope of best possible estimates.
Theorem 1. will be proved by a fairly complicated induction process and to
make this as painless as possible we will restate Theorem I. in a very much more
complicated form but which will be more suitable for our induction process.
Theorem 1'. Every G„n) contains a circuit (k = 1). For k > 1 put
g{n,k)
f{n,k) + {24k-n){k-\) for
\24k
An, k)
for n 5 24k.
294 GRAPH THEORY

On circuits of a graph
5
Then if 3k?=n ?=24k — 1 every Gg(„tk) contains k independent circuits, and if n^24k
and l0 ~g(n, k) then every G/"' J) contains k independent circuits except only if
l0=g(n, k) and g1"' contains 2k— 1 vertices of valency n — \ and n — (2k— 1) vertices
of valency 2k— I.
Since g{n,k)=f{n,k) for n^24k Theorem Y implies Theorem 1.
Theorem Y is trivial for k=\. It is also trivial for k > 1 if 3k^n^6k since
by a simple calculation
g(n,k)mfy
and for n^3k the complete graph contains k independent circuits.
First we prove two Lemmas.
Lemma 1. Let n^6k and assume that G(n) contains 2k vertices xi,x2, ...,x2k
of valency v(xt)^n~k (1 ==/==2/c). Then G("> contains k independent quadrilaterals.
Denote by yu ..., y„-2k the other n—2k vertices of G(n). Consider a maxim a
system of independent quadrilaterals of the form
(*2i-i> yn-u x2i, y2i), 1==/==/, /rSfc.
We shall show l — k. Assume l~=-k. Each of the vertices x2l+1 and A"2; + 2 are
connected with at least n — k vertices. Thus every vertex except possibly 2k vertices are
connected with both x2i+1 and x2l+2, i. e. these are at least n—2k^4k of them
which are connected with both x2l + 1 and x21+2. Since 2k + 2l^4k — 2 there are
two further vertices y2i + 1 and y2i + 2 which are connected with both x2l + 1 and
x2i + 2- Thus the quadrilateral (x2! + 1, y2!+u x2l + 2, y2l + 2) is independent of the
others, which contradicts our maximality assumption, which proves Lemma 1.
Lemma 2. Let n = 2k and assume that every vertex of G(n) has valency ^2k,
then G{n) contains k independent edges.
Lemma 2 can be proved from first principles in a few lines as follows:2) Let
e; = [a'2j-i, x2i], I ?=/==/ be a maximal set of independent edges. Assume t<k
(otherwise there is nothing to prove). But then since n^2k there are two vertices
of G(n) jj and y2 distinct from the xh 1==/9=2/. y1 and y2 can be joined only to
the xt 1==/9=2/ (by our maximality assumption), and by the same assumption
if yt is connected to an endpoint of et 1==/=5/ then y2 can not be connected to
the other endpoint. Thus v(y1) + v(y2)^2t<2k which contradicts v(y^)^k. This
contradiction proved the Lemma,
Now we prove Theorem Y by induction. Let k> 1 and assume that Theorem Y
holds for k — \ and assume that it holds for every 6i<m<n. (We already remarked
that it trivially holds for 3k == m == 6k). Then we shall prove it for n, and if we have
succeeded in this the proof of Theorem Y and therefore Theorem 1 will be complete.
1) If /o>(-,) G, will denote the complete graph of n vertices.
2) This proof is due to G. Dirac (written communication).
295 CIRCUITS

6
P. Erdos and L. Posa
Assume first that our Gg"„yk) contains 2k vertices of valency ^n—k.
Then by Lemma 1. our graph contains k independent quadrilaterals and thus
Theorem 1' is proved in this case.
Henceforth we can assume that G contains at most 2k —I vertices of valency
^n—k. If all the other vertices have valency <2k then the number of edges of
G is at most
{2k - 1) (w - 1) + (2k - 1) (n - 2k + 1) __ fi
and equality can occur only if G contains 2k — 1 vertices of valency n — 1 (i. e. these
vertices are connected with all the vertices of the graph), and G contains no other
edge, otherwise it would contain another vertex of valency ^2k). Thus the structure
of our G is uniquely determined (G can have this structure only if n^24k) and
Theorem 1' is proved in this case too.
Therefore we can now assume that G has a vertex — say x0 — of valency I'
satisfying
2k^l' <n-k.
Let xx, ..., xv, 2k^r<n—k be the vertices of G connected with x0 by an
edge. Assume first that in the graph G(xlt ..., xr) there is a vertex — say xx —
of valency <k. It may be assumed that xx is not connected by an edge to any of
the vertices xr+1, ..., xt-, where r^k. Define the graph G1 with n — 1 vertices as
follows: Omit the vertex x0 and all the /' edges incident to it, and add the edges
[xlt xr+i] l^i^l' — r (i.e. x1 is connected in Gx to all the vertices to which x0
is connected in G [except of course xj). Clearly
v(Gi)ev(G)-i
or
(3) vtGj^Cn, fc)-fc=£(n-l,fc).
Thus by our induction hypothesis G1 contains k independent circuits (from the
first inequality of (3) it follows that v(Gt) >/(n — \,k) thus Gt cannot have 2k — 1
vertices of valency n — \ and n —2k vertices of valency 2k — I, i. e. the second
alternative of Theorem V. is excluded). But then G must also contain k independent
circuits. To see this let Cu ..., Ck be the k independent circuits of Glt at most one
of these circuits — say C\ — contains one or two of the new edges [x1,xr + i],
l^i^l' — r, (if none of these circuits contains any of these edges, then Cu ..., Ck
are k independent circuits of G). If C1 contains only one of the new edges — say
[x1,xr+1] — then we obtain C\ by omitting [x1,xr+1] from C1 and replacing it
by [x0, x-^] and [x0 xr+1]. If C1 contains [xlt xr] and [x1,xr+1] then in C* these
are replaced by [x0, xr] and [x0,xr+l]. In any case C*, ■■■,Ck are k independent
circuits of G. Thus Theorem I' is proved in this case too.
Assume next that all the vertices of G(x1> ..., xr) have valency ^k. Then
(since /' ^2k) by Lemma 2. G(xt, ..., xr) contains k independent edges et —
= [x2i-i, x2H, 1 ^i = k. Assume first that each of the e,- are contained in at least
k — \ triangles {x2;-i, x2i, y^), 1=/ = ^-1, l^i^k where the yj'' are all
different from x0,x1, ..., x2k.
296 GRAPH THEORY

On circuits of a graph
7
In this case Theorem V easily follows since G contains k independent triangles.
To see this observe that it immediately follows from our assumptions that there
are k—l independent triangles (x2i-t, x2i, yt), l^i^k — I. (x0, x2k-l, x2k) is the
A>th independent triangle.
Henceforth we can thus assume that — say e1 = [xi, x2] — is contained in
at most k — 2 triangles in the graph (G — x0 — x3 —... — x2k).PutG2 = (G — x0 — xt — x2).
Now we estimate v(G2) from below, by estimating from above the number of edges
of G incident to x0, xt and x2. v(x0)-^n — k by our assumption. In G(xl7 ..., x2k)
the vertices xt and x2 are incident to at most 4k — 3 edges. Finally every vertex
of (G — x0 — x1 — ... —x2k) is connected with at most one of the vertices xx and x2,
except possibly k—2 vertices which might be connected with both. Thus we obtain
at most n—k — 3 further edges. Thus the total number of edges incident to x0, xx
and x2 is at most
n - k- 1 + 4k - 3 + n - k - 3 = 2n + 2k - 7.
Hence for k > 2
v(G2) ^v(G)-2n-2k+7=(n- 3) (2k -3)-2{k-1)2+ k +
<4) +g(n,k)-f(n,k)>g(n-3,k-l)
since clearly
g(n, k)-f{n, k)^g{n-3,k-l)-f(n-3, k- 1).
For k = 2, we obtain
{4') v(G2)==v(G)- 2/7 + 3==/7-3.
Thus (4) and (4') imply that by our induction hypothesis G2 contains k—l
independent circuits. These and (x0, xu x2) are together k independent circuits
contained in G. Theorem V is now proved.
If A: =2 then the assertion of Theorem 1 holds for all «56. The reader can
verify it for n = 6 and then adapt our induction process to prove it for n > 6.
Perhaps the following result is of some interest.
Theorem 2. Let n^4k, then every G"2k-i)n-(2k-if+i which contains no
triangle contains k independent circuits.
For k = 1 the Theorem is trivial. We use induction and assume that it holds
for k—l. Let (x1, ..., Xil) = C1 be the shortest circuit of G and denote by xh + 1, ..., x„
the other vertices of G. No two non-neighbouring vertices of C\ can be connected
by an edge (for otherwise C\ would not be the shortest circuit of G). Assume first
that /i>4. Then every xr, l^-^r^n can be connected to at most one vertex
of C1 (for otherwise Ct would not be the shortest circuit). Thus the vertices
xl,...,x,l are incident to at most n edges. Let C2 =(xli + 1, ..., -f/l + /2) be the
shortest circuit of (G — x1 — ... —xtl) and C3=(xh + h + 1, ..., xli + ll + h) the shortest
circuit of (G—xl — ... —xll + l7) etc. Thus we obtain the circuits Cu ..., Cr of length
4-==/] ^ ... ^/r and we assume that the graph (G — x1 — ... — Xil + ... + ir) contains
no circuit. If r^k our Theorem is proved. Assume r<k. By our previous
argument we obtain that in (G — x1 —... —.vIl + ... + i.) the vertices of Ci+1 are
297 CIRCUITS

8
P. Erdos and L. Posa
incident to at most n — ^l, edges. Since (G —xt —... —x/l + ... + /) has no circuit
r
it has fewer than n — ^l, edges. Thus the total number of edges of G is less than
t = i
n+ 2 (n-/,-...-/,)^(r+l)n- 2 l,^kn- 5< (2k- i)n -(2k- 1)2
t-i t-i
for n^4k, an evident contradiction.
Assume next that ^=4. Then every vertex xr, 4-=r^n is connected with at
most two of the vertices xu x2, x3, xA. Thus the number of edges incident to xlt
x2, x3, xA is at most
4 + 2(n-4) = 2n-4.
Hence
v(G-xl-x2-x3-xA)^v(G)-2n + 4 = (2k~l)n-(2k-l)1-2n+5 =
= (2k-3)(n-4)-(2k-3)2 + \.
By our induction hypothesis (G — x1—x2 — x3 — xA) contains k—\
independent circuits which together with (x1( x2, x3, xA) gives k independent circuits of G.
Thus the proof of Theorem 2. is complete.
Now we show that Theorem 2. is best possible. Let G be a graph whose
vertices are xit ..., .\"„, and whose edges are [.v;, x}] where 1 ^ i^2k— 1 -=/^/7. Clearly
G has (2/c -- \)n — (2k — 1)2 edges and does not contain k independent circuits.
If k > 1 then this graph is the only G(("l-i)n-(21--1>2 which contains no triangles
and does not contain k independent circuits, we leave the proof to the reader.
G. Dirac [2] proved that for n^4 every G'zV-i contains a topological complete
quadrilateral (i. e. it contains four vertices .v,, .v2, x3, .y4 any two of which are
connected by pairwise disjoint paths). We shall give a simple proof of this theorem
by our method. For « = 4 the theorem clearly holds. We will assume that it holds
for «—1 and prove it for n. Our G2V-2 clearly contains a vertex x0 of valency not
exceeding 3. If v(x0)<3 then v(G — x0) ^ 2n — 4 and thus by our induction
hypothesis (G — x0) and therefore G contains a topological complete quadrilateral.
Assume that v(x0) = 3 and let x1,x2,x3 be the vertices connected with .v0 by an
edge. If [a-!, x2], [x1,x3], [x2,x3] are all edges of G then G contains the complete
quadrilateral {x1, x2, x3, .v4}. Thus we can assume that one of these edges — say
[xt, x2] — does not occur in G. Add the edge [\\, x2] to (G — x0), thus we obtain
a graph G' having n—\ vertices and 2n — 4 edges. By our induction hypothesis
G1 contains a topological complete quadrilateral {yu y2, y3, yA}. But it is immediate
that {y\, y2, y3, yA} is a topological complete quadrilateral of G. To see this observe
that the new edge [xt, x2] can occur in at most one of the connecting paths and
there it can be replaced by [x1, x0] and [x0, x2\. G. Dirac showed by simple
examples that not every G2n-3 contains a complete topological quadrilateral, e. g. the
vertices are Xj, ... , x„ the edges [Xj,Xj], 2^j^n, [x2,xj\, 3^j^n.
One could perhaps conjecture that for 11^ 5 every G;3„_s contains a complete
topological pentagon, but the above proof breaks down and we can not even show
that there exists an absolute constant C so that for n ^ 5 every (¾ contains
complete topological pentagon.
298 GRAPH THEORY

On circuits of a graph
9
Theorem 3. Every G"+a contains two weakly independent circuits.
In other words g(2)=4 (see the introduction). We use induction on n. Our
Theorem clearly holds for n = \. We will assume that it holds for n—l and prove
it for n. If our graph contains a circuit of four or fewer edges, then our Theorem
is immediate, since by omitting the edges of this circuit a G{„"lA-< remains with
n + 4 — i^n, thus it contains another circuit thus giving our two weakly
independent circuits. Thus we can suppose that our graph contains no circuit with fewer
than five edges. If our graph contains a vertex of valency one we omit this vertex
and obtain a GJT+Y' which by our induction hypothesis contains two weakly
independent circuits. If a_0 is a vertex of valency two and xu x2 are the vertices connected
to x0 by an edge then we define Gt as the graph which we obtain from (G — x0)
by adding the edge [xt, x2]. Clearly Gt has n—l vertices and n + 3 edges and thus
our Theorem again follows. If all vertices of G have valency S3 then it has at least
■jn edges, or ^n ^ n + 4, which implies n ^ 8. But it is well known and easy to show
that every graph with fewer than 10 vertices every vertex of which has valency ^3
contains a circuit of at most 4 edges (for 10 vertices this is false as is shown by the
well known Petersen graph). This completes the proof of Theorem 3.
Theorem 4. For every k > i
(5) c^k logk<g(k)<c2k log k
where c\ and c2 are suitable absolute constants.
First we prove the upper bound in (5), (no attempt will be made to get a good
estimation for c2). We shall use induction with respect to k. For k = 2 the
inequality follows from Theorem 3. Assume that it holds for k—\, we shall prove it
for k. As in the proof of Theorem 3. we can assume that every vertex of our graph
has valency S3. But then
v(G)s=-}n
<Jn + [c2h log k]
or
(6)
n^lcjc log k.
First we prove
Lemma 3. Let n = 2. Every graph G(n) every vertex of which has valency S3
log n
contains a circuit of at most 2
log 2
edges.
If our graph contains a loop or a circuit of two edges our Lemma is trivial.
Thus assume that such circuits do not occur in our graph. Let xt be any vertex
of G(n). If G(n) contains no circuit of ^2/ edges, then all the vertices which can
be reached from x\ in t or fewer edges are all distinct. Since every vertex of Gin)
has valency s3a simple argument shows that in t steps we can reach at least
vertices if t ~-
as stated.
log n
Jog2.
I + 3 + ...+32'-1^2'+[ >«
Thus G(n) contains a circuit of length not exceeding 2
log n
Jog2.
299 CIRCUITS

10
P. ErdSs and L. Posa
than
From (6) and Lemma 3. our graph contains a circuit of length not greater
log n
Jog~2
log (2c2k log k)
. I^g~2 .
: \c2 log k
for sufficiently large c2. If we omit the edges of this circuit we obtain a graph of
at most n vertices and more than
n + [c2k log k] —— log k > n + c2 (k — 1) log k
edges. By our induction hypothesis our new graph contains k—\ weakly
independent circuits, thus together with our first circuit we have our required k weakly
independent circuits, which completes the proof of the right side of (5).
To prove the lower bound in (5) we need
Lemma 4. There exists a constant c3 >0 so that for every m there exists a G2„
which contains no circuit of length less than c3 log m.
The proof of the Lemma is implicitely contained in a paper by Erdos [3], but
for the sake of completeness we give it here in full detail.
Consider all graphs of m labelled vertices having 2m edges. The number / of
these graphs clearly equals I \ J I. Denote these graphs by Gx, ..., Gt and denote
\ 3m/
by/(G,) the number of distinct circuits of length not exceeding [c3 log w] contained
in G;. We are going to estimate
M =
r IRGd
from above. A simple combinatorial argument shows that the number of graphs G;
which contain a given circuit of k edges equals
(7)
V 3w — k J
The number of circuits of length k is clearly less than
(8) *!('?H"A-
Thus from (7) and (8) we obtain by a simple argument
k
1 [c, logm] I { ^-)
(9)
M-
] [r,logm]
T J3 m\3m-k
[c3log m]
y nr
A-= 3
In
(")
(3w-I)
[(")-']
...(3m-
(m\
U.)
-k + l)
-k+\
[c3log m]
k= 3
-■m'1
300 GRAPH THEORY

On circuits of a graph
11
if c3 is a sufficiently small absolute constant (again no attempt is made to get a
good estimate for c3 since as in the previous cases there seems no hope at present
to obtain the best possible value for c3). From (9) we obtain that at least one of
our graphs — say GL — contains fewer than m1'2 circuits of length -= [c3 log w].
Omitting one edge from each of these circuits we obtain a graph of m vertices and
more than 3m — m"'2 s 2m edges which contains no circuit of length less than c3 log m,
which proves Lemma 4.
Assume first k>k0 and let cx >0 be a sufficiently small absolute constant
and put m = [cjc \ogk]. Then by Lemma 4. there is a GT,n which contains no circuit
of length <c3 log m. Therefore our Gzm contains at most
2m
1 <k
c3 log m
weakly independent circuits, if c1 is sufficiently small. On the other hand our graph
has 2m = m + [clk logk] edges, which completes the proof of the left side of (5)
for k>k0. But clearly for 2^k^k0 g(k)^g(2)=4, thus if ct is sufficiently
small (5) holds for all k^2 and thus Theorem 4. is proved.
If we have already constructed a Gm+[c,fciogA-] which does not contain k
weakly independent circuits, we can construct such a G„ + [Cl*iog*] for every /7=-/77
by adding a path of n —m new vertices and edges to our GmliCrkiogkj ■
Finally we consider the following question: Let m^n and consider a graph
G"n\ Define h(G'm) as the length of the shortest circuit of our G,^. Put
Ch
/(/?, m) = max h(G
where the maximum is taken over all graphs G^\
Trivially/(/?,/?) =n and it is not difficult to show that
/(/7,77+1):
2/7 + 2
The determination, or even the estimation, of/(//, m) for general n and m seems
a difficult problem.
Theorem 5. Put m = n + d, J>I. Then we have
,,m fl , ,, (n+d) log d
(10) /(/7, /7 +*/)< C4 ^
and to every constant C>0 there exists an A(C) depending only on C so that
01) f{n,n+d)~AiC)^W.
(11) shows that for d-^Cn (10) gives the correct order of magnitude for /(n, m).
301 CIRCUITS

12
P. Erd6s and L. Posa: On circuits of a graph
From Theorem 4. every G„"+d contains c5 d/log d weakly independent circuits,
thus at least one of them has length not exceeding
{n+d)\ogd (rr+d) log d
^d " °A d
for sufficiently large cA, which proves (10).
n
We shall only outline the proof of (II). Assume first -r^.d<Cn. By the same
method as used in Lemma 4. we can construct a Gn+d the smallest circuit of which
has more than c6 log n edges (cb depends on C and tends to 0 as C tends to infinity),
which implies (II) by a simple calculation.
Assume next ¢/-=—. By Lemma 4. there exists a GS2 all circuits of which have
¥1
length ^ c3 log d. Put on each edge of this graph —7- — 1 vertices of valency 2.
Thus we obtain a graph of m = n vertices and m + d edges the smallest circuit of
which has length not less than
(12)
n
id
1 1 , , (n + d) log d
- 1 )Cj, logf/>c7 '- ^-.
By adding a path of n — d new edges and vertices to this graph we obtain a
Gn% the shortest circuit of which satisfies the inequality (12), thus (II) and
therefore Theorem 5. is proved.
It would be easy to strengthen Lemma 3. as follows: Let C-*°° then there
exists an ec which tends to 0 as C tends to infinity so that every G[c„^ contains
a circuit of length less than cc log n, but we are far from being able to determine
the exact dependence of cc from C.
Bibliography
[1] P. Erd6s and T. Gallai, On maximal paths and circuits of graphs, Ada Maih. Acad. Sci. Hun-
gar. 10 (1959), 337-356.
[2] G. Dirac, In abstrakten Graphen vorhandene vollstandige 4-Graphen und ihre Unterteilun-
gen, Math. Nachr. 22 (1960), 61-85.
[3] P. ErdOs, Graph theory and probability, Canad. J. Math. 11 (1959), 34-38.
(Received September 30, 1961.)
302 GRAPH THEORY

ON THE MAXIMAL NUMBER OF INDEPENDENT
CIRCUITS IN A GRAPH
By
G. DIRAC (Hamburg) and P. ERDOS (Budapest), member of the Academy
1. Introduction
In a recent paper [I] K. Corradi and A. Hajnal proved that if a finite graph
without multiple edges contains at least 3Ac vertices and the valency of every vertex
is at least 2k, where A: is a positive integer, then the graph contains k independent
circuits, i. e. the graph contains as a subgraph a set of A: circuits no two of which have
a vertex in common. The present paper contains extensions of this theorem. In a
recent paper [2] P. Erdos and L. Posa proved, among other things, that if a finite
graph with or without loops and multiple edges contains n vertices and at least
n +4 edges, then the graph contains two circuits without an edge in common. The
present paper contains analogous results for planar graphs.
We adopt the following notation: Ok denotes a graph consisting of Ac independent
circuits, kO denotes a graph consisting of Ac or more circuits no two of which have
an edge in common. If (^ is a graph then ^((^) denotes the set of vertices of (|, ^,-((^)
denotes the set of vertices of & having valency i in (^ (i being a non-negative integer),
^9¾ i ((?)> ^95 i (^) denote the set of vertices of 6j having valency tk i and ^ i,
respectively, and $ ((|) denotes the set of edges of iS. The valency of the vertex x in the graph
q will be denoted by v{x, 6f). |^(^)| will be denoted by V(§), |<?(6})| by £((|) etc.
In this notation the theorem of Corradi and Hajnal quoted above states
that if i| is a finite graph without multiple edges and if V(§)^3k and ~fs2k-1(^) = 0,
then £^3 Ok; and the theorem of Erdos and Posa quoted above states that if S
is a finite graph and E(§)^ V(§) + 4, then i|d2o.
2. Concerning the existence of two independent circuits
in finite graphs without multiple edges
Theorem I. Let <3 denote a finite graph without loops or multiple edges.
I. 7/K((f)s6, Vs2((%) = 0, and KS4(<^)S4, then (§3 O2.
II. If ¥{§) = 7 and VSA(fy ^ 6, then § 3 O 2.
III. If F(6|) = 8, Fs4(i2)s6, and if § does not contain a vertex having valency
4 joined to two vertices having valency I then § 3 O 2.
IV. IfV(q)^9 and V^A{fy SiF(^) + 2, then^O1-
V. IfV(q)*9 and VSA(§)-Vs2(§)^4, ///en <^3 O2-
303 CIRCUITS

80
G. D1RAC AND P. LRDOS
VI. If V {§)=!, VSA{q) = 5, and V1{§)=Vi{q) = \, then $-=> Z2.
VII. If K,4(6f) = 5, K3(6f) = 2, and Ka2(Oj) = I, then q^ O2.
Definition. A graph barely satisfies the conditions of I, III, IV, or Theorem 3,
if it satisfies the conditions of I, III, IV, or Theorem 3, respectively, but when any
one of its edges is deleted the remaining graph no longer does so.
It is easy to see that
(1) If a graph satisfies the conditions of I or III or IV then it contains a subgraph
having the same vertices which barely satisfies them.
Proof of I. Suppose first that K(of) = 6. Let the vertices of of be denoted by
g,,g2, ..., g6. The following two alternatives will be considered separately: (i) of
contains two vertices of valency £4 not joined by an edge, (ii) Each pair of vertices
of valency =4 are joined by an edge.
Assuming that (i) holds, it may be supposed that i'(g,,^)s4, t(g2,^f)£4
and (g1,g2)<iq- Then gL and g2 are both joined to each of g3, g4. g5, g6.
^0(4-^1-^2)=0 because Kfi2(of)=0, and Ke2(of -g1 ~g2) s2 because
K_.4(of) = 4; it follows that § —gL — g2 contains two edges without a common vertex,
and hence of— 02.
Assuming that (ii) holds, it may be supposed that g|,g2,g3,g4 each have
valency g4. Each of them is joined to at least one ofg5 and gh and g5 and gb are
each joined to at least two of gt, g2, g3, gA. The three alternatives that g5 is joined
to exactly two, three or four ofgi, g2, g3, gA will be considered in turn: If (g5,gi) Gof,
Us-gzK^h (gs,g3)ri^ and (#5, £4)^ then (g6,,?3)€of and (g6,g4)6of, so in
this case & contains the two independent circuits [gi, g2, gs] and [g3,g4.-. gtl- 'f
(£s-£iK^ (85-82)^^, (g5,g3)tq and (g5,g4Mof, then (gb,gA)^q, and it may
be supposed that (,?6.giK^f (f°r y(&6> ^) — 3), in this case of contains the two
independent circuits [g2.g3,g5] and [gl,gA,g6]. If (g5, giKSi for /=1,2,3,4,
then it may be supposed that (g0,giKef and (g(,,g2)^q (for c(gb, of) ^3) and in
this case of contains the two independent circuits [g3, g4. g5] and [gl, g2, g(l. Hence
o]3 O2 if (ii) holds. So I is true in the case K(of) = 6.
The proof of I will now be completed by reductio ad absurdum. Assume that 1
is untrue. Then by (1) there exists a graph 6f0 with the following properties: of0
barely satisfies the conditions of I and of0 ^) G2, and all graphs which satisfy the
conditions of I and contain fewer vertices then of0 contain two independent circuits.
K(oT0)^7 because 1 is true for graphs with six vertices.
It is easy to see that
(2) If a graph barely satisfies the conditions of I then at least one of the end-
vertices of every edge has valency 3 or 4, and at least one Vertex has valency 3.
304 GRAPH THEORY

ON THE MAXIMAL NUMBER OF INDEPENDENT CIRCUITS IN A GRAPH
81
Let d denote a vertex of 6^ having valency 3, and let dx,d2, d3 denote the
three vertices of £j0 to which d is joined. The three alternatives £(6j0 (dt, d2, d3)) ^ 1,
E(£\0(d{, d2, d3)) — 2, E^oidi, d2,d3)) = 3 will be considered separately.
Assuming that E(§0(di, d2, d3)) ^ 1, it may be supposed that (dl, d3) 16j0 and
(d2, d3) i §0 . Let i|' = (6j0 — d) U (dl, d3) U (d2, d3). ej" satisfies the conditions of I.
Hence 6^3 02 because of the minimal nature of ij0. It follows that 6j03O2.
((di,d3) can be replaced by dtj (d, d{)j{d, d3) if necessary, alternatively d3'<j
U (t/i, d3) U (d2, d3) can be replaced by dU (d, dt) ij (d. d2) if necessary. This argument
is used in [2].) This contradicts tJ01)O!.
Assuming that E{6ji0{dl, cl2, d,)) = 2, it may be supposed that (dl,d2)£§0>
(dt, d3)d^0 and (d2,d3)$^0. There are two alternatives: v(d1,(S0)^5 and
i)(rfi,(^0)S4. If v (dl, 6^,) S 5 then (<?\0—d)U(d2, d3) satisfies the conditions of I,
and therefore contains two independent circuits because of the minimal nature of
0j0; it follows that §0 3 O2 (replacing (d2, d3) by dU(d, d2)U(d, d3) if necessary;
this argument is used in [2]), in contradiction to <^0 ■$> Q2. If v(dt, ^0) = 4 then a
contradiction is arrived at as follows: 6f0 — d—dl — d2—d3J) C , so if it has v
connected components then it contains at most K(£^0) — 4 — v edges. Hence £(6j0) =
=1 V($0) -4-v +v(d2, §0) +v(d3, 6j0) + 2 (since i-(c/,, ^)==4). On the other hand,
summing the valencies of all the vertices of §0, 2E(§0)^v(d2,§0)+v(d3, £^0)^-
+ 8 + 3(K(6j0)-4). From the two inequalities it follows that V(§0)^v(d2, §0)-
+ v(d3, 6j0) —2v. But d2 and d3 are each joined to at most one vertex of each
connected component of >S0 —d—d1—d2 —d3 because <S03>O2, so v{d2, 6j0) £ v +2
and v(d3, 6^0) = v + 2. Hence K(6j0)^4 which is a contradiction.
The only remaining alternative is that Ei^^d^, d2,d3))—3. By (2) it may be
assumed that (:(^,6^) = 4 and v(d2,6^,)^4. §0—d—dl—d2—d3J>C so
E(q0~d^-dl-d2-d3)^V(q0)-5. Consequently E{§0)^V(q0) + b(d3,qo). On
the other hand, summing the valencies of all the vertices of 6^, 2£(6j0) = v (d3, (6J0) +
+ 12 + 3(K(6j0)-4) = 3K(6j'o)+y(£/j,6jo). From the two inequalities it follows
that t(d3, tfo)— V(G0), which is absurd because 6|0 contains no loops or multiple
edges.
The hypothesis that I is untrue leads to a contradiction, therefore I is true.
Proof of II. If K£2(6j) = 0 then 6,3 O2 by I. If K^2(6j) ^0 then K-=2(6J) = 1.
Suppose that Vs2((S)=l and let v denote a vertex having valency ^2. Then
K(Cj-u) = 6, Vs2{§-v)=0 and Ka4(6j-u)="4. Therefore Cj-U302 by I.
Proof of III. By (1) in order to prove III it is sufficient to prove it for graphs
which barely satisfy its conditions. Let S denote a graph which barely satisfies the
conditions of III.
(3) K3 3(^)=1.
305 CIRCUITS

82
G. D1RAC AND P. ERDOS
For if K53((|)=0 then the graph obtained by deleting any one edge from (3
satisfies the conditions of III.
IfV0(q)^l ther,qz>02.
For if a is a vertex of i| having valency 0 then K((| — a) = 7 and V^A(§ — a) ^6,
so fi — a 3 O2 by II. In the remainder of the proof of III suppose that Ko(6?) = 0.
If V2{§) ^ 1 then it follows that 6j 3 O2.
For suppose that c is a vertex of 6J having valency 2 and is joined to the vertices
ct and c2. Either (ct, c?2) $ i| or (c1; c2)€^. If (c1( c2)$(| then let (3/ = (^-c)U
U^i.Cj). ¥(§')=! and K34(i|)s6, consequently Cj'=>02 by II. It'follows
that Sz)02. ((Ci, c2) can be replaced by cU(c, ct) U(c, c2) if necessary. This
argument is used in [2].)
Suppose that (c1,c2)£i|. If v(ct, ^)S5 and u(c2,^)s5, then Ke4((|-c) =
= K?4(6j:)s6 and V{§-c) = l, so ^-£3 O2 by 11. There remains the alternative
that v(ct, 3,) = 4 or v(c2, ^) = 4, let it be assumed that v(clt ^)=4. It follows that
0fz)O2. For suppose on the contrary that (|:p Q2. Then £(£| — c —ct — c2)S4
because S^-c-c^ -c2 £ O- Hence £(¢£)^4+1)(^, <|)-2 + u(c2,1¾)-2 + 3 =1
^7+o(c2, (3). On the other hand, summing the valencies of all the vertices of 3,
2£(£j)&5-4+u(c2,6j!)+2 = 22+v(c2,§) since u(c, 6j!) = 2 and K0(cj)=0. From
the two inequalities it follows that v(c2, <Sp = 8 which is absurd because (| contains
no loops or multiple edges. Hence 33 O2 if V2(<3) = 1. In the remainder of the
proof of III suppose that K2((^)=0.
//" K3(6j)Sl tfjen it follows that i|3 O2.
For suppose that d is a vertex of (3 having valency 3 and is joined to the vertices
dltd2 and d3. The two alternatives E(^3(d1, d2, d3))^l and £(3(^,^,^)) = 2
will be considered separately.
Assuming that E(<q(dL, d2, d3))^l, it may be supposed that (d^d^fy^ and
(d2,d3)$G. Let (3/ = (£j - J) U (^, d3) U (d2, d3). Clearly ¥(§') = ! and K»4((^)S
i= Kg4(6j) i= 6, so (|' 3 O 2 by II. It follows as in the proof of I that (| 3 O 2.
Assuming that £(3(^,^2,^))^2, it may be supposed that (rfi,rf2)€<3 and
(dlt d3)^q. If any edge incident with d is deleted from (% then the remaining graph
does not contain two vertices of valency 1, therefore the remaining graph contains
only five vertices of valency £4, since <3 barely satisfies the conditions of III.
Consequently v(d1,(2i) = v(d2,(3i) = v(d3,i3i) = 4. It follows that (|3 O2. For suppose
on the contrary that <Sf O2. Then E(S — d—d1 — d2—d3)^3 because c| — d—^ —
— d2 — d31p O. Hence £(6^)^3 + 5 + 5 = 13. On the other hand, summing the
valencies of all the vertices of 6j, 2£((|)s;6-4 + 3 = 27, since v(d,(3i) = 3. This
contradiction proves that 6j 3 O2 if K3(6p^l.
It thus remains only to consider the case in which K;(3) = 0 for / = 0, 2 and 3.
By (3) there are then two alternatives: KS4(<|) = 7 and V1(<S)=l, or else Ks4((|) = 6
and K,(Cj) = 2.
306 GRAPH THEORY

ON THE MAXIMAL NUMBER OF INDEPENDENT CIRCUITS IN A GRAPH
83
Assume first that K^4(Cj) = 7 and (/^)= 1. Let 6 denote the vertex of i| having
valency 1. Then V(§-b) = 7 and Kg4(i|-<b) =-6. Therefore by U q^bz^Q2.
Assume next that Kj,4(6|) = 6 and V^) = 2. Let b and b' denote the vertices
of fi having valency 1. Either b and 6' are both joined to the same vertex, or they
are not. If b and 6' are both joined to the same vertex bt say, then it follows from
the conditions of ]11 that v(b1, (3) = 5. Therefore S — (b, bt) satisfies the conditions
of III, in contradiction to the definition of q. Hence b and b' are not both joined
to the same vertex. Therefore K((§ - ft - ft') = 6, Vn2(q -b-b') = 0 and
Ka4((^-ft-ft')S4. Consequently by 1 q~b-b' 3 O2. Ill is now proved.
Proof of IV by reductio ad absurdum. Assume that IV is untrue. Then by (1)
there exists a graph S0 with the following properties: S0 barely satisfies the conditions
of IV and i^0JO2, and all graphs which satisfy the conditions of IV and have fewer
vertices than (30 contain two independent circuits.
(4) At least one end-vertex of every edge contained in S0 has valency 4, and
For, since 30 barely satisfies the conditions of IV, if any edge is deleted from
qo the number of vertices of valency £4 is decreased; and if ^3(^0)=0 then if
any edge is deleted from <20, the remaining graph satisfies the conditions of IV,
which contradicts the hypothesis that S0 barely satisfies the conditions of IV.
By summing the valencies of all the vertices of (^0 we have that
(5) If Ko((fo)=0 then £(<20)Sl±K((f0) + 3, if Ko((fo)=0 and K2(i|0)^l, then
£((f0) = HK((|0) + 3|, and if V0(q0)=Q and K3(^0)Sl. then £(§0) = U K(^0) + 4.
It will now be proved that Vi(§0)=0 for / = 0, 2, 3.
(6) v0(q0)=o.
For suppose on the contrary that the vertex a of (|0 has valency 0. If V{qo) = 9
then K(Cj0 - a) = 8 and K§4((|0 - a) ^ 7, so by III §0 - a 3 O 2, contrary to 6f0 :p O 2
If K(S0) = 10, then S0 — a satisfies the conditions of IV, and so (|0 — a 3 O 2 because
of the minimal property of (|0 whereas (|0JO2. Hence K0(6j0) = 0.
(7) K2(Cf0)=0.
For suppose on the contrary that the vertex c of (50 has valency 2 and is joined
to the vertices c1 and c2. The two alternatives (c1,c2)$qo and (c1( c2)€(fo will
be considered in turn and a contradiction will be derived in each case. If (Cj, c2) ¢¢3,,
then let (3/ =(§„ -c) \J(cit c2). Clearly KB4«3/) = K£4((f0). If K((f0) = 9 then
K((|') = 8 and Ke4(<2/)=7, consequently ^O2 by III, if K(Cj0) = 10 then ¢3/
satisfies the conditions of IV, so (3/ 3 O 2 because of the minimal property of (|0.
From i|'3 O2 it follows as in the proof of III that <|0 3 O2, whereas (|0 ^ O2.
Suppose next that (ct, c2)€(|0. By (4)^(^, (^0) = ^(^2, ^o) = 4- £(<|o ~c~ ci ~c2) =
307 CIRCUITS

84
G. D1RAC AND P. ERDOS
SK(C|0)-4 because (ij0 -c-c^^ — c2 t> C It follows that £(iij) S K(\|0) + 3, which
contradicts (5). Hence K2((^0)=0.
(8) K3(fr]o) = 0.
For suppose on the contrary that the vertex d of i5j0 has valency 3 and is joined
to the vertices d1,d2 and d3. The two alternatives E(^0(d1, d2, d3))7^l and
E(6^0(d1, d2, ¢/3)) = 2 will be considered in turn and a contradiction derived in each
case.
Assuming that E(6l0(dt, d2, d3))^l, it may be supposed that (dlf d3)§§0
and (d2)rf3)€Cjo- Let cj''= (if0 - d) U (rft, d3) U (rf2, rf3). Clearly K,4«f)£ K£4(^0).
If K(\5,0) = 9 then K(6,') = 8 and K»4(6j')S7, and consequently ifj" 3 O2 by 111;
if K(50)5l10 then 6" satisfies the conditions of IV, and consequently 6|' 3 O 2
because of the minimal property of ^0 . From 6j" 3 Q 2 it follows as in the proof
of I that oj,03 02, whereas ej^^O2. There remains the alternative that
E^id,, d2, rf3)) s2. By (4) «;(<*!, <§<>) = v(d2, §„) = v(d3, $,) = 4. £(^0 -d-d,-
-d2-d3)^V(§0)-5 because c]0-d-^-t/2-t/3 :£ O- It follows that £(e]o) =
sk(C|o) + 5, which contradicts (5). Hence K3(^o)=0.
From the conditions of IV it follows that V^J§0)s^7 and KS4(e]0) - Ks3((^0) S
£4. By (6), (7) and (8) K(^0) = K^o) + Kg4(i|0) and by (4) each vertex of valency
1 is joined to a vertex of valency 4. Let iij" denote the graph obtained by deleting
all vertices of valency 1 from C,. Clearly K(Cj") = V^A{S}0) a 7, Ke4(ef") = Ke4(C]0) -
-Ki(6|0)s?4 and Ks2(^") = 0- Therefore by I 6|"3 02, whereas cJ0jO2. This
contradiction proves IV.
Proof of V. If K3(ef)=0 then K(e])= Ks2(c]) +KS4(ej) and so V-^ifys
= i '/(^|) + 2; hence by IV iJjQ2. Suppose that K3(Cj)^l, and let *,,...,*„
denote the vertices having valency 3 in cf. Let j^, ..., yu be w distinct vertices none
of which belong to e^, and let ^j* = iij U {j,, ..., >'„} U (a-! , y{) U ... U (xu, >•„). Clearly
K(^*)= K(^+w= K(l^)+ K3(cf); K^*) = Ks2(e,)+ K3(§); KK4(cf) = Kg4(§) +
+ K3(CJ). Hence KS4(^*)is Ks2(e]*)+4, since K=4(Cj) =? Ka2(6j)-f- 4. Also K3(e]*) = 0.
It follows that K§4(^*) ^i K(c]*) + 2. Hence by IV e,* 3 O2, consequently cj 3 O 2.
This proves V.
Proof of VI. Let the vertices of l^ be gt, g2, ..., g7, where t-(g1,^,) = 2 and
y (#2, ^) = 3. Let g8,g9,gio be three distinct vertices not belonging to (3 and let
e]° = Lfu{g8g9g10}U(g1,g8)U(g2,g9)U(g1,g10).ThenK(Lf)=10andK34(^)=7,
so c]J 3 O 2 by IV, and consequently cJ^O2.
Proof of VII. Construct § + from ^ by adding a new vertex not in t| and
joining it by an edge to a vertex of V3(§). Then KS4(i^+) = 6, K3(i^+) = 1, and
^==2(4^)=2. Hence by V 6}+ 3 O2, consequently i^DQ2.
Remarks concerning Theorem 1. 1. IV and V are equivalent. For V has been
deduced from IV, and IV clearly follows from V.
308 GRAPH THEORY

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85
2. If K(Cj') = 9 then I is a consequence ofW.
3. I, ..., VII are best possible. In order to demonstrate this we define the graphs
s/,st',%,<£. and ® as follows: cf(st) = {x1,x2, x3, ylt ..., yu} (w^3), <f(j^) =
= ((^,^)(^,^3)(^3,^)(^,^)} (/=1,2,3; j=l,...,u). *f(s/') = "V(jf)\J
U{Zl...z„}, (f(^') = ^(^)U{(j1, z,)...(yu,zu)}. °?m = {fif2-f6}, <?(«) =
= {(fi,fj)}V(f6,fi)V(f6,f5)-(fi,fs) 0=1,-, 5; 7=1,.-,5). f(£)=T(#) +
+/7, *(£) = *(#)+ (A,/,)- T (^)=^(^+/8, ^(^)=^(^) + (/,,/8). K(J/)^
&6, K^2(j^) = 0,and K,4(ja0 = 3 but ^ J O 2. K(j^')=9, Ka4(j^") = i V(^') + H
but J*" JO2. V(S) = 6, K4(^) = 5and K2(^) = l, but JJO2. K(<2) = 7, K4(S) = 5,
K3(<2)=1 and ^(£)=1 but £¢02. K(^) = 8, K4(^) = 6, ^(^)=2 but 3 JO2.
J?/, J1, l? and LfT show that the conditions of I cannot be relaxed. (2, ^ and
stf', respectively, show that the conditions of II, III and IV cannot be relaxed, stf
and stf' show that the conditions of V cannot be relaxed. <B and <3\ respectively,
show that the conditions of VI and VII cannot be relaxed.
For finite graphs without multiple edges which contain at least one loop a much
weaker condition ensures the existence of two independent circuits.
Theorem 2. If 8 is a finite graph containing at least one loop but no multiple
edges, and if V(§) & 3 and K£4(^) = Ke2(iS,), then i|jQ2.
Proof. Let m denote a vertex of iff which is incident with one or more loops.
If 8j— m contains a loop, then (^z> O2. In what follows suppose that iij — m contains
no loop. Clearly Ki3(cj — w)s K_^4(^)—1, and Ksl(e] — m)^ Ks2(^)- Therefore
K-sG-j -'«) + 1 = ^W^j) = ^2(^) ^ ^siO-j - 'n)- Als« V(G - w)& 2. It follows that
ij-mD 3, for if 3f is a finite graph without circuits and with at least two vertices,
then K£1(?0& K53("Jl') +2 as may easily be verified. Hence cjz) O2.
Remarks. 1. The conditions of Theorem 2 cannot be relaxed. The example of a
graph consisting of a vertex incident with four loops joined to a vertex of valency 1
shows that the condition V((8) = 3 is essential. The example of a graph consisting
of a vertex incident with at least two loops joined to every vertex of a path shows
that the condition Ka4(^)^ K32(i^) is essential.
2. No result analogous to Theorems 1 and 2 can be obtained for graphs which
may contain multiple edges. This is shown by the example of a graph consisting of a
vertex incident with any number of loops joined to the vertices of a path by any
number of edges.
3. Concerning the existence of three or more independent
circuits in finite graphs without multiple edges
Theorem 3. If 8^ is a finite graph without loops or multiple edges and k is a
natural number s 3, and if Ks2t(u) — K:;24- 2(^) = k2 + 2k — 4, then 0j z> O'.
N. B. Theorem 1. V states that this is true for k = 2 if K(iSj)^9.
309 CIRCUITS

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G. DIRAC AND P. ERDOS
Proof of Theorem 3 by reductio ad absurdum. Assume that Theorem 3 is
untrue. Let «; denote the least value of k for which the assertion of Theorem 3 is
false, and among the graphs (| such that K?2l((^)- ^2,-2(^) = ^2 + 2* — 4 and
§~X> Oy let i|0 be one with the least possible number of vertices and barely
satisfying the conditions of Theorem 3 with k = x. This will be shown to lead to a
contradiction by a method amounting to an induction process starting from the particular
case of the theorem of Corradi and Hajnal with x2 + 2x — 4¾ V(§)^x2 +2k —3
(note that x2 + 2x - 4 5 3x + 2 if x & 3).
(1) At least one of the end-vertices of every edge of(S0 has valency 2x or 2x— 1.
For, since £J0 barely satisfies the conditions of Theorem 3 with k = x, by deleting
any edge from i|0 either Vs2y{§0) is decreased or Vs2x~2(^jo) is increased.
(2) ^2,.,(^)51.
If K((^0)s?x2+2?<:-2 then ^2,-1(^0) = 1 because cj0 barely satisfies the
conditions of Theorem 3. If x2 + 2x-4^ K((|0)S x2 +2x.- 3 then Ks2x-i(^o) = l
can be deduced with the help of the theorem of Corradi and Hajnal : if Ks2z-i(^o)=0
then, since x2 + 2x — 4^3% + 2 because ^^3, fi0DO', which contradicts cj0 £ O*.
(3) Ko(^o)=0.
For if a is an isolated vertex of i^0 then Vs2y.(§o ~a) ~ ^2^2(¾ ~~a) >
> x2 + 2x — 4 and (^0-«J O", which contradicts the minimal property of (^0.
(4) ^(^0)=0.
For if <b is a vertex of 6^ having valency 1 then V'* 2x(^o — b) - Ks 2y - 2(^0 — *) =
^*2+2k—4 and i^o-^JO*, which contradicts the minimal property of S0.
(5) £Jo contains at least one triangle.
For suppose on the contrary that §0 contains no triangle. By (2) 6^ contains
a vertex of valency ^2*: —1, let e denote such a vertex and let c(e, §0)=c; by (3)
and (4) 2^u^2« —1; let e1; ..., e„ denote the vertices of c|0 to which e is joined.
No two of el, ..., e„ are joined by an edge because ^0 contains no triangle by
hypothesis. Let e{^{q0-e) + {ev,ev^l) + ...+{ev,el). Clearly V*2J,§') 5 Kft2x(iSjV> and
^^-2(^)=^^2,-2(6,0), so Ka2x(^')-Ks2x_2((5')s«2+2x-4, hence J-O"
by the minimal property of §0. It follows as in the proof of Theorem 1.1, that i|03 C,
whereas §0 :£ Ok. This contradiction proves (5).
Let xl, x2, ,v3 be the vertices of a triangle contained in §0 and let i|" = fi — x, —
- x2 - x3.
(6) fJO-1.
Because £j 0 i> C "•
Let 5 denote the set of those vertices of S" which are in c|0 joined to all three
vertices x,, x2 , A'3.
310 GRAPH THEORY

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(7) |s|==2x-2.
Because by (1) at least two of x,, x2, x3 have valency ^2x in 3^0.
It is easy to see that
(8) f*2x(q0) ets2«-2«?*) ^1 (snt2k-3(6|")) u {,x1 a-2,v3}
and
(9) tfi2x_2(^0) 5 *p s2v_ 4((f) - (. n T2,_4(Lf)).
From (7), (8) and (9) and the conditions of Theorem 3 it follows that
(10) K£2K-2(^")-^W4(f')S
SKe2^0)-Ka2x-2(iJo)-2x-lS(«-l)2+2(K-l)-4.
It follows from (10) that * = 3, because if k^4 then by (10) and by the minimal
property of x, §" zj Q"1 contrary to (6). If x= 3 then K(^0)= H from the conditions
of Theorem 3 with fc=3. The two alternatives V(30) — ll and K(^0)—12 will be
considered in turn: If V{3\0) = H then Ks5(^0) = 0 from the conditions of Theorem 3
with k = 3, hence in this case, by the theorem of Hajnal and Corradi [I], 3[0 3 C 3
contrary to hypothesis. If K(C,0)=12 then K(^")&9, and by (10) K?'4(c}")-
~~ ^s2(^") —^, so in this case, by Theorem 1. V, c^"3 O2, which contradicts (6).
The assumption that Theorem 3 is false therefore leads to a contradiction.
Remarks. 1. Theorem 3 is probably not best possible. On the other hand if
Vs2ki^) - V^2k-1^=12k-\ then it is possible that ^JO' whatever the values
of/r(s3) and of V(fy. For example let T(6j) = {xl,'..., x2k-,, >-,, ...y„, -^....-,,}
(u^Uk+l)) and ^) = ((^,^),(.^,,),(^,.-,,)} (/=1,...,2^-1; ./=1,....
2* - 1; i */; // = 1,...,1/). Kft24($ = i K($ +,-(2*- 1).
2. It is worth noting, that in the proof of Theorem 3 only the particular case
of the theorem of Corradi and Hajnal in which k2 + 2k — 4 ^ K(cj') ^ k2 + 2k — 3
was used. By a simpler method not using the theorem of CorrAdi and Hajnal
at all the weaker result can be proved that corresponding to any integers k ^2 and
c^ 1 there exists a number n(k, c) such that if the graph 3 satisfies the conditions
V,.2^,(^)^0 and V{3)^n(k, c), then ^3 O4.
In the case of finite graphs without multiple edges which contain loops much
weaker conditions than those of Theorem 3 ensure the existence of k independent
circuits, as Theorem 4 shows. In any concrete case it is best to delete all vertices
incident with loops (together with all edges incident with at least one of them) and
apply Theorem 2 or Theorem 3 to the remaining graph however, because Theorem 4
takes into account all the most unfavourable situations.
Theorem 4. Let 3 be a finite graph without multiple edges and k a natural
number ^3. If exactly k— 1 vertices of § are incident with loops, and K(^) = A:+1
and K.»t + 2((ij) — V^k{3\)^k — 2, then 3^ Z> Ok. If exactly I vertices of 3^ are incident
with loops, where l^k-2, and K (6))^/ + 9 and Vf_2k-i(^) - V~2k-1-2(^ =
s(k~l)2 +2k-1-4, then ^zjO".
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G. DlRAC AND P. ERDOS
Proof. Let L denote the set of those vertices of § which are indicent with loops.
Suppose first that \L\=k-l, V(^)^k + l and V*k+2(§) - V^k{§) ^k -2. Then
clearly Vs3(q~L)^Vsk + 2($)-k+ land Vs ,($-£)==» Kgt($. Hence Ks 3(e]- L)-
-^-=1(^-^)-+1, consequently, since Vs\(§— L)^2, t| — LdQ. Therefore
4 - J ■
Suppose secondly that \L\=l^k-2, V(^)'^l + 9 and Vs2k-i(§)-
-V^2k-,-2(q)^(k-2)2+2k-l-4. Clearly K£2,_2,(c] - L) s K£2,_,(6J) -/ 'and
^-21-2(^-^)5^.,-2(¾). Hence KB2,t_0(i§-L)-K^^-^-L)^
S(>t — /J2 + 2(/t — /) — 4. It follows by Theorem 1. V (V(§-L)^9) or Theorem 3
that cJ-Lz) Ok~'. Therefore Cj => 3*.
4. Concerning the existence of independent circuits in
finite planar graphs without multiple edges
Theorem 5. Let § denote a finite planar graph without loops or multiple edges.
1. If K(Cj)a6, Vr,2(^)=0 and l"34(ej)^2, then $^02.
11.// K(J,)^8 and Ks4(3) ==1 K(cJ) +'l|, f/jen cj=)32.
111. // K(6,)s8 aw/ K?4(4)-Ks2(Cj)^3 Men §'dO!.
7/" K=.2t(-ij)— 13 24-2(Cj)s 5/:-7 n7;er<? £: /5 a« integer ^3, ///<?« Oj 3 3*.
Proof of I. Suppose first that K(Oj') = 6 and that cj barely satisfies the conditions
of I. Then clearly K3(^)S 1. Let ^(^)=-(^,^---^6} ar»d suppose that v(gl, cp = 3
and that g, is joined to g2 , g3 and g4 in oj\
(1) (8,,8,)^-
(2) f(g5,C|) = 3 o/- c(g6, cj) = 3.
(3) £(e,(g2,g3,g4))^l.
For if (1) or (2) or (3) were untrue then (g,-, #,-) £^ for / = 2, 3, 4 and 7= 1, 5, 6
and so £ would not be planar.
The three alternatives £(c7(g2, g3, g4))= 1, 2, 3 will now be considered in turn.
Assuming that E(^(g2, g3, g4)) = 1 it will be supposed that (g2, g3) 3i}- Then
(?4,.?5) €^j and (g4,^)^^- '" this case by (1) ep [gt, g2, g3] U [g4, g5, g6].
Assuming that if(^(g2, g3, g4)) = 2 it will be supposed that (g2,g3)3i) and
(82, 84) £ 4 • Then either u (g3, S{) s= 4 or u(g4, 3) §•: 4, because if v(g3, cj) = u (g4, cj) = 3
then r(g5,iip§:4 or i'(g6,cpS4 since K^4(cj)^2; and if e. g. u(g5,cj)^4 then
(#3,^5), (g4.?s) ¢^ so (g3,ge), (^geK^j, therefore y(g6, ^)==2 whereas
V~2(^)=0. If e. g. y(g4,ej)^4thenC|=)[g1,g2,g3]U[g4,g5,g6].
Assume finally that £(^(g2, g3, g4)) ==3. At least one of g2>.?3,?4 is joined
to both of g5,g6 because v(g5,^)^3 and v(gb,^)^3; suppose that (g4,g5Kej
and (#4,^)^, then ^ z> [g,. g2 , g3] 3 [g4,g5, gj. I is thus true if K(cJ) = 6.
312 GRAPH THEORY

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89
The proof of 1 will now be completed by reductio ad absurdum. Assume that I
is untrue. Then there exists a planar graph i50 with the following properties: c^o
barely satisfies the conditions of 1 and cj0 :p Q2, and all graphs which satisfy the
conditions of I and have fewer vertices than ^0 contain two independent circuits.
V(&0)^7 because 1 is true for planar graphs with six vertices.
(4) At least one of the end-vertices of each edge of 0^0 has valency 3 or 4, and
at least one vertex ofS0 has valency 3.
Because S0 barely satisfies the conditions of 1 and V{S^0) ^7.
Let d denote a vertex of cj'o having valency 3 and let dl,d2, d3 be the three
vertices of S0 to which d is joined. The three alternatives E(^0(dt, d2, d3))^l,
E(S0(dl, d2, d3)) = 2, E(^0(dl, d2, c/3)) = 3 will be considered separately.
E(<50(dl, d2, d3)) ^ 1 leads to a contradiction exactly as in the proof of Theorem
1. 1.
£(^0(^15^2,^3)) = 2 and (dt, d2), (dt, d3)^Si0 and v (dY, ^0) S 5 leads to a
contradiction exactly as in the proof of Theorem 1. I. If v(dl, S0) ^4 then a
contradiction is arrived at as follows: \fS — d — dl — d2—d3 has v connected components, then
£(i50) — ^(^o)"^— v + v(d2, cj0) + fc(rf3, ej0) as before. On the other hand, summing
the valencies of all the vertices of^jV 2E(<Sj0)^v(d2, ^0) + v(d3, iij0)+ 3(K(cJ0) — 2).
From the two inequalities it follows that V(^0)^v(d2, ^0) + v(d3, Sj0) + 2 — 2v.
v(d2, i|o)S v + 2 and v(d3, 0j0) ^ v + 2, since d2 and d3 are each joined to at most
one vertex of each connected component of S,0 — d—dl —d2 — ¢/-, because S0 ~$> O2.
Hence V{S0)^6 which contradicts V(^0)^7.
The only remaining alternative is that £(c^0(rfi, d2, d3)) = 5. By (4) it may be
assumed that D(dl,i^0)^4 and D(d2,\^0)^4. 0^ 0 — d—dl—d2—d33)0 so
E(S0 — d-dl —d2 —d3)^ V(£0) — 4 — v where v is the number of connected
components of cj0 — d — dx — d2 — d3. Consequently £(6[0) ^ V(S,0) + v(d3l ^0) + 1 — v. On
the other hand, summing the valencies of all the vertices of^0> 2£(i^0)gi'(rf3, ^0) +
+ 4 + 3(K(^0)-2). From the two inequalities it follows that V(^0)^v(d3,S}0) +
+ 4 — 2v. But d3 is joined to at most one vertex of each connected component of
<Sj0—d—dl—d2— d3 because ^0 £ O 2, so v (d3, ^0) Sv + 3. Hence V(tf0) ^7-vl6.
But K(^0)S7. With this contradiction the proof of ] is complete.
Proof of II.
(5) If% is a graph without loops or multiple edges and V(X)=6, ^4.(¾) = 5-
and K2C3C) = 1, then % is not planar.
For suppose on the contrary that % is planar. Let b denote the vertex of "jC
having valency 2 and let bx, b2 denote the two vertices of H to which b is joined.
In % — b every vertex other than bx and b2 has valency 4. Consequently (bt, b2)0C
since % is planar. Therefore ("J( — b)\J(bl, b2) is a planar graph without loops or
313 CIRCUITS

90
G. DIRAC AND P. ERDOS
multiple edges containing five vertices each having valency 4. This contradiction
proves (5).
Now suppose first that K(Cj) = 8. Then Ka4(t|)s6. 3 does not contain a vertex
of valency 4 joined to two vertices of valency 1, because if it did then the graph
obtained from 3 by deleting the two vertices of valency 1 together with the edges
incident with them would contradict (5). Therefore c^^Q2 by Theorem 1. HI.
The proof of II will now be completed by reducfio ad absurdum. Assume that
II is untrue. Then there exists a planar graph 3^ with the following properties: Cj0
barely satisfies the conditions of II and 3,01pO2, and all planar graphs which
satisfy the conditions of II and have fewer vertices than ^0 contain two
independent circuits. It has just been proved that
(6) K(i$0)==9.
As in the proof of Theorem 1. IV
(7) At least one end-vertex of every edge contained in £0 has valency 4, and
K=-3(^0)S:1.
By summing the valencies of all the vertices of ^0 we have that
(8) If V0(3j0) =0 then E(3l0) ^ U V{$0) + 2{, if K0(^0) =0 and V2(3l0) ^ 1 then
E(3}0) a U K(^0) + 2J, and if K0(^0) = 0 and V2{3}Q) ^ 1 then E(3l0) ^ 1 J K(^0) + 3±.
It will now be proved that K,-(Cj0) =0 for /=0, 2. 3.
(9) v0{q0)=o.
For suppose on the contrary that the vertex a of Crj0 has valency 0. Then (^0 —a)
satisfies the conditions of II (by (6)) and therefore li^ — a z> O2 by the minimal
property of Cj0.
(10) K2(^0)=0.
For suppose on the contrary that the vertex c of 30 has valency 2 and is joined
to the vertices c, and c2. The two alternatives (c,, c2) ¢^ and (c,, c2)(z<S[0 will be
considered in turn and a contradiction will be derived in both cases. If (ct, c2H^|
then (ifj'o — r) U (r,, c2) satisfies the conditions of II (by (6)) and therefore contains
two independent circuits (because of the minimal property of ^0). it follows as in
the proof of Theorem 1. Ill that ^0 z> O2, whereas 0^0^O2. Suppose next that
(fi.^KCjo- By (7) u(cx,q0) = i-(r2,^0) = 4. Ei^-c-ci-cz) =i V($0)-4
because e,0 -r- rt -c2 f G . It follows that £(0^,) d K(^0) + 3. By (8) £(^0) =
^ \i K(^0) + 2|. From the two inequalities it follows that K(^0)Sl, which
contradicts (6). Hence K2(^o)=0.
(11) K3(^o)=0.
For suppose on the contrary that the vertex d of i^0 has valency 3 and is joined
to the vertices dl,d1 and c/3. The two alternatives E(^3l0(dl, d2, d?))^\ and
314 GRAPH THEORY

ON THE MAXIMAL NUMBER OF INDEPENDENT CIRCUITS IN A GRAPH 91
E((S0(dl, d2, ¢/3))^2 will be considered in turn and a contradiction derived in
both cases.
Assuming that E((30(dl, d2, c/3))sil, let it be supposed that (dlt d3)§§0 and
(d2,d3)^0. Then {^0-d)U(dl, d3)U{d2, d3) satisfies the conditions of 11 (by (6))
and therefore contains two independent circuits (because of the minimal property
of (|0), it follows as in the proof of Theorem 1. 1 that i^0 id O2, whereas ^0 t> O2.
There remains the alternative that E(^0(dl ,d2, c/3)) =2. By (7) v(dl, §0) = v(d2, ^o) =
= 1)((/3,^)=4. E(§0-d-dY-d2-d3)^V(§o)-5 because qo-d-dY-d2-
-i/jJO. It follows that £((§<,) si K(if0) + 5- By (8) £(i£0) ^ U(K(^0) + 3^. From
the two inequalities it follows that K(i^0) si 7, which contradicts (6). Hence K3(C[0) =0.
From the conditions of II it follows that Ks4(i^0) ^ 6 and K£4(i|0) - ^3=3(^0)-^.
By (9), (10) and (11) V(§0)= K^Cjo) + Ks4(Cj0) and by (7) each vertex of valency 1
is joined to a vertex of valency 4. If tS0 does not contain a vertex having valency 4
joined to at least two vertices having valency 1, then let 61" denote the graph
obtained by deleting all vertices of valency 1 from 6^. Clearly K(^")^6, Ke4(^"')^3
and Kg2(i|") = 0. Therefore by I 6J"':d02, whereas i£0 £ O 2'. This contradiction
proves II in the case considered.
If i50 contains a vertex, say d, having valency 4 which is joined to two vertices
of valency 1, say dt and d2 , then let d3 and </4 denote the remaining to vertices of
<5j0 joined to d. Let df" = (§0 -^) U (¢/,, d2) U (¢/,, d3) U(dt, t/4). i|" satisfies the
conditions of 11 (by (6)) and therefore contains two independent circuits (because
of the minimal property of G0), it follows as in the proof of Theorem 1. 1 that
G03 O2, whereas G0 $> O2- This contradiction proves II in the remaining case.
Proof of III. For k = 2 HI follows from II in the same way as Theorem 1. V
follows from Theorem 1. IV. For k^3 HI will be proved by reductio ad abswdum.
Assume that III is untrue. Let x denote the least value of A: for which the assertion
of 111 is false, and among the planar graphs to which the assertion of III with k = y.
does not apply let iS0 be one with the least number of vertices and barely satisfying
the conditions of III with k=x.
(12) At least one of the end-vertices of every edge of (S0 has valency 2x or 2¾ — 1.
For i|0 barely satisfies the conditions of III with k — x.
(13) ^,^(^)51.
Because x^3 and G0 is planar.
(14) Ka ,(^,)=0.
For if x is a vertex of valency ==1 in ^0 then V-~2k(&0 —x)— V^2k~2(^jo—x) —
^5x~7 and &0 — xj) 0*, which contradicts the minimal property of ^0.
(15) S0 contains at least one triangle.
Replace x2 + 2x — 4 by 5x —7 in the proof of (5) in the proof of Theorem 3.
315 CIRCUITS

92
G. DIRAC and p. erdos
Let xl,x2,x3 be the vertices of a triangle contained in k30 and let <3" =
(16) fjO^'.
Because ^JO'.
Let s denote the set of those vertices of iij" which are in ^0 joined to all three
vertices x,, a2, x3 .
(17) \s\==2.
Because iS0 is planar.
From (17) and the conditions of III it follows as in the proof of Theorem 3
that
(18) ^^^-^^4^^,4)-^:,^((1,)-555(^ 1)-7.
It follows from (18) that y. = 3, beca use if >i = 4 then, by (18) and the minimal
property of x, ^"idO'-1 contrary to (16). If x = 3 then K56(^0) - K«4(^0) §s8.
In this case obviously K(^0) — H because ^0 is planar. Hence K(^j"')^8. Therefore,
from the first part of 111, oj"' ^ O2, wich contradicts (16). The assumption that III
is false therefore leads to a contradiction. Theorem 5 is now proved.
Remarks. 1. The condition of I cannot be relaxed, V(<5)^6, V^,2(S)=0 and
K^4(iip = 1 does not necessarily imply that the planar graph ^ contains two
independent circuits. This is illustrated by the planar graphs consisting of a circuit together
with a vertex not belonging to the circuit joined to every vertex of the circuit.
2. If Cj is planar and K1?4(^) 5+ V($) + H then K(^)^7, as the reader can
easily verify. If K,4(^) ^-V K(^)+1] and V{§)=7 then the planar graph ^ does
not necessarily contain two independent circuits. For example if ~~f(X) = {xl, ..., x-,}
and 8 (X) --= {(Xl, x2), (.vt, x3), (xt, x4), (a^, x5), (x2, x3), (x2 , xA), (x2 , x5), (x3, x4),
(x3, .v5), (x4, x6), (x$, x-,)} then A" is planar and contains no loops or multiple
edges, and VA{X) = 5, r,(I)=2,butIJ02.
Jf ^ is planar and V(^) =8 and K»4(^) = | V(^)+ 1, then <S does not necessarily
contain two independent circuits, as is shown by any graph Y obtained from X by
subdividing an arbitrary edge through inserting a new vertex.
3. X shows that if C, is planar, Ks4(^)- V,i2{^) = 3 and K(Cp = 7 then iij does
not necessarily contain two independent circuits. X — x-, shows that if ^ is planar,
^4(^)^^2(^) = 3 and K(^) = 6 then ^ does not necessarily contain two
independent circuits. Y shows that if ^ is planar, VSA(§) — Vs2(£) = 2 and K(t^) = 8
then Sf does not necessarily contain two independent circuits. Ill is surely not best
possible for k^3.
4. The remarks after Theorem 2 apply equally to planar graphs, because the
counterexamples described there are planar.
316 GRAPH THEORY

ON THE MAXIMAL NUMBER OF INDEPENDENT CIRCUITS IN A GRAPH
93
5. Concerning the existence of two circuits without a common
edge in finite planar graphs
Theorem 6. If 3 is a finite planar graph which may contain multiple edges and
loops, and if E(SS) =? K(Cp + 3, then C, 2 2 O•
Note. If & is finite and contains a loop and if E(£[) 5 K(Cp + 2, a loop being
counted as two edges, then cf32Q, whether ^ is planar or not. For if / is a loop
then £(6, - /) S K(C,), hence ^ - / => Q, therefore ^ 3 2 C . If £(e,) = K(^) + 1 then
CJ1 need not contain two circuits without a common edge, a simple example is a
path with a loop incident with one of its edges.
Proof of Theorem 6 by reductio ad absurdum. Assume that Theorem 6 is
untrue. Then there exists a finite planar graph ^0 such that £(^0) = V(^0) + 3,
and k50^2O, and Theorem 6 is true for all graphs with fewer vertices than ^0-
Clearly V(§0)^3.
(1) <S0 contains no circuit having fewer than four edges.
For if C is a circuit with fewer than four edges contained in S0 then
£(^0-£(C))s K(Cjo), so §o ~ <? (O i? O and consequently ^032C contrary to
hypothesis.
(2) KS2(^0)=0.
^0(^0) = 0 because if a is an isolated vertex of ^0 then £(^0 —a) =
= K(<5o -- fl) + 4, socJo — fl=?20 from the minimal property of 0^, whereas i?0 5 2 O •
^1(^0)= ^2(^10)=0 can be proved by the same argument as is used in the proof
of Theorem 3 in [2].
It follows from (2) that 2£(^0) =? 3 V(§0). Consequently, since £(^0) = K(^0) + 3.
K(C|o)=6\ On the other hand it follows easily from (1) and (2) that K(^0) = 6-
Hence V(<S0) = 6. From this and (1) and (2) it follows easily that ^0 contains a
circuit having exactly four edges and vertices.
Let the vertices of S0 be denoted by gt, g2 , ..., g6 and suppose that £0 contains
a circuit whose vertices are gi,g2,g3,#4 in this order. By (1) g5 and g6 are each
joined to at most two of gl5 ..., g4, therefore by (2) (g5, g6) ¢^ and by (1) it may
be supposed without loss of generality that (gl, gb), (g3,g6), (g2,gs), (g^gsK^o-
So 6j0 contains the circuit [gl, g2, g3, g4, g5, g6] and the edges (g^gj, (g2,g5),
(g3, g6). Therefore £0 is not planar. This contradiction proves Theorem 6.
Remarks. 1. If K(Cf)^3 and £(^)^K(^) + 2 then ^220, as the reader
may very easily verify.
2. If K(^)^4 and £(^)= V(^) + 2 then the planar graph ^ need not contain
two circuits without a common edge. This is illustrated by the graph containing
317 CIRCUITS

94 G. D1RAC AND P. ERDOS: ON THE MAXIMAL NUMBER OF INDEPENDENT CIRCUITS IN A GRAPH
four vertices, each pair of distinct vertices joined by one edge (this graph is
3-fold connected) and by all graphs obtained from this graph through subdividing
edges by the insertion of new vertices (these graphs are all 2-fold connected) — in
other words by the complete 4-graph and the topological complete 4-graphs.
MATHEMAT1SCHES seminar
DER UNIVERSItSt HAMBURG
MATHEMATICAL INSTITUTE,
eOtvOs LORAND UNIVERSITY,
BUDAPEST
(Received 27 March 1962)
References
[11 K. CorrAdi and A. Hajnal, On the maximal number of independent circuits of a graph
(to appear).
[2] P. Erdos and L. Posa, On the maximal number of disjoint circuits of a graph, Publ. Math.
Debrecen, 9 (1962), pp. 3-12.
318 GRAPH THEORY

ON THE STRUCTURE OF LINEAR GRAPHS
BY
p. erdOs
ABSTRACT
Denote by G(n; m) a graph of n vertices and m edges. We prove that every
G(ii; [n2/4] + J) contains a circuit of I edges for every 3 55 I < c2n, also
that every G(n:[n2/4]+ 1) contains a ke{un,u„) with u„ = [qlogw] (for
the definition of ke(u„, u„) see the introduction). Finally for t > to every
G(n; [tnl!2]) contains a circuit of 21 edges for 2 f£ I < c3i2.
G(n\ m) will denote a graph of n vertices and m edges, K(p) will denote the
complete graph of p vertices, and K(p,p) will denote the complete bipartite graph
of 2p vertices. More generally K(p1, •••, pr) denotes the r-chromatic graph where
there are px vertices of the ;-th color and any two vertices of dirferent color are
adjacent. Ke(pl,---,pr), pl f£ p2 f£ ••• ^ pr, will denote a K(ply •••, pr) where
two vertices of the first color are adjacent, i.e. Ke(pl, ■■■ ,pr) is a K(pv,---,pr)
with an extra edge. The vertices of G will be denoted by x, x1? v, •••;
the edge connecting x and y will be denoted by (x, y). (G — Xi — ■■■ — xr)
denotes the graph G from which the vertices x{,---,xr and all edges which are
incident to them have been deleted. v(x), the valency of x, is the number of edges
adjacent to x. C, will denote a circuit having I edges. c,,c2,--- denote suitable
positive absolute constants, [f] is the greatest integer not exceeding t.
A special case of a well known theorem of Turan [1] states that every
G(n;[n2/4] +1) contains a K(3) (i.e. a triangle). Dirac and I observed
(independently) that every G(n; [n2/4] + 1) contains for every 4^Hna
subgraph G(k; [fc2/4] + 1) and in fact Dirac proved a more general theorem [2].
In the present paper we continue the investigation of the structure of the graphs
G(n; [n2/4] + 1) and we are going to prove the following theorems:
Theorem 1. Put [^ log n] = «„. Every G(n; [«2/4] + 1) contains a KL,(un,un).
Remark. The structure of KJun,un) is clearly uniquely determined. It is the
G(2u„; u2 + 1) which contains a K(u„,u„) as a subgraph.
Theorem 2. Every G(n; [n2/4]+1) contains a C, for every 3 f£ 1^ c2n.
Theorem 3. Let i>t0, then every G(n;\_in3/ ]) contains a C?J for every
2^ l<c3t2.
Received September 28, 1963.
This work was done while the author received support from the National Science Foundation,
N.S.F. G.88.
156
319 CIRCUITS

ON THE STRUCTURE OF LINEAR GRAPHS
157
Apart from the value of ct Theorem 1 is best possible. In fact we can show
the following
Theorem 4. To every e > 0 there is a c(e) so that for every n there is a
G{n\ [(2)(1 — e)]) which does not contain a K([c(e)logn], [c(e)logn]).
We suppress the proof of Theorem 4 since it uses the methods used in [3].
A theorem of A. H. Stone and myself [4] implies that every G{n; [en2]) contains
a K([c,(E)logn], [c,(e)logn\). The exact determination of c(e) and Ci(s) seems
difficult.
1 would expect that the exact determination of c2 in Theorem 2 will be difficult.
Theorem 3 is best possible in the sense that E. Klein [5] showed that there is a
G(n;[cAn2~\) which contains no C4. For t > t0 perhaps every G(n\ [tn312])
contains a C2; for every 2 ^ /< c5tnl/2 ; if true, then apart from the value of cs
this is easily seen to be best possible.
By the same method as used in the proof of Theorem 1 we can prove
Theorem 5. To every k there is an n0 = n0(k) and a ck so that, for n > n0,
G(n; [«2/4] + k) always contains a K([q,log«], [qlogn]) and k further ed%es.
We suppress the proof of Theorem 5. Put rk = [<\logn]. For k >1 the structure
of ourG(2r4; r42 + k) is of course not uniquely determined. Perhaps the following
result holds: Let n > 8. Then every G(n; [«2/4] + n - 1) contains a K([clogn],
[c log n\) and two edges which have no vertex in common and all four vertices of
which have the same color. It is easy to see that a G(n; [n2/4] + n — 2) does not
have to have this property. To see this consider a K([n/2], [(n + 1)/2])
where further one vertex of each color is adjacent to all the vertices of our graph
i.e., the vertices of our G(n; [n2/4] + n — 2) are x{, ■■■ , xk; yt, ■■■, yt
k = [n/2], 1= [_(n + 1)/2] and its edges are
(x„yjy, l£i£k,l£j£l and (^,),0^.):2 g i g k, 2½)½ I.
Put
m(n,p) = ypbiyt"2 " r^ + (2). n = (p-1) t + r,l £r £ p - 1.
Turan proved that every G(n; m(n,p)) contains a K(p) and Dirac and I [2] observed
(independently) that it contains a K(p + 1) from which one edge is missing . By
very much more complicated methods I can prove that for n > n0(p,k) G(n; m(n,p))
contains a achromatic subgraph K(k,--,k) and one further edge (i.e., a
Ke(k,---,k)); for p = 2 this is a weakened form of Theorem 1.
Now we prove Theorem 1. First we need two Lemmas.
Lemma 1. Every G(n; m) contains a subgraph G(N,M) every vertex of
which has valency greater than [m/n~\. Further
(1) M^m-(n-N)
320 GRAPH THEORY

J58
p. erdOs
[September
(The Lemma of course means that every vertex of G(N,M) has valency in
G(N,M) greater than [/«/«]).
If every vertex of G(/i,/n) has valency >\n/m~], there is nothing to prove.
Hence we can assume that G(n,m) has a vertex xi of valency Sj [w/n]. If
G(n; m) — xt has a vertex x2 withu(x2) ^ [m/n] we considerG(n; m) — Xi — x2-
We repeat this process and obtain a sequence of vertices xi,---,xk so that the
valency of xt in {G(n; m) — xl — ■■■ — x,-i) is f£ [m/ n] for every 1 f£ ;' f£ /< — 1,
but every vertex of
(2)
(G(n; m) - xi —
x,) = G(N; M)
has valency > [m/nj.
Clearly M > 0 for otherwise, since {G(n; m) - xY - ■■■ - *„-,) has only one
vertex and thus no edges, we can put in (2) k ^ n — 1 and by our construction
we would have
< m
m^(n- 1)
an evident contradiction. Further by our construction (k — n — N)
M^m-(n- N)
which proves (1), and the proof of Lemma 1 is complete.
Lemma 2. Let m>[n2/4]. Then every G(n;m) contains a Ke(2,k) where
k = [c5n].
Lemma 2 is known [6].
Now we can prove Theoiem 1. In fact we shall prove the stronger statement:
To every e > 0 there is a r, = c,(e) so that every G(n; [n2/4] + 1) contains a
Kc([C,logn], [«'-'])•
By Lemma 1 our G(n; [n2/4] + 1) contains a subgraph G(N, M) every vertex of
which has valency > [
computation
[n2/4] + l
] = [n/4]. Further (1) implies by a simple
(2)
M>
n
+ 1 - (n - JV)
n
T
>
" jv2 "
_~4~_
Further since every vertex of G(N,M) has valency > [n/4] we have
(3)
JV >
By (2) Lemma 2 can be applied to G(N,M) and by Lemma 2 and (3) we obtain
that G(JV,M) contains a Ke(l,k) with k=[csn/4j. Let the vertices of our Ke{2,k)
be (we choose cs < 1/3)
321 CIRCUITS

1963]
ON THE STRUCTURE OF LINEAR GRAPHS
159
(4) xl,x2;yu ---,^,
c5n
~4
<
- 1
Denote by z,, ---,2, the other vertices of G(N,M). Each y has by Lemma 1 valency
> [n/4] (in G(N,M)), hence each .Vj. 1 < i % k is connected with more than
,5, H-* + i>ir
z's. ((5) follows immediately from (4) since the number of x's and y's is k +2 <
[n/8] + 1 and in the worst case y, is connected with all of them).
Let z(j'\ 1 < ; ^ /|, /; > n/8, be the z's adjacent to v,. Form all the («„ — 2)-tuples
(iin = [c, lognj of Theorem 1) of these vertices for each i',lgigi= [c5n/4j.
By a simple computation we obtain (we use ( ) > (a/ b) )
k
(6) i L\) ^(["/82:l)>csB
,, vw„ —2/ — 4 \ «/„-2 / 4 \ 8(«,-2)
Further trivially
/ n \ n""~2 n""-2^""'2 / 3n \"""2
() W-2J < (un~2)\ < (un~2)-r,~2 <\un-2j
Hence from (6) and (7)
« i (.:-2) >-^-(..-2)2^^--(..-2)
for every e >0 if c, = c,(e) is sufficiently small. The number of the z's is clearly
less than n, hence the number of the («„ — 2)-tuples formed from z'.s is less than
I . ). Thus from (8) there is a (u. — 2Vtuple which occurs more than
\"n -2/
n1"1 times — in other words there is a set of «,. — 2 z's which are adjacent to the
same [n1 ~E] y's. If we adjoin to these z's x, and x2 (which are adjacent and are
adjacent to all y's) we obtain that G(N;M) and hence our G(n: [n2/4] + 1)
contains a Ke(«„, n'~E) for every e>0 if c, = c,(e) is sufficiently small. This
completes the proof of our assertion and hence Theorem 1 is proved.
Proof of Theorem 2. As in the proof of Theorem 1 our G(n; [n2/4] + 1)
contains a Ke(2, [c5n/4]), c5<!l/3, having the vertices xux2, yt,---,yk,
k = [c5/i/4]. Each of the k vertices yt,--,yk are adjacent to more than n/8
z's (we use the notations of Theorem 1). Consider now the bipartite graph whose
322 GRAPH THEORY

160
p. erdos
vertices are y l,---,yk\ z,, ---,2,. and whose edges are the edges (yt,Zj) of G(n; m).
This bipartite graph has fewer than n vertices and more than
4
edges. Hence by a theorem of Gallai and myself [7] it has a path of length c2n
(the length of a path is the number of its edges). Since our graph is bipartite every
second of its vertices is a y. Now since x, and x2 are adjacent and they are adjacent
to each of the j/'s we immediately obtain that our G(n;[n2/4] + 1) contains a
C, for each 3 ^ k 5£ [c2"j, which proves Theorem 2.
Proof of Theorem 3. By Lemma 1 G(n; [tti3/2]) contains a subgraph G(JV; M)
every vertex of which has valency ^[fn1/2]. Let x be one such vertex and let
)>i,"-,yk, k = \\tnlll~\ be some of the vertices adjacent to x and denote by
z,,--- the other vertices of G(N,M). Every y has valency ^ [fnI/2], thus
since the number of j's is i[tn1/2] there are at least \ [tn1/2] z's adjacent
to each y. Hence the bipartite graph whose vertices are yu ■■-,yk: z,, ••• and whose
edges are the edges (y-„Zj) of G(n,m) has at least
M[^1/2] = i[^1/2J2>y«
edges. The number of its vertices is clearly < n. Thus by the theorem of Gallai
and myself [7J it has a path of length > 2c3 t2 and as in the proof of Theorem 2
every second vertex of this graph is a y. Since x is adjacent to every y this path
together with the vertex x gives the required circuits C27, 2 :g I ^ c^i2, which
proves Theorem 3.
References
1. Turan, P., 1941, Mat. Lapok, 48, 436-452 (Hungarian), see also Turan, P., 1955, On the
theory of graphs, Coll. Math., 3, 19-30.
2. Dirac, G., will appear in Acta Math. Acad. Sci. Hungar
3. Erdos, P., 1947, Some remarks on the theory of graphs, Bull. Amer, Math. Soc, 53
292-294; see also Erdos, P. and Renyi, A., 1960, On the evolution of random graphs, Publ. Math.
Inst. Hung. Acad, 5, 17-61.
4. Erdos, P. and Stone, A. H., 1946, On the structure of linear graphs, Bull. Amer. Math.
Soc. ,52, 1087-1091.
5. Erdos, P., 1938, On sequences of integers no one of which divides the product of two
others and on some related problems, Izv. Nauk. Inst. Mat. Mech. Tomsk., 2, 74-82.
6. Erdos, P., 1962, On the theorem of Rademacher-Turan, Illinois J. of Math., 6,
122-127.
7. Erdos, P. and Gallai, T., 1959, On maximal paths and circuits of graphs, Acta. Math. Acad.
Sci. Hungar., 10, 337-356.
University of Michigan,
Ann Arbor, Michigan, U.S.A.
323 CIRCUITS

ON INDEPENDENT CIRCUITS CONTAINED IN A GRAPH
p. erdos and l. posa
A family of circuits of a graph G is said to be independent if no two of the
circuits have a comnion vertex; it is called edge-independent if no two of them
have an edge in common. A set of vertices will be called a representing set
for the circuits (for the sake of brevity we shall call it a representing set),
if every circuit of G passes through at least one vertex of the representing set.
Denote by 1(G) = k the maximum number of circuits in an independent
family and by R(G) the minimum number of vertices of a representing set.
Dlrac and Gallai asked whether there is any relation between 1(G) and R(G)
(trivially R(G) > 1(G)). B. Bollobas (unpublished) proved that if 1(G) = 1,
then R(G) < 3 and the complete graph of five vertices shows that R(G) < 3
is best possible.
Consider now all graphs with 1(G) = k. Denote by r(k) the maximum value
of R(G) for all graphs with 1(G) = k. It is not immediately obvious that
r(k) is finite and the theorem of Bollobas states that r(\) = 3. The value of
/-(2) does not seem to be known. We are going to prove the following
Theorem. There are absolute constants C\ and c-i such that
(1) c, k log k < r(k) < c-2 k log k.
We cannot determine
lim r(k)/k log k
and in fact cannot even prove that the limit exists.
First we prove the lower bound in (1). In fact we shall prove a somewhat
stronger result. Denote by E(G) the maximum number of edge-independent
circuits of G. We shall show that for every k there is a graph G with 1(G) = k
and
(2) r(k) > CiE(G) log E(G).
(2) is stronger than the lower bound in (1) since clearly E(G) > 7(G) = k.
We shall prove (2) by a probabilistic argument and cannot at present give
an explicit example of a graph satisfying (2). Our proof will be very similar to
the one used in (1, 2, and 3).
First we introduce a few notations. Vertices of G will be denoted by
Xi, ■ ■ ■ , yi, ■ ■ ■ ; circuits will be denoted by Ct; the subgraph of G spanned by
the vertices Xi, . . . , x, will be denoted by G(xl} ..., x,); G(n; m) will denote
Received January 10, 1964.
347
324 GRAPH THEORY

348
1>. ICRDOS AND I.. POSA
a graph of n vertices and m edges; 11(G) denotes the number of edges of G;
the edges of G will be denoted by e,, or by (.v,, xf)\ and G — eL — . . . — e„,
will denote the graph from which the edges ei, .-., e,„ have been omitted. The
length of a circuit 6\ is the number of its edges.
Consider all graphs G(n; lOOw) with n labelled vertices Xi, ■ ■ ■ , x„. The
number of these graphs is clearly
(3) \2/ = A„.
First we state two lemmas.
Lii.MMA 1. All but o(A„) graphs G{n; 100») have the property that for every
choice x, ..., I,-,,, p = [w/2], of p vertices,
(4) n(G(x(1,.. . ,.r,,) > 2w.
Lemma 2. Put I = [(log w)/100]. All but o(An) graphs G(n; lOOw) have fewer
than n circuits of length </.
Assume that the lemmas have already been proved. Then we prove (2) as
follows. By Lemmas 1 and 2 for n > n0 there is a G(w; lOOw) which satisfies
(4) and for which the number of circuits of length not exceeding I is less than
n. Denote by C,, 1 < i < m < n, these circuits, and let e< be an arbitrary
edge of d. The e's are not necessarily different. Put
G' = G — ej — . . . — e,„.
Clearly each circuit of G' has more than 1 edges and since G' has at most
lOOw edges, we evidently have
(5) E(G') < 100»// < 20,000 n/(log n).
On the other hand
(6) R{Q') > w/2.
To prove (6) observe that if x,, ..., xk, k < w/2, would represent all circuits
of G', then G'(xi+I, ..., xn) would not contain any- circuits, hence would have
fewer than n — k edges, or
U(G'(xk+i, ..., x-„)) < n — k < n.
But we evidently have by (4) and m < n in — k > w/2)
II(G'(.rt+I, ..., xn)) > n(G(xk+l, ..., xn)) — m > 2w — w = w,
an evident contradiction. Hence (6) is proved, (o) and (6) easily imply (2).
To see this, let w be the largest integer with 20,000w/(log w) < k. For our
graph G' we have by (5) and (G)
E(G') < k, R{G') > cklogk
325 CIRCUITS

INDEPENDENT CIRCUITS IN A GRAPH
349
and by perhaps adding to G' some (at most k) independent circuits we clearly
obtain a graph C with
/(C) = k, £(C) < 2k, Rid') > c,£(C) log £(C),
which completes the proof of (2), if (5) and (6) are assumed.
Thus to complete the proof of (2) we only have to prove our lemmas. To
prove Lemma 1, observe that the number of graphs G{n; lOOw) which have
p vertices xn, ..., xip with
n(G(xi, xtp)) < 2»
is at most (1, pp. 35-6)
\p/ k.« \ l J\ 100w _lJ \ 2« A 98« I
since
and a simple computation shows that the terms in the sum (7) are increasing
for I < 2». Now e2" > (2w)2"/(2w!) and p = [n/2] imply that
PM&Mi
(8)
\2w
and for n > na we obtain by a simple computation and (3)
=(1+0(1)),4„(r^r.
From (7), (8), and (9) we have
In < (1 + o{\))\A„(f)98"2002" = o(A„).
which proves Lemma 1.
Now we prove Lemma 2 (1, p. 36). The number of graphs G(n; lOOw) which
contain a given circuit (xt, x2), (x2, x3), ..., (xr_r, xr), (xr, xt) clearly equals
$-).
dOOw - rl
■bib GRAPH THEORY

350
P. ERDOS AND L. POSA
A circuit is determined by its vertices and their order. Thus there are
n(n — 1) . . . (w — r + 1) < nr such circuits. Therefore the expected number
of circuits of length r < I = [(logw)/100] is less than
( W ) E n ( \2/ " ) < (1 + 0(1)) X) nT \(n\) = o(n).
\100«/ 3<r<i \l00»-r/ 3<'<l \\2//
Therefore by a simple and well-known argument the number of graphs
G(n; 100«) having n or more circuits of length not exceeding I is o(An), which
proves Lemma 2 and hence the proof of (2) is complete.
To complete the proof of our theorem we now have to prove that
r{k) < Ci k log k. We are going to use two theorems, the first, due to ourselves
(1, p. 9), which states: There exists an absolute constant c3 so that every
Gin, n + /) contains at least c3 //log I edge-independent circuits.
Assume now that every vertex of our graph has valency <3. Then clearly
it contains c3 //log I independent circuits; since if two circuits are edge-
independent and not independent, then every common vertex of the two circuits
must have valency 4.
The second theorem is due to T. Gallai (4). Let G be a graph. Designate
some of its vertices, say xu . . . , xu, as principal vertices; the other vertices,
ju . . . , y, of G, will be the subsidiary vertices. A path is called a principal
path if its end points are principal vertices and it contains no other principal
vertices. (A circuit having only one principal vertex is not allowed.) Denote by
Knax the maximum number of independent principal paths (two principal
paths are called independent if they have no vertex [principal or subsidiary]
in common). nm]n denotes the smallest integer such that there are nimn vertices
representing all the principal paths—in other words there are k = nmln
vertices xh, . . . , xlk (principal or subsidiary) so that every principal path
contains one of the Xi/s and one cannot find fewer than k vertices with this
property. Gallai's theorem asserts that
(10) nmln<<2rmax.
Now we are ready to prove the right-side inequality of (1). Assume that in G
the maximum number of independent circuits is k and let
(7) Ch 1 < i < k,
be a maximal system of independent circuits of G. Omit all the edges of d,
1 < i < k, but retain the vertices of C,. Thus we obtain the graph G\. Let the
principal vertices of G\ be the vertices of d, 1 < i < k, all other vertices being
subsidiary ones. Consider now a maximal system of independent principal
paths of G\. The circuits d and the maximal system of independent paths
define a graph G* every vertex of which has valency not exceeding three.
(G* is a subgraph of G but not of d.) Let m denote the number of vertices of
G*. Then clearly the number of edges of G* is
327 CIRCUITS

INDEPENDENT CIRCUITS IN A GRAPH
351
(11) m + ymas
since each principal path gives an excess of 1 of the number of edges over the
number of vertices. Thus by our theorem G* (and therefore G) contains at
least
C3 T7max,/l0g Fmax
independent circuits. Hence
(12) T^lf^ < ^ or Fmax < c4 k log k.
'Og ' max
Now let j\, . . . , y, be a minimal system of vertices representing all the principal
paths of Gi. By (12) and Gallai's theorem
(13) t < 2c4 k log k.
For some i, 1 < i < k, there may exist a circuit Dt which has one (and only
one) common vertex xt with Ci} which is independent of Cj{\ < j < k, j ^ i)
and does not pass through any of the y jt 1 < j < t. But for a given i there
cannot be two such -D/s, say Dn and Di2, whose unique common vertex with
Ct is Xii and .Ti2, where xtl and xi2 are distinct. To see this, observe that if
Dtl and D,, are independent, then the k + 1 circuits
Cj(l < j < k, j 9^ i), Dilt Di2
would be independent, which contradicts the maximality property of k. If
Dii and DH are not independent, then their union contains a principal path
connecting xtl and xti; hence it contains one of the vertices y3{\ <j < t),
which by assumption represent all principal paths; but this contradicts our
assumption that Dil and DH do not contain any of the y3(l < j < t).
If Ct is such that there is a Dt corresponding to it, adjoin their common
vertex xt to the y's; otherwise choose any vertex of Ci} denote it by x{, and
adjoin it to the y's. Some of the xt's might have already occurred amongst the
y's; but in any case the system
(14) yj(l <j <t), xt (1 < i < k)
contains at most
2c4 k log k + k < Co k log k
vertices. Our proof will be complete if we show that the system (14) represents
every circuit of G. Let C be any circuit of G. We have to show that it contains
at least one of the vertices (14). The circuits C, are clearly represented by the
vertices (14); thus we can assume that C ^ Cu 1 < i < k. If C contains
at least two of the vertices of C{, 1 < i < k, then C contains a principal path
of G, and hence one of the vertices yj} 1 < j < t. If C contains only one of the
vertices of Ct and does not contain any of the yj (1 < j < t), then it contains
328 GRAPH THEORY

352
P. ERDOS AND L. PoSA
Xi, 1 < i < k. Finally, C cannot be disjoint of all the 6','s because of the
maximally property of the Ct, 1 < i < k. This completes the proof of our theorem.
It would be easy to obtain explicit inequalities for c, andc2 but they would
be very far from being best possible.
References
1. P. Krdos and L. Posa, On the maximal number of disjoint circuits of a graph, Publ. Math.
Debrecen, .9 (1962), 3-12.
2. P. Erdos, Graph theory and probability, Can. J. Math., 11 (1959), 34-8.
3. On circuits and subgraphs of chromatic graphs, Mathematika, 9 (1962), 170-5.
4. T. Gallai, Maximum-minimum Siitze and vei'allgemeinerte Faktoren von Graphen. Acta Math.
Hung. Acad. So'., 12 (1961), 131-73; cf. pp. 158 and 101.
University of British Columbia, Vancouver,
and Budapest, Hungary
329 CIRCUITS

Chapter 7
Assorted Graph Theory
Paper [410] consists of a self-contained proof of an interesting elementary
result.
Paper [401] is improved on in ES. Further results on [401] have been
gotten by Shelah (University of Jerusalem, unpublished).
The conjecture in [480] was settled in "On cliques in graphs" by Joel
Spencer in Israel J. Math., 9(1971)419-421.
Papers in Chapter 7
[401] (with A. Renyi) Asymmetric graphs
[410] (with P. Kelly) The minimal regular graph containing a given graph
[459] (with F. Harary and W. T. Tutte) On the dimension of a graph
[480] On cliques in graphs
[520] Uber die in Graphen enthaltenen saturierten planaren Graphen
331 ASSORTED GRAPH THEORY

ASYMMETRIC GRAPHS
By
P. ERDOS and A. RENY1 (Budapest), members of the Academy
Dedicated to T. Gallai, at the occasion of his 50,h birthday
Introduction
We consider in this paper only non-directed graphs without multiple edges
and without loops. The number of vertices of a graph G will be called its order,
and will be denoted by N(G). We shall call such a graph symmetric, if there exists
a non-identical permutation of its vertices, which leaves the graph invariant. By
other words a graph is called symmetric if the group of its automorphisms has degree
greater than 1. A graph which is not symmetric will be called asymmetric. The
degree of symmetry of a symmetric graph is evidently measured by the degree of
its group of automorphisms. The question which led us to the results contained
in the present paper is the following: how can we measure the degree of asymmetry
of an asymmetric graph?
Evidently any asymmetric graph can be made symmetric by deleting certain
of its edges and by adding certain new edges connecting its vertices. We shall call
such a transformation of the graph its symmetrization. For each symmetrization
of the graph let us take the sum of the number of deleted edges — say r — and the
number of new edges — say s; it is reasonable to define the degree of asymmetry
A [G] of a graph G, as the minimum of r + s where the minimum is taken over all
possible symmetrizations of the graph G. (In what follows if in order to make a
graph symmetric we delete r of its edges and add s new edges, we shall say that
we changed r + s edges.) Clearly the asymmetry of a symmetric graph is according
to this definition equal to 0, while the asymmetry of any asymmetric graph is a
positive integer.
The question arises: how large can be the degree of asymmetry of a graph
of order n (i. e. a graph which has n vertices)? We shall denote by A («) the maximum
of A[G] for all graphs G of order n (n=2, 3, ...). We put further A{\) = + °°. It
is evident trurt A(2) = A(3) = 0.
Now let G denote the complementary graph of G, that is the graph which
consists of the same vertices as G and of those and only those edges which do not
belong to G; then we have evidently
Lemma 1.
(1) A[G] = A[G].
As a matter of fact the complementary graph of a symmetric graph is evidently
also symmetric (i. e. (1) holds if A[G]=Q) and if a transformation T, consisting
in deleting r edges and adding s new edges, makes G symmetric, then the
transformation T, consisting in adding those s edges which are deleted by T and deleting
those r edges which are added by T, is clearly a symmetrization of G, and thus
Lemma 1 follows.
333 ASSORTED GRAPH THEORY

296
P. ERDOS AND A. RENYI
We shall need also the following evident fact:
Lemma 2. If a graph G is not connected and its components are G,, G2, ..., Gc
then we have
(2) A[G]^ min A[G;].
1 sisc
Let us mention further that a graph containing more than one isolated point
is symmetric.
Now we can prove that .4(4)=0 and /1(5)=0. Let us first consider A (4).
Clearly any not connected graph of order 4 is symmetric by Lemma 2, further by
Lemma 1 we may restrict ourselves to graphs
\/ fo order 4 having not more than— 2 = 3
edges, because if the graph has more than 3
edges, the complementary graph has less
than 3 edges. But the only connected graphs
Fig. l of order 4 with not more than 3 edges are
the path and the star shown on Fig. 1
which are clearly symmetric. Thus .4(4)=0. Now we show .4(5)=0. Again we
can restrict ourselves to graphs of order 5 which are connected and which contain
■ 5 edges. These belong however all to one of the 8 types
not more than
i®
shown on Fig. 2 which are evidently all symmetric.
Fig. 2
(We have drawn the graphs so that each is symmetric with respect to its
vertical axis.)
Now we shall show that .4(6) = 1. Here again we may restrict ourselves to
consider connected graphs having not more than
m
7 edges.* Among
these we find four asymmetric types, shown by Fig. 3.
All have their degree of
asymmetry equal to 1. As a
matter of fact, each can be
made symmetric by deleting
the edge which is indicated
by a thick line. It is easy to
see that any of these graphs
can also be made symmetric
A
Fig. 3
* Here and in what follows [x] denotes the integral part of the real number x.
334 GRAPH THEORY

ASYMMETRIC GRAPHS
297
A\
n
by adding a suitably chosen edge, as shown on Fig. 4, where the edge to be
added is indicated by a dotted line.
However it is not true in general that if a graph can be made symmetric by
omitting one edge, it can also be made symmetric by adding one edge. For instance
Fig. 5 shows a graph of order 10 which can be made symmetric by omitting one
edge (that which is drawn by
a thick line) but can not be
made symmetric by adding one
new edge. (Of course if by
omitting an edge an involutory
symmetry is produced, then the
same symmetry can be
produced by adding (instead of
omitting) a suitably chosen edge.)
In § 1 we shall show (Theorem 1) by a simple argument that the asymmetry of
n — 1
a graph of order n can not exceed —-— if n is odd, while if n is even the asymmetry
it
can not exceed — --1; in § 2 we prove (Theorem 2) that this estimate is
asymptotically best possible, that is for any e>0 there can be found an integer n0(e) such
that for any n > «0(£) there exists a graph G„ of order n for which A [Gn] > ^- (1 — e).
Fig. 4
In other words we have
(3)
lim
A{n)
1
2'
We can prove still more, namely that there
exists a positive constant C such that
A(n):
Cyn logn.
Fig. 5
However we do not know whether there exist
graphs G„ of even resp. odd order n for which
" i a r^ i n — \
1 resp. A [Gn] = —-—; we can prove
A[Gn]
2 *' l "' 2
that this is impossible if n = 3 mod 4 and we guess that this is impossible for all n.
In view of (3) it is reasonable to introduce the quantity
A[G]
(4)
a[G]
[N(G)
1
for any graph G with N(G)^3, and call it the relative asymmetry of G. It follows
from our results that for any graph G with N(G)^3 one has
(5)
0^a[G]sl.
335 ASSORTED GRAPH THEORY

298
P . ERDOS AND A. RENYI
The proof of Theorem 2 is not constructive, only a proof of existence. It uses
probabilistic considerations. This method gives however more than stated above:
it shows that for large values of n most graphs of order n are asymmetric, the degree
it
of asymmetry of most of them being near to —.
An other interesting question is to investigate the asymmetry or symmetry
of a graph for which not only the number of vertices but also the number N of
edges is fixed, and to ask that if we choose one of these graphs at random, what
is the probability of its being asymmetric. We have solved this question too, and
n
have shown that if N = -- log n +o)(n)n where co(n) tends arbitrarily slowly to
+ oo for n — + <=°, then the probability that a graph with n vertices and N edges
iftY1
chosen at random (so that any such graph has the same probability [2) to
\N !
be chosen) should be asymmetric, tends to 1 for n — + °°. This and some further
related results will be published in an other forthcoming paper.
In § 3 we deal with (denumerably) infinite graphs, more exactly with random
infinite graphs r defined as follows. Let Plt P2, ..., P„, ... be an infinite sequence
of vertices. Let us suppose that for each/ and k(J^k) if E-k denotes the event that
Pj and Pk are connected by an edge, then the events Ejk are independent and each
has the probability V . We prove the simple but surprising fact that r is symmetric
with probability 1 (Theorem 3).
Thus there is a striking contrast between finite and infinite graphs: while „al-
most all" finite graphs are asymmetric, „almost all" infinite graphs are symmetric.
In § 4 we deal with the asymmetry of graphs of order n in which the total
number N of edges is fixed.
In § 5 we deal with some related unsolved problems.
Our thanks are due to T. Gallai for his valuable remarks.
§ 1. Proof of the theorem that the asymmetry of a graph
of order n can not exceed
In this § we prove
Theorem 1.
A{n)i
n-\
Remark. Of course Theorem 1 implies that if n is odd, then A(n) ^-
and if n is even, we have A («) si — — 1.
2
Proof. Let G be an arbitrary graph of order n. We may suppose «S6. Let
Pl, P2, ..., P„ be the vertices of G and let us denote by vk the valency of Pk in G
(i. e. vk is the number of edges having Pk as one of their endpoints.)
336 GRAPH THEORY

ASYMMETRIC GRAPHS
299
Let further vJk (j ^k) denote the number of vertices Ph of G (h -^j, h^k) which
are connected in G both with P- and with Pk. Let us put further vjj=0. Clearly
%- = vJk and
(1-1) 2 Z»jt= 2vh(vh-l).
As a matter of fact, both the left hand side and the right hand side of (1. l)are equal
to the number of (ordered) pairs of edges of G which have one common endpoint.
Let us choose now two distinct vertices P- and Pk of G (j^k) and let us put
(1.2) AJk = Vj + vk-2vjk-25Jk
where 6jk = 1 or 0 according to whether P}- and Pk are connected by an edge in
G or not. Let us put further Ajj = 0. Evidently AJk is the number of vertices of G
which are in different relation with Pj and Pk (i. e. which are either connected with
Pj and not connected with Pk or connected with Pk and not connected with P-).
Clearly by omitting all edges connecting Py (resp. Pk) with some point of G which
is not connected with Pk (resp. Pj) we obtain a graph G' in which P- and Pk are
connected with the same points. Thus G' has the symmetry consisting in the
interchange of Py and Pk and leaving all other points unchanged. But G' is obtained
from G by deleting AJk edges. Thus G can be made symmetric by deleting AJk edges.
It follows that
(1. 3) A [G] =§ min A,-
z z*lk
J = 1 t = 1
7** Jk n(n—1)
On the other hand we have
(1.4) 2 2^ = 22^-1-4
j=l A:=l (= I
As a matter of fact, the left hand side of (1. 4) is equal to the number of ordered
triplets (Pj, Pk, Pj) of vertices such that G contains exactly one of the two possible
edges PjPt and PkPt; if we fix P, then that among Pj and Pk which is connected
with P; can be chosen in vl ways and the other in n — 1 --vt ways; this proves (1. 4).
((1. 4) could also be deduced from (1. 1) and (1. 2)).
As clearly
(1.5) „,(„_!_„,) ' "_
we obtain
2
n(n-l)2 .
if n is odd
(1.6) 2 2 «i(«-l -"i)
\
2
-1)2-1]
f n is even.
337 ASSORTED GRAPH THEORY

300
P. F.RDOS and a. renyi
It follows from (1.3), (1.4) and (1. 6)
( —-— if n is odd
(1.7) A(G)
n(n — 2)
j n(n
' ~2(n
\ 2(«-l)
Now we have evidently
n(n-2) n
2(h-I)"*" 2
Thus it follows from (1.7) that
n-\
if n is even.
if n =- 1.
if « is odd
4r — 1 if n is even
i ,
(1.8) ^= >„
V2
and thus for every n
(1.9) ^[G]S^-_Lj.
As (1. 9) holds for every graph G of order n, Theorem 1 is proved.
The problem arises, for which odd values of n does there exist a graph G of
order n such that
(1.10) min^ = -^--.
As by (1. 3) and (1. 6) we have for odd n
22 &* n_.
«-1
«-1
it follows that (1. 10) can hold only if A]k = —-— for all j^k. It follows from
(1. 5) that in this case we have also v, = —-— for /= 1, 2, ..., n. Now if « = 3 mod4
n — 1 .
then —-— is odd, and as in any graph the number of vertices having an odd valency
is even we obtain a contradiction. Thus (1. 10) can hold for an odd n only if
n = 1 mod 4.
We shall call a graph G of order n = 1 mod 4 for which (1. 10) holds a zl-graph.
For n = 5 the cycle of order 5 is a zl-graph. For n = 9 a zl-graph is shown by
Fig. 6.
A simple way to describe the zl-graph shown by Fig. 6 is as follows: let the 9
vertices be labelled by ordered pairs of numbers {a, b) where a and b may take
on independently the values 0, 1,2. Let us connect the vertices labelled by (a, b)
and (a, b') if and only if either a = a' or b = b'.
338 GRAPH THEORY

ASYMMETRIC GRAPHS
301
We can construct a zl-graph of order p, if p is an arbitrary prime for which
p= 1 mod 4, as follows: Let Pl5 P2, ..., Pp be the vertices of G and let us connect
the vertices P} and Pk if and only if k —j is a quadratic residue mod p. In this case
clearly each vertex P}- has valency
p-\
We show that for each j^k we have A
Jk
= =-—. This follows immediately from the following well-known property of
quadratic residues observed first by Lagrange (see [1] and [2]): If rlt r2, ...,rp-i
2
are all quadratic residues among the numbers 1, 2, ..., p— 1, then among the num-
Flg. 6 Fig. 7
bers r, + d\l= 1,2, ...
are exactly
where d is any of the numbers 1, 2, ...,p— 1, there
which are congruent to a quadratic non-residue mod p. As a
matter of fact ASk is equal to the number of those integers h (// = 1, 2, ..., p) for which
h —j is a quadratic residue and h — k a non-residue, or /; — / a quadratic nonresidue
and h—k a residue. Putting d = k—j this means that Ajk is equal to te sum of
the number of non-residues among the numbers rt + d 1/=1,2,...,=-— and
the number of non-residues among the numbers r, — d (/= 1, 2, ...,^—r— and
thus Ajk = 2(^1) =^ .
Thus there exists a zl-graph of every order n which is a prime of the form 4k + 1.
Clearly the zl-graph of order 5 mentioned above is the same as that obtained by
the above general construction for p = 5. For p = 13 the zl-graph obtained by our
construction is shown by Fig. 7.
We can construct also a zl-graph of order n = p2 if p is a prime of the form
p = 4/: + 3. The construction is as follows: let us label the venices by the pairs
of numbers (a,-b) where O^a^p— 1, OsiS^-1. Let us connect the vertices
339 ASSORTED GRAPH THEORY

302
P. ERDOS AND A. RENYI
labelled with (a, 6) and (a, b') if either a = a, or {a — a'){b — b') is a quadratic
rebel
ip-l)2
p2 -1
sidue mod p. In this case each vertex (a, b) has the valency —-— , because it is
connected with the p— 1 vertices (a 6') where 6'^ b and with the vertices
' (p-lu
(a', b') such that a — a' and b — V are both quadratic residues mod p and the
2 J
vertices (a',b') such that a —a' and 6 — 6' are both quadratic non-residues mod p
and 2 - —- +p—l =—- . Further denoting by v the number of vertices
which are connected with one of the vertices (a, 6) and (a', b') but not with the
2 T
other, we always have v —■ -—- - - . This follows from the theorem (due to Lagrange)
according to which if r,, r2 ..., rp__ i is a complete set of quadratic residues mod p
then exactly —- among the numbers r}+ d\j — 1, 2, ....—— are congruent
to a quadratic residue mod p (see [3]). Mr. A. Heppes (oral communication) has
constructed by a similar but different method a zl-graph of order p2 for every odd
prime p.
We can construct a zl-graph of order p' where p is an odd prime, and r an
arbitrary positive integer such that p' = 1 mod 4 (that is, if p = 1 mod 4 then r is
arbitrary, while if/? = 3 mod 4 then r has to be an even number), as follows. Let us label
the p' vertices of the graph by the elements of a Galois-field GF(pr). Let us connect
two vertices labelled by U and V(U£GF(pr), V£GF(pr)) if and only if U- V = C2
where C is some element of GF(pr). Now J. B. Kelly [2] has proved that for any
GF(pr) with pr = 1 mod 4 by denoting by A the subset of those non-zero elements
which are squares, and by B the subset of those elements which are not squares,
pr— 1
it follows that any non-zei'o element d can be represented in exactly ——— ways
in the form d = a — 6 where ad A and b£B. Thus it follows (exactly as in the case
/=1) that our graph Gisa A- graph. Thus there exists a /J-graph of order n if
n =pr = 1 mod 4 where p is a prime. We do not know whether there exists a zl-graph
of order n if n = 1 mod 4 and n is not a prime-power.
Let us mention that all zl-graphs which we have constructed are symmetric;
for instance the zl-graph of order 9 shown by Fig. 6. has the automorphism which
carries over the vertex labelled with (a, 6) into the vertex labelled with (a , b') where
a = a+ 1 (mod 3) and 6' =6+1 (mod 3). The J-graph of order p where p is a
prime of the form 4A:+1 constructed above has the symmetry which carries p,
into pv where /' = /+1 mod/;.
Thus while there exist at least for certain odd values of n graphs for which
n — \
mm AJk = ———, we do not know any graph of (odd) order n for which A[G] =
We guess that this is impossible.
2
It is possible that the following stronger conjecture holds also: all A graphs
are symmetric.
340 GRAPH THEORY

ASYMMETRIC GRAPHS
303
Finally we should add the following remark: Let C3(G) denote the number
of triangles contained in a graph G. A. Goodman [4] (see also [5] and [6]) has
determined the minimum of C3(G„) + C3(G„) for all graphs of order n. For n = \
mod 4 his result is as follows:
(1. 12) mm(C3(G„)+ C3(G„)) = "-(^-^~ 5) .
Let us call a graph of order n for which the minimum in (1. 12) is attained, a
Goodman-graph. Now it is easy to see that any /J-graph is at the same time a Goodman-
graph (but not conversely). This can be proved as follows: If G„ is a /J-graph of
order n, then the number of triangles contained in G„ and containing the edge PQ
is equal to the number of vertices connected with both P and Q, and thus is equal
n — \ n—l , n—5 . , , , ,. . , „ . n(n —1)
to — 1 =—— - . As the total number ot edges of G„ is and
2 4 4 4
n (n — 1)(/1-5)
each triangle is counted in this way three times, C3(G„) — — —. Cle-
4o
arly if G„ is a zl-graph then (7„ is a zl-graph too; thus it follows that C3((7n) + C3(Gn) =
n (n — 1) (n — 5^
—2T ' i- e" that (L 12) holds for G"'
§ 2. The asymmetry of a random graph of order n
In this § we prove the following
Theorem 2. Let us choose at random a graph F having n given vertices so that
0)
all possible 2 graphs should have the same probability to be chosen. Let & =-0 be
n{\ — e)
arbitrary. Let P„(e) denote the probability that by changing not more than —
edges of T it can be transformed into a symmetric graph. Then we have
(2.2) lim P„(e) = 0.
n-. +«.
Corollary. For any e with 0<e<l there exists an integer «0(e) depending
only on e, such that for n>n0(e) there exist graphs G of order n with A[G] >
n(l-e)
>- .
2
Remark. Clearly it follows from Theorem 1 and the corollary of Theorem 2 that
A(n) 1
(2. 3) lim
n-. +-
n
The same method yields also —A(n) = 0(/nlogn) but we shall not prove this
in detail.
Proof of Theorem 2. As the proof is not simple, we first give a sketch of the
proof.
341 ASSORTED GRAPH THEORY

304
P. ERDOS AND A. RENYI
Let us denote by P„(e, q) (q = 2, 3, ...) the probability that a random graph
ft
of order n can be transformed by changing in < —-(1 --e) of its edges into a graph
admitting a permutation /7 as an automorphism, where /7 is a permutation which
leaves exactly / = n — q of the n vertices of the graph unchanged, but which can
not be transformed into a symmetric graph by changing less than m of its edges.
Then we have
(2.4) PB(e)S Z Pn(e,<7).
We shall estimate P„(e, q) as follows
(2.5) Pn(e,q)^A^-3>*:c»;<<.
2(2)
where A„ q is the number of ways a permutation /7,, leaving exactly / = « — <jr
of the n vertices of the graph invariant, can be chosen; Bn< q is an upper bound
for the number of graphs which are invariant under such a permutation Ilq and
C„ q is an upper bound for the number of graphs which can be transformed into
a graph admitting a given permutation TIq by changing mS-(l —e) of its edges,
and can not be transformed into a symmetric graph by changing less than m of
its edges.
We shall deal first with the terms for which q lies in the range
■i
(2. 6) \n~Si q-^ n
then with the terms for which q lies in the range
■i _
(2.7) 53E<7<^«
and finally with the terms corresponding to ¢ = 2, 3 and 4 separately. We shall
show that the sum figuring on the right hand side of (2. 4) tends to 0 for n — + °°;
this clearly implies the assertion of Theorem 2.
Let us go now into the details.
Let /7 be an arbitrary permutation of order n having the cycle-representation
(2.1) /7= (fl, ,, ..., fl,iCl)(fl2,l, .-., «2,c2)--.(«r,l> ■■-. «r,cr)
where at j (1^/^C;; lsi^r) are the numbers 1,2, ..., n in some order. Thus
cl,c2, •■■, cr are the cycle-lengths of 77. The permutation /7 can also be
interpreted as a one-to-one mapping of the set {1, 2, ..., n] onto itself. Let /7, interpreted
this way, map k into Ilk (k= 1, 2, ..., n). We shall denote by /7s the mapping
obtained by applying the mapping TI s times. Clearly /7a; ■ = au j+1 for/= 1, 2, ..., c,
where aj,ci + i stands for a, 1. Let us calculate first the probability that a graph
r of order n chosen at random should admit /7 as its automorphism. By choosing
a graph r at random we mean that n vertices Px, ..., P„ are prescribed and we choose
342 GRAPH THEORY

ASYMMFTR1C GRAPHS 305
some set of edges connecting these vertices at random, so that each of the IS2' pos-
sible choices has the same probability 2 w.
Thus the random choice of r is equivalent with a sequence of L independent
random decisions concerning all possible U edges, so that with respect to any
possible edge the probability of including it into P is equal to \ . An equivalent
way of characterizing the random choice of P is as follows: let us put ej ^ = 1 if*
the edge PjPk is contained in T and e-_ k = 0 if not (1 =j<k^n). Then the random
choice of P means that the e^ k with/<fc are independent random variables each
taking on the values I and 0 with probability \ . Let us put efc, J=eJ t for j<k.
Now r admits the automorphism /7 if and only if for any pairy, k (j^k) one has
fi/jy, nk = fiy, k • Let us calculate now how many of the values ey k can still be chosen
arbitrarily. An easy argument shows that if j belongs to the a-th cycle of 77 (of
length ca) and k to the 6-th cycle of/7 (of length ch) (where a^b) then the sequence
of equations
8j,k = rhj,nk = sn2j, rt?k= ■■■ = Ensj,ii'k = ■ ■ ■
contains [ca,cb] different terms where [A, B] denotes the least common multiple
of A and B. Thus among the ca-cb values Sj k where j belongs to the a-th cycle
and k to the 6-th cvcle of /7 we can choose only -r—?—-—- = (c„, ch) values inde-
ka, cb]
pendently, where (A, B) stands for the greatest common divisor of A and B; all
other such &jk are then determined (a^b\ l^a = c; l^i^r).
By a similar argument we get that among the e}i k with bothy and k belonging
_2
integral part of .v. Thus there are exactly
(2.8) 2,-"<b'r «=iL2J
different graphs of order n which admit the automorphism /7, having the cycle
representation (2. 7), and the probability of P admitting the automorphism 77,
i. e. of PIP =-P is
(2.9) P(77r = r)=2,Sfl<^r «=iL2J w.
Now let us fix a graph G for which 77G = G and count the number of such graphs
which can be transformed into G by changing m of its edges.
/hi
Clearly the in edges to be changed can be chosen in (2) ways. Thus the
I m
number of graphs which can be transformed into one admitting the automorphism
77 by changing m edges can not exceed
(2. 10) I V ' 101--^(-^ ,,= 1 L2J
to the a-th cycle of 77 we can choose
independently, where [x] denotes the
343 ASSORTED GRAPH THEORY

306
P. ERD& AND A, RENYI
Now let us suppose that among the cycle-lengths ca of II there are exactly / = n—q
which are equal to 1; we may suppose cx =c2 = ■■■ =c, = 1 and ci+i&2 for
i = 1, 2, .,. r—l. Then we have
and thus
(2.11) (r_/)s| = Y and **m-
As (ca, c6) ^ min (ca, cb) S -~r~^, it follows
(2. 12)
Thus
(2. 13)
2 {Ca,Cb)+ 2
1 ^a<6<r a= 1
2 (Ca> Cb) +
1 =sa<6^r
.I[tK
+ (/■-/)'
2 J v ' 2
M<^
Thus the probability of choosing at random such a graph which can be transfor-
n
med by less than (1 — e)— changes into one admitting U as an automorphism does
not exceed
/2_ „2
hO(nlogll)
(2. 14) 2 4 .
As the number of permutations /7 with a fixed I is less than /(« — /)! =
= 2°<"!°s") we have
S/4
+ 0(n log n)
(2.15) 2 P„(e,<7)^2 2
4
^ n ^: q ^ n
4 _
Now we consider the permutations with 5 ^q =5^«. Concerning these we have
to use much careful estimations.
4 __
Let us fix first the value of q (5s^<l/«). The number of permutations which
leave n — q = / elements unchanged is clearly less than \q\^n^". In estimating
the number of ways in which the m edges to be changed can be chosen, we may
restrict ourselves to those edges, which connect either the q points which do not
remain unchanged by /7 among themselves, or edges connecting such points with
the invariant ones. Thus an upper bound for the number of choices of the
344 GRAPH THEORY

ASYMMETRIC GRAPHS
307
m < —(1—e) edges is given by
(2. 16)
2
'l) + c!(n-q)\
;n<—(1 -e)
I
nq
?(.-.)
2W(st)"« + 0(log")
/
where a =:
1
2?
and H(cc)
Now if <7 ^ 5 then a ^
(2. 17)
It follows that for 5^q-
1
10
1 1
a log2 - + (1 - a) logjj-^ .
:1 1
--=-:— and thus (as /f (x) is increasing for 0 < x < —
H(a)<H
:0,47.
:j/«
and thus
(2.18)
P„(e,<?) = 2
0(Vn]ogn) +
n2_(_j7_gj2 /n—q-
(VK>°-- «•
V P„(e, ?)^2-°'03«" + o*''").
«=5
Thus it remains only to consider permutations i7 for which <7 = 2, 3 or 4, i. e.
which interchange not more than 4 points and leave all others untouched. Let us
start with the case q = 2. The number of such permutations is clearly L • The
number of graphs of order n admitting such a permutation as an automorphism is
In- 1\
(as in this case c, =c2 = ...=c„_2 = 1, c„-l=2) 2 2 , and thus the
probability that a random graph admits an automorphism interchanging two points is
■s 2-" + 0*!°s"). Now, if a graph G* can be transformed by changing m of its edges
into a graph G admitting the permutation II interchanging P- and Pk and leaving
all other points invariant, we may suppose that all edges changed have either Pi
or Pk (but not both) as one of their endpoints.
It is clear that we may restrict ourselves to count those graphs, which can be
transformed into G by deleting edges, because any graph G* which can be
transformed in a graph, which is invariant with respect to the permutation interchanging
Pj and Pk, by changing (i. e. deleting or adding) m edges, can also be transformed
in such a graph by deleting m edges. Thus the number of such graphs G* belonging
to a fixed G does not exceed
2
'»<y (1-e)
n-2
m
H(-2-?)" + 0(logn)
As however H(x)<l for x^j it follows that
(2.19) P„(£,2)S2-*)"
345 ASSORTED GRAPH THEORY

?08
P. ERDOS AND A. RENY1
where c(e) is a positive constant depending only on e. Let us consider now the
case <7 = 3. In this case clearly cx — c2 = ••• = c„_j = 1 and c„_2 = 3 and
therefore the number of graphs admitting such an automorphism is
As further we can select the 3 points which are moved by the permutation in (,1
ways, and the permutation itself in two ways, the probability that a graph admits
as an automorphism a permutation cyclically interchanging 3 points and leaving
all others unmoved, does not exceed
3y
Now if such a graph G is fixed, all graphs which can be transformed into G
n
by changing m - — (1 — e) edges (and can not be transformed into any other graph
admitting the same automorphism by changing a smaller number of edgesj are
obtained if we choose m among the w - - 3 points left unchanged by the permutation,
and select one of the three edges connecting this point with the 3 points moved
by the permutation and delete or add this edge according to whether it is or is not
contained in G. Thus the number of graphs which can in this way be transformed
mto C is - (^)3- = 0(2-3^-1 Besides this, we may change some
m< :, (1 -e)
of the 3 edges between the 3 points moved by the permutation. As the number
of ways doing this is 8 we obtain that
(2.20) P„(e,3) = 0" ^3 '
Let us consider finally the case q = 4. Here two cases have to be distinguished:
either c, = cz = ... = c„_4 = I and c„_3 = 4 or c, = c2 = ... = c„_4 = 1
and £■„__, = c„_2 = 2. For the first case we obtain
,„i»>")+i[fH"23)+2
for the second
M<"'.
2 (c.,o+ 2|f H'\2
Thus the probabilities of a graph admitting such an automorphism are
:6f")2\V)*2~C>) = 2-3" + 0(log"> and -:=6^12^2 )+3~G) = 2~2^ 0<i°g">? res.
(4j" - " a,,u =u|4|
pectively. As regards the number of graphs which can be transformed into a given
graph G invariant with respect to a fixed permutation of the mentioned types by
n
changing not more than (I - s) edges, we obtain in the first case an upper bound
346 GRAPH THEORY

ASYMMETRIC GRAPHS 309
of order
y y /"" ■* )/w —4 —' ]g(4ra-2!^21'82" + b"og")
„(f-o ~!n\ I )\m-2l )
and in the second case an upper bound of order
1-,,,-::-
It follows that
z„fLA / /U-/r "■
(2.21) P„(e, 4)=£2-°-b's" + 0ll°s">.
Collecting the estimates (2. 15), (2. 18), (2. 19), (2. 20) and (2. 21) in view of
(2. 4) Theorem 2 follows.
§ 3. Symmetries of infinite graphs
Let r„ denote a random infinite graph which has the vertices P„ (n--1,2, ...)
and which is such that denoting by Ei<k the event that P- and Pk are connected
by an edge (j^k) the events E-^ (/, k = 1, 2, ...; ./'-=&) are independent and
P(E! fc) = 2 • We shall prove that with probability one f„ admits non-trivial
automorphisms. We can construct such an automorphism as follows.
Let us denote by A(k) the index of the vertex into which the automorphism
carries over the vertex Pk. We put .4(1) = 2 and A (2)=1.
Now let us consider P3. This vertex can be in 4 possible relations with P, and
P2 (connected with both; connected with Pi but not with P2; connected with P2
but not with PL; connected neither with P, nor with P2). Let A (3) be the least
integer (if there exists any) for which PA(i) is in the same relation with P2 and P,
as P, with P, and P2 and put A(A(3)) = 3. If A(n) is already defined for any finite
number of values of n, for instance if A(>ij)=n'j and A(n') = A(nj) (j= I, 2,..., s)
where »,, n2, ..., «s, n{, «2, ..., n's are different integers, let m denote the least
integer for which A(m) is not yet defined. Let us define A(m) as the least integer
different from m and from all values n for which A(n) is already defined, for which
PAimy is in the same relation with P„* as P,„ with P„ , and in the same relation with
P„. as P,„ with P„ (J =1,2, ..., s), and put /1(/1 (m)) = m.
In this way a non-trivial automorphism of P„ is constructed step-by-step,
provided that the construction can always be continued. But it is easy to see that
with probability 1 the construction can always be continued. This follows from the
following
Lemma 3. Let ;',, /,, ..., ik, jt,j2, ---Ji be arbitrary different natural numbers.
Then with probability 1 the number oj vertices P„ which are connected in f„„ with
each of Pn,Ph, ..., Pik and not connected with Pji,Pjl. ..., Pj, is infinite for every
choice of the indices /,, /2, ..., ik, y, ,j2, ..., jl (k, 1=1, 2, ...).
347 ASSORTED GRAPH THEORY

310
P. ERDOS AND A. Ri.NYl
Proof of Lemma 3. The probability of the event E„ that P„ is in the required
relation with all vertices P,l( ..., Pik, P}1, ..., Pjt is clearly equal to -^-^, further
these events E„ are independent. Thus by the Borel-Cantelli lemma E„ takes place
for an infinity of values of n with probability 1. As the union of a denumerable set
of sets of probability 0 has probability 0 too, with probability 1 in T there are
infinitely many vertices connected with the vertices Ph,...,P;k and not connected
with the vertices PJl5 ..., P}-t simultaneously for all choices of the indices (\, i2, ■ ■■, 4>
j\, ...,/;. This proves Lemma 3.
Thus we have proved that with probability 1 r„ admits a non-trivial
automorphism, which moreover is involutory (i. e. A{A(«)) = « for every n).
This is what we wanted to prove. It can be seen from the proof that r„ admits
with probability 1 an infinity of nontrivial automorphi?ms. As a matter of fact,
instead of putting A(\)=2- we could have prescribed A(1) = k with an arbitrary k.
It is easy to see, that our result remains also valid if instead of supposing that
the edge PjPk is contained in f„ with probability £, we suppose only that this
probability pjk is contained between the limits S and 1—5 where 0<(5<1,
admitting that this probability should depend on j and k. The result holds also if
pJk is not bounded away from 0 and 1 but is such that the series
2 Ph,nPh,n--Plk,n(l -/^1,-.)---(1 -Pl„n)
is divergent for every choice of the integers )',, ..., ik, jlr -..,.1,.
§ 4. Asymmetry of graphs of order n
with a fixed number N of edges
In this § we consider only such graphs of order n which contain exactly N
edges. If the valencies of the vertices P,, ..., P„ are denoted by vi, ..., v„ then we have
by supposition
(4.1) 2v, = 2N.
For such a graph we have by Cauchy's inequality
1 / " Y 4N2
(4.2) 2^(2 0,) =~.
Thus it follows from (1.3) and (1.4) that for such a graph
AN 8N2
(4-3) A[G]^- #—TV
n nz\n— 1)
348 GRAPH THEORY

ASYMMETRIC GRAPHS
311
Thus we have proved the following
Theorem 3. Ij a graph G oj order n has JV = A« edges 0 < A -=—-—I then
2A
(4.4) A[G]*4kll /j_1
[The maximum of the right hand side of (4.~4) is clearly attained if A = ——-,
in which case it is equal to —-—
We can prove with the same probabilistic method as applied in § 2 combined
with methods of our paper [7] that the estimate (4. 4) is asymptotically best possible
if together with n — + «= we have A-*- + «= in such a way that lim -. = + ■»:
„^+0Jog n
moreover A[G] is near to 4A 1 - for most graphs of order'« having N = Xn
\ n—\)
edges.
The meaning of the condition lim -. = + °° is that as we have shown in
„^+„,log n
[7] in a random graph of order n and having N=Xn edges the valencies of all ver-
tices are asymptotically equal with probability tending to 1 for n -*°° if -. ► + «=.
log n
§ 5. Further remarks and unsolved problems
The following problems, closely connected with that considered in § 4 can be
raised: for a fixed positive integer k, and n >~2k+ 1 determine the least value F(n, k)
such that there exists a graph G of order n, having N=F(n, k) edges and asymmetry
A[G]=k; further the least value C(n, k) such that there exists a connected graph
G of order n, having N= C(n, k) edges, and asymmetry A[G] = k. We can not give
a full answer to these questions, only some partial results. We^prove first
Theorem 4. We have C(6, 1) = 6 and C(n, l)] = [n — l for nS7.
Remark. As shown in the introduction each graph of order ^5 is symmetric,
thus C{n, 1) is defined only for n^6.
Proof of Theorem 4. For n = 6 there are, as we have seen, in the introduction,
four types of asymmetric graphs, each having the asymmetry 1; as shown by Fig. 3
among these there is one having 6 edges, the others have 7 edges or more. Thus
C(6, 1) = 6. As any connected graph G of order n has at least n— 1 edges, clearly
C(n, 1)£«-1 for n^7 with equality only
if there exists an asymmetric tree of order n. (
Now it is easy to see that for any ns7 there
exists an asymmetric tree of order n; such a
tree for n=l is shown by Fig. 8; for any
n'^1, such a tree T„ can be obtained as fol- Fig. 8
349 ASSORTED GRAPH THEORY

312
P. ERD6s AND A. RENYl
lows: Let T„ consist of the vertices P1; ..., Pn and the edges PtPi + l (.(=1,2, ...,
n — 2) and of the edge Pn-3Pn.
Thus Theorem 4 is proved. Let us add that the asymmetry of a tree can not exceed 1.
As a matter of fact, let T be an arbitrary tree; we may suppose that T has at least 3
vertices, as a tree of order 2 is evidently symmetric. Let us consider a longest path
with any fixed starting point Pl in T and let P2 be the endpoint of this path. Let
Pz be the (unique) vertex which is connected with P2 in T. Then two cases are possible.
Either P, = Pl, in this case Tis a star with center Pt, and thus is evidently symmetric;
or P3 ?£ P1, then again two cases are possible. Either P3 has valency 2; in this case
let P4 be the unique vertex connected with P3 besides P2; by omitting from T the
edge P3P4 we obtain a graph which has the symmetry interchanging P2 and P3.
If P3 has valency larger than 2, then any vertex P, connected with P3 which is not
on the path PiP2 has valency 1 because otherwise the path PlP2 would not be the
longest. In this case the tree itself is symmetric as it is invariant under the permutation
interchanging P2 and P,. As Pl has been chosen arbitrarily, we have incidentally
proved the following
Theorem 5. Let T be a tree of order n ^ 3; let us select one of the vertices of T,
say P1. Then either there exists a nontrivial permutation /7 of the vertices of T which
does not move Pt and under which T is invariant, or one can transform T into a graph
having such an automorphism by omitting one of its edges.
We do not know the exact value of F{n, 1). It is an interesting question also
what is the total number of non-isomorphic asymmetric trees of order n? We can
not answer this question; we can prove however that in a certain sense „almost
all" trees of order n are symmetric, if n is large. This is a consequence of Theorem 6
below. Before formulating this theorem we introduce the following definition. If a
graph G contains two vertices Pt, P2 of valency 1 which are connected with the
same vertex P3, we shall say that G contains the cherry P1P3P2.
.A graph containing a cherry is evidently symmetric, as it is invariant under the
permutation which interchanges the two vertices of order 1. Thus our assertion that
almost all trees of order n are symmetric if n is large, is contained in the following
Theorem 6. Let us choose at random a tree from the set of all possible trees
which can be formed from a given set of n labelled vertices, so that each of these trees
should have the sanv~ probability to be chosen. Let y„ denote the probability that the
random tree contains at least one cherry. Then we have
lim yn = \.
Proof of Theorem 6. Let P1, .... P„ denote the vertices of our random tree
T„. Let us put e(il, i2,j) = l (i1, i2,j are different natural numbers not exceeding
n) if P^PjP^ is a cherry in the random tree, i. e. if Ptl and P,-2 have the valency 1
in Tn and if both are connected in Tn with P}; let us put 6(^, i2,j) = 0 otherwise.
Taking into account that according to a well-known theorem of A. Cayley [8]
the total number of trees which can be formed from n given labelled vertices is equal
350 GRAPH THEORY

ASYMMETRIC GRAPHS
313
to n"~2, we obtain that
(n —2V~4
(5. 1) M(Eiui2j) = 'K-^r—
and
/ (n-6)"~6 if iui2,i3, U J\>h
\ n"~2 are all different,
(5.2) Af(.(f„ /,,7,).(,-3, UJ2)) = (n„5rb tf .^=7 and ^,^
V n""2 are different,
I (»-3)"-4 if i1=i3, i2=U and ^=/2
(5. 3) Af(e(/1, /2,706(^, U,j2)) = J """2 or »'i ='4. '2 =»3 and 7\ = /2
(0 otherwise.
Let r„ denote the number of cherries in Tn. Then by (5. 1), (5. 2) and (5. 3)
we obtain
M(r.) = ^3 + 0(1)
and
n2
M{ri) = —b + 0{n)
and thus
D\rn) = 0{n).
It follows by the inequality of Chebyshev that for n^kn0
Thus Theorem 6 is proved.
Now we prove the following
,-„ = oi
Theorem 7. Any connected graph of order n having n edges is either symmetric,
or its asymmetry is equal to 1.
Remark. By other words we have C(n, 2) =-« for n&7 (As we have seen any
graph of order ^6 is either symmetric or has the asymmetry 1.)
Proof of Theorem 7. Any connected graph of order n having n edges has as
well known the following structure: it contains exactly one cycle, and any vertex
of this cycle may be the root of one or more trees. Now suppose that contrary to
the assertion of Theorem 7 there exists a graph G of order n having n edges, for
which A[G]^2. In such a graph any tree attached to a vertex of the single cycle
of the graph consists of a single edge only, because otherwise by Theorem 5 we
would have A[G]^l. Let us call such an edge a ,,thorn". We can exclude the
case when to a vertex two or more thorns are attached, because two thorns make
a cherry which admits a symmetry. Now if to a vertex P of the cycle a thorn PQ
is attached, then necessarily a thorn has to be attached to both neighbouring verti-
351 ASSORTED GRAPH THEORY

314
P. ERDOS AND A. RENYI
ces of the cycle too, because if P' would be a neighbour of P which is not the
starting point of a thorn, then if P" is the other neighbour of P' in the cycle by omitting
the edge P'P" we would obtain a graph containing a cherry QPP'. Thus either a
thorn is attached to all vertices of the cycle or to none of them. As in both cases
the graph has a cyclic symmetry, we obtained a contradiction, which proves
Theorem 7.
It can be shown by a similar argument that C(n, 2)=-n + l.
Our last result is a lower estimate for F(n, 3). We prove
An 3
Theorem 8. We have F(n, 3)s—- —.
Proof. Let G be a graph of order n having N edges for which A[G] = 3. Clearly
G can contain only a single vertex having the valency 1. Let n2 be the number of
vertices of G of valency 2 and n3 the number of vertices of G of valency S3.
Clearly two vertices Pi, P2 of valency 2 can not be connected by an edge, because
if Pi and P2 were connected by an edge, and Px would be connected besides P2
with P[ and P2 besides Px with P2,'then omitting the edges PiPi and P2P2 the
resulting graph would admit the symmetry consisting in interchanging Pt and P2.
Further no vertex with valency ^3 can be connected with more than one vertex
with valency 2; as a matter of fact if Pi and P2 were vertices with valency 2 connected
with a vertex P3 with valency ^ 3, then omitting the two edges connecting the vertices
Pi and P7 with vertices different from P3 the resulting graph would contain the
cherry PiP3P2.
It follows that «3&2/z2 —1. As on the other hand n2 + n3^n —1, we obtain
3«3^2«--3. Now we have
8n-9
2N=2n2 + 3n3 ^2n—2 + n3 S—-—
and therefore N^. = ——— .
6 3 2
Finally we mention a further unsolved problem: is it true that C(n, k) = F(n, k)
for k^21
Remarks, added on November 8. 1963. Prof. R. C. Bose kindly informed us
that in a forthcoming paper he introduced a class of graphs, called by him
strongly regular graphs, which contains the class of zl-graphs discussed in
the present paper, as a subclass. A graph of order n is called strongly regular
with parameters nl,p1,p2 if each vertex of Gn is joined with pother vertices,
further any two joined vertices are both joined to exactly pl vertices and any
two unjoined vertices are both joined to exactly p2 other vertices. Clearly a
digraph of order n=\ mod 4 is a strongly regular graph with parameters n^ =
n-\
= Pi=P2= 2 ■
The notion of strongly regular graphs is closely connected with the
concept of an association scheme with two associate classes introduced by R. C. Bose
and T. Shimamoto in their paper: Classification and analysis of partially balanced
352 GRAPH THEORY

ASYMMETRIC GRAPHS
315
incomplete block designs with two associate classes (Journal of the American
Statistical Association, 47 (1952), pp. 151-184).
We should like to add further that the ,d-graph of order p constructed on
p. 301 is identical with the graph constructed by H. Sachs on p. 282 of his
paper: tJber selbstkomplementare Graphen (Publicationes Mathematicae, 9 (1962),
pp. 270-288). This paper was not known to us at the time when our paper was
written. As shovn by H. Sachs, this graph is isomorphic with its complementary
graph.
MATHEMATICAL INSTITUTE,
e0TVOS LORAND UNIVERSITY,
BUDAPEST
(Received 19 July 1962)
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[5] I. L. Sauve, On chromatic graphs, American Math. Monthly, 68 (1961), pp. 107—111.
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Hung. Acad. Sci., 5 (I960), pp. 17-61.
[8] A. Cayley, A theorem on trees, Quarterly Journal of Pure and Applied Math., 23 (1889), pp.
376-378.
[9] A. Renyi, Some remarks on the theory of trees, Puhl. Math. Inst. Hung. Acad. Sci., 4(1959),
pp. 73-85.
10] A. Renyi, On connected graphs, Publ. Math. Inst. Hung. Acad. Sci., 4 (1959), pp. 385-388.
353 ASSORTED GRAPH THEORY

THE MINIMAL REGULAR GRAPH CONTAINING A GIVEN GRAPH
P. Erdos, University College, London, and P. Kelly, University of
California, Santa Barbara
Let G be an ordinary graph of order n which is not regular and whose
maximum degree is v>0. Let H denote any regular graph of degree v which contains
a subgraph isomorphic to G. We seek the minimal order possible for H. Let x,
denote the degree of the ith vertex in G, so v — Xi is the "deficiency" of that
vertex; let <r = '£(v — Xi) be the sum of the deficiencies and d be the maximum
deficiency
Theorem. The necessary and sufficient condition that m-\-n be the minimal
order possible for H is that m be the least positive integer such that: (1) w^or/o;
(2) m2 — (v + l)m-fcr^0, (3) m^d and (4) (m-\-n)v is an even integer. The
maximum value of m is n, and for each n>3 there exists a graph G such that m = n.
Proof. Necessity. It is known that finite graphs H exist, so there is a minimal
solution, say a graph H of order m-\-n, and {m-\-n)v is clearly an even integer.
Let G' be the subgraph of H isomorphic to G and let A be the subgraph
induced on the vertices of H not in G'. Then in H there are <x joins between the
subgraphs G' and A. Since each of the m vertices of A receives at most v of these
joins, mvl^a, and clearly m^d.
Denote by m(A) the number of joins in A. The sum of the degrees of the
vertices of A, as points of A, must be mv — a, hence
(i) m(A) = %{mv-a).
Then from m(m— 1)/2 g; m(A), it follows that
(ii) m2-(v + l)m+<rtO,
so all four conditions are necessary.
To establish the sufficiency, let m be the least positive integer satisfying
conditions (1)-(4). Define a graph H by beginning with G and m extra
independent points ai, ai, ■ ■ ■ , am. Let plt pz, ■ ■ ■ , pk denote the points of G with
positive deficiencies di, • ■ ■ , dk. Let the completion of G be done in the
following way. First, px is completed by joins to the points a,\, az, ■ ■ ■ , adl in
succession. Then pi is completed by joins to successive points ait starting with aivix,
which is taken cyclically to be ax if d\ = m. These completions are possible
because m>:d. The degrees attained by points of A in this construction cannot
differ from one another at any stage by more than one. So this is also true when
the points of G are all complete.
Now let a/m = h-\-r/m, where h and r are nonnegative integers and where
r <m, and h<v if r>0. Then when the vertices of G have been completed the
354 GRAPH THEORY

1963]
MATHEMATICAL NOTES
1075
set ft of vertices au t= 1, • • • , m, consists of r points of degree h-\-\ and m — r
points of degree h. Since there are as yet no joins between points in ft, any point
of the greatest remaining deficiency v — h can be completed if v — h^m—1. But
condition (2) can be written in the form
(iii) v — cr/m ^ m — 1,
from which it follows that
(iv) v — h^m— 1+r/m.
Because 0^r/m<l, while v — h and m— 1 are integers, (iv) implies that
(v) v — h^m—1.
Thus there are in ft sufficient points so that each point individually can be
completed.
Finally, the collective completion of all the points in ft will be possible if
the sum of the deficiencies is an even integer, that is, if
(vi) r(v — h— l)+(w — r)(v — h) = mv — cr
is even. But
(vii) mv — cr=mv— [nv — 2m(G)] = (m-\-n)v — 2[nv — m(G)].
By assumption (m-\-n)v is even, hence mv — cr is even and the completion of all
points in ft is possible.
Since <r<nv, the condition m^cr/v cannot force m>n. Similarly mi— (v + l)w
+or^O always holds for m = v-\-\, and v + l = n. Condition (3) cannot force m to
exceed n —1. The maximum possible value m = n, satisfying conditions (1) and
(2) cannot be increased by condition (4), since (m-\-n)v ={n-\-n)v is necessarily
even. Thus in all cases m^n.
If n>3, let G be the graph obtained from a complete graph of order n by
deleting one join. Then v = n— 1 and cr = 2, and the condition
(viii) m} — nm+2 ^0 implies that w^n.
The second author received support for this work from grant NSF-G23718.
355 ASSORTED GRAPH THEORY

ON THE DIMENSION OF A GRAPH
Paul Erdos, Frank Harary and William T. Tittte
Our purpose in this note is to present a natural geometrical definition
of the dimension of a graph and to explore some of its ramifications. In
§1 we determine the dimension of some special graphs. We observe in §2
that several results in the literature are unified by the concept of the
dimension of a graph, and state some related unsolved problems.
We define the dimension of a graph G, denoted dim O, as the minimum
number n such that G can be embedded into Euclidean w-space En with
every edge of G having length 1. The vertices of G are mapped onto
distinct points of En, but there is no restriction on the crossing of edges.
1. Some graphs and their dimensions. Let Kn be the complete graph
with n vertices in which every pair of vortices are adjacent (joined by an
edge). The triangle K3 and the tetrahedron Ki are shown in Figure 1.
^A *A
The dimension of K3 is 2 since it may bo drawn as a unit equilateral
triangle. But clearly, dim if4 = 3 and in general dim Kn = n — 1.
By Kn — x wo mean the graph obtained from the complete graph Kn
by deleting any one edge, x. For example if3 —x and Ki — x are shown
in Figure 2.
K3 - x: 9 • •
Fig. 2.
From this figure, we see at once that dim (K3 — x)= 1 and that
dim (if4 — x) = 2 since it can be drawn as two equilateral triangles with the
same base. By a similar construction it is easy to show that in general
dim (Kn — x) = n — 2.
The complete bicoloured graph K„hn has m vertices of one colour, n of
another colour, and two vertices are adjacent if and only if they have
^Mathematika 12 (1965), 118-122]
<t>
356 GRAPH THEORY

On the dimension of a graph
119
different colours. We shall see how to determine the dimension of Km n
for all positive integers m and n. In Figure 3 are shown three of these
graphs, each of which we will see has a different dimension.
1,4"
'2,4-
3,3"
1¾. 3.
Which of the graphs Km have dimension 2? Since K11 = K2,
dimi^ i = l, and as shown in Figure 3, dim/i'14 = 2. Obviously, for
every n>\, AimK^ n = 2. There is also one other complete bicoloured
graph with dimension 2, namely the rhombus K2 a- Again from the figure,
we see that dimK2i=3 and in general that dim/^2?l = 3 when w^3.
Finally, it is easy to show that the dimension of every other graph Km n
not already mentioned in this paragraph is 4, including the famous 3 houses-
3 utilities graph K3 3. The proof is due to Lonz, as mentioned in a paper
by Erd5s [2], and proceeds as follows.
Let {mJ be the m vertices of the first colour and let {vt] be the n vertices
of the second colour. We assign coordinates in Ei to ut — (xt, yt, 0, 0)
and Vj = (0, 0, zs, «',,) in such a way that xt2 + y*~\ and zf + wf = i. Then
every distance (2(^,^)=1, proving the assertion.
In the next two illustrations of the dimension of a graph we use the
operations of the "join" and the "product" of two graphs G-', and G2.
Let V± and V2 be their respective vertex sets. The join Gt + G2 of two
disjoint grajihs contains both of them and also has an edge joining each
vertex of G1 with each vertex of G2. The cartesian product Gt x G2 of Gt
and G2 has Fj x V2 as its set of vertices. Two vertices u— (u1, u2) and
v= (vv v2) are adjacent in Gx x G2 if and only if u1 = v1 and u2v2 is an edge
of G2 or u2 = v2 and uxvx is in Gx. Let Pn denote the polygon with n sides.
By the wheel with n spokes is meant the graph Pn + K1; see Figure 4,
P4 + Kr
p5 + Kr
p6 + kx:
Fig. 4.
What is the dimension of a wheel? We already have one example since the
smallest wheel P3 + K1 = Ki has dimension 3. From Figure 4, we see that
dim (Pi + K1) = dim (P5 + ifj) = 3 and that dim (Pe + K1) = 2. By making
357 ASSORTED GRAPH THEORY

120 Paul Erd5s, Frank Haeaey and William T. Tutte
expeditious use of the unit sphere, the reader can verify that for all n > 6,
dim (Pn +1(^) = 3. Thus we observe that the dimension of the w-spoked
wheel is 3 except for " the odd number 6 ".
The n-cube Qn is defined as the cartesian product of n copies of K2;
see Figure 5. Since Q1 = K2, dim Q±= 1. Since Q2 = K2i2 = P4, dim Q2= 2.
V
^3-
Kg. 5.
The 3-cube Q3 is drawn twice in Figure 5. Its first appearance might
suggest that its dimension is 3. But its second depiction (in which two
pairs of edges intersect) shows that dim ¢3 =2. Similarly, for all n>\,
dim Qn = 2.
A modest generalization of this observation asserts that for any
graph 0, dim (GxK2) equals dimG, if dim G ^2, and equals dimG+1,
if dimG = 0 or 1.
Fig. 6.
The well-known Petersen graph is shown in Figure 6. What is its
dimension? It is easy to see (especially after seeing it) that the answer
is 2; see Figure 7.
Fig. 7.
358 GRAPH THEORY

On the dimension or a graph
121
By the way, note that the dimension of any tree is at most 2. A cactus
is a graph in which no edge is on more than one polygon. Since the
definition of dim G allows edges to intersect, it is easily seen that the dimension
of any cactus is at most 2.
In this section we have evaluated the dimension of a few special graphs.
But for a given graph G, we know of no systematic method for determining
the number dim G. Thus the calculation of the dimension of a given graph
is at present in the nature of mathematical recreation.
2. Some theorems on dimension. In the theorems of this section we
use the following concepts : the girth of a graph, the chromatic number
of a graph, and the chromatic number of a Euclidean space. The girth
of a graph G is the number of edges in its smallest polygon (if any). The
chromatic number x(G) of G is the least integer n such that the vertices of
G can be coloured using n colours so that no two adjacent vertices have
the same colour. The chromatic number x(En) °f a Euclidean space En
is the smallest number of point sets into which En can be partitioned so
that in no set does the distance 1 occur.
Theorem 1. For any graph G, dimG^2x(G).
The proof of this theorem is a single generalization of the argument
used in §1 to establish that dimKnun sg 4; see [2]. The next two theorems
do not deal with the dimension of a graph, but will be used in later proofs.
Theorem 2. (ErdOs [1]). There exists a graph with arbitrarily high
girth and arbitrarily high chromatic number.
Theorem 3. (ErdOs [4]). If G is a graph with n vertices and girth
greater than C logn, for C sufficiently large, then x(G) ^ 3.
Corollary. Under the above hypothesis, dim G ^ 6.
It is possible that the above hypothesis implies dim G sg 3 or even
dim O ^ 2, but we could not decide this question.
Theorem 4. (ErdOs [3]). Among all graphs with n vertices, q edges,
and dimension 2k or 2k + 1,
lim max— =|( 1 )
n=oo wr \ k J
The following question was posed by ErdOs [2]: What is the maximum
number of edges among all graphs of dimension d which have n vertices?
The next theorem gives the answer for d = 4.
Theorem 5. (ErdOs, unpublished). Among any n points of 2J4 the
distance 1 between pairs of points can occur at most n + [w2/4] times, and this
number can be realized if ws 0 (mod 8).
359 ASSORTED GRAPH THEORY

122
On the dimension of a graph
We now turn to some results concerning the chromatic number of a
Euclidean space. The brothers Moser [6] called for a proof of the inequality
X(E2)>3. Hadwigor [5] found the following inequalities.
Theorem 6. 4 ^(2¾ ^7.
Corollary. If dim 67 = 2, then ^(67)^7.
Klee (unpublished) proved the next theorem.
Theorem 7. For every positive integer n, x(En) is finite.
This result has some consequences for the dimension of a graph, but
they arc not as sharp as Theorem 1.
Corollary 1. If dim 67 is large, so is x(67).
Corollary 2. There exist graphs with arbitrarily high dimension and
girth.
One might think that a graph of sufficiently high dimension must
contain a complete subgraph Kn of specified order n > 2. That this is not
necessarily so follows from the last corollary.
Unsolved problems.
I. Call a graph G critical of dimension n if dim G = n and for any proper
subgraph H, dim.Il<n. For example, Kn^1 is critical of dimension n.
Characterize the critical w-diniensional grajMis, at least for m=3 (this is
trivial for n— 2).
II. Let 67 have n vertices and assume that every subgraph H with h
vertices has dimension at most in. How large can dimG be? (For
chromatic number instead of dimension, Erdos investigates tlii.s in [4].)
References
1. P. Erdos, '" Graph theory and probability", Canad. J. Math., 11 (1059), 34-38.
2. P. Erdos, '' On sots of distances of n points in Euclidean space ", Publ. Math. Inst. Hung.
Acrid. Sci., 5 (1960), 165-169.
3. P. Erdos, '' Some unsolved problems ", Pub/. Math. Inst. Hung. Acad. Sci., G (1.961),
221-254, esp. p. 244.
4. P. Erdos, " On circuits and subgraphs of chromatic graphs", Mafhnnatil-a, 9 (1,962),
170-175.
5. H. Hadwiger, " Ungoloste Problemo No. 40", Ehmente tier Math., 16 (1961), 103-104.
6. L. Moser and W. Moser, "Solution to Problem 10", Canad. Math. Butt., 4 (1961),
187-189.
Mathematical Institute, Budapest, Hungary.
University of Michigan, Ann Arbor, U.S.A.
University of Waterloo, Waterloo, Canada.
[Received on the 1th of January, 1965.)
360 GRAPH THEORY

Reprinted from
ISRAEL JOURNAL OF MATHEMATICS
Volume 4, Number 4, December 1966
ON CLIQUES IN GRAPHS
BY
P. ERDOS
ABSTRACT
A clique is a maximal complete subgraph of a graph. Moon and Moser
obtained bounds for the maximum possible number of cliques of different
sizes in a graph of n vertices. These bounds are improved in this note.
Let G(n) be a graph of n vertices. A non empty set S of vertices of G forms a
complete graph if each vertex of S is joined to every other vertex of S. A complete
subgraph of G is called a clique if it is maximal i.e., if it is not contained in
any other complete subgraph of G.
Denote by g(n) the maximum number of diiferent sizes of cliques that can
occur in a graph of n vertices. In a recent paper [1] Moon and Moser obtained
surprisingly sharp estimates for g(n). In fact they proved (throughout this paper
log n will denote logarithm to the base 2) that for n §: 26
(1) n - [logn] -2[loglogn] - 4 <; g(n) ^ n - [logn]
In the present note we shall improve the lower bound on g(n). Denote by
logtn the /c-times iterated logarithm and let H(n) be the smallest integer for which
logH(n)n < 2. Let n, = [n — logn —//(h)] and for ;'> 1 define nt as the least
integer satisfying
(2) 2"i + «j-l^n,._1.
Now we prove the following
Theorem. g(ri) §: n — log n — H(n) — 0(1).
H(n) increases much slower then the fe-fold iterated logarithm thus our theorem
is an improvement on (1). It seems likely that our theorem is very close to being
best possible but I could not prove this. In fact I could not even prove that
lim (g(n) — (n — logn)) = oo.
n = oo
The proof of our theorem will use the method of Moon and Moser [1]. We
construct our graph G(n) as follows: The vertices of our G(n) are x1,---,x„l;
yl,---yn2> zi,,,zm. where nx = [n — logn — H(n)], n2 is denned by (2) and
m = n — n^ — n2- Clearly m ~ H(n) + 0(1). Any two x's and any two y's are
joined. Further for 1 ^j < n2 yj is joined to every xt except to the xt satis-
fying
Received November 15, 1966.
233
361 ASSORTED GRAPH THEORY

234
P. ERDOS
2J'_1 + 7'-2< i^2J'+j-l
and yll2 is joined to every xt except to those satisfying
2"2"1 + n2 - 2 < i ^ «! (nj ^ 2"2 + n2 - 1).
Now we use the vertices zk, 1 ^ /c ^ m, zk is joined to yj for 1 ^_/ ^ nk + 2 and
to the x,- for 1 ^ i ^ "k + i ■ No two z's are joined. This completes the definition of
our G(n).
Now we show that our G(n) contains a clique for every
(3) nm + 2<t<Lnl
and since by m = H(n) + 0(1) and (2) nm + 2 is less than an absolute constant
independent of m,(3) implies our Theorem.
Assume first n2 = ' = "i- F°r ' = ni the set of all y's and for ( = «, the set of
all x's gives the required cliques. For «, < t < n2 we construct our clique of t
vertices as follows: We distinguish two cases. If n1 — t < 2"2_1 we consider the
unique binary expansion
«! - t = 2h + ••• + 2Jr , O^ii < ••• <jr< n2 - 1.
If 2"2_1 5? /tj — r < 2"2 (this last inequality always holds by the definition of n1
and n2) we consider the unique binary expansion
«! -r-(«! -2"2"1 -n2 + 2) =2"2_1 + n2 -2 -r = 2J1 + ••• + V,
O^ii < ■■■jr<n2-l.
In the first case consider the clique determined by yjlt-",yjr and all the x's
which are joined to all the yju, u= 1, ••-,/", in the second case we consider the
clique determined by ^-,,---,^,., y„2 and all the x's joined to y„2 and to all the
Vj,, u = 1,---, r. A simple argument shows that this construction gives a clique
having t vertices. (To see this observe that yjt 1 ^j < n2 is joined to n1 — 2J_1 —1
.v's and y„2 is joined to 2"2_1 + n2 — 2 x's and no x is joined to every vertex of
our clique since no z is joined to y„2 or to an x which is not joined to yni).
Let now ns + 2 + 1 ^ ' ^ ns + 1 + 1, 0 < s ^ m. If t = ns + 2 + 1 then the complete
graph having the vertices zs, yu ••-, vn> + 2 is a clique of size t (no x is joined to all
these vertices), if t = ns + 1+ 1 then the complete graph having the vertices zs,
.v-!, ••• x„s+, is a clique of size t (no j> is joined to all these vertices). If
'is + 2 + l< t ^ ns + 1 we consider the graph spanned by the vertices zs,yl,---,yn ,
xu ---,-^ + , (zs 's joined to all these x's and y's) and argue as in the case s = 0.
This completes the proof of (3) and of our Theorem.
It would be easy to replace 0(1) by an explicit inequality, but I made no attempt
to do so since it is uncertain to what extent our Theorem is best possible.
Reference
1. J. W. Moon and L. Moser, On cliques in graphs, Israel J. of Math. 3 (1965), 23-28.
Technion—Israel Institute of Technology,
Haifa
362 GRAPH THEORY

tlber die in Graphen enthaltenen saturierten planaren Graphen
Herrn Herbert Grotzsch zum 65. Geburtstag am 21. Mai 1967 gewidniet
Von P. Erdos in Budapest
(Eingegangen am 19. 4. 1907)
In dieser Arbeit bezeichnet 0(n: I) stets einen Graphen mit n
Knotenpunkten und I Kanten (Schlingen und mehrfache Kanten sind aus-
geschlossen). Bekanntlich hat ein plauarer Graph mit n Knotenpunkten
hochstens 3 n — 6 Kanten und ein plauarer 0(n; 3 n — 6) gibt immer eine
Triangulation der Ebene (Kugelflache).
/ \nn \
Nach einem bekannten TuRANschen Satz [7] enthalt ein Gin; + 1J
immer ein Dreieck, also einen saturierten planaren Graph mit drei
Knotenpunkten, aber ein G I n; I ) muG kein Dreieck, also auch keinen saturierten
planaren Graphen enthalten. In einem Gesprach stellte mir einmal Dirac
folgende Frage: MuG ein O
H?
/) fur stoBo Werte von I einen
saturierten planaren Graphen mit vielen Knotenpunkten enthalten? Folgender
Satz gibt eine partielle Antwort auf diese Frage (cx, ... sind absolute
positive Konstanten):
Satz 1. Set f(n) > 0 eine beliebigeFunktion. JederG («; I + f(n))
ent-
C[/(w)
halt einen saturierten planaren Graph mit mehr als Knotenpunkten.
n
Abgesehen von dem Werte von cx ist diese Schranke sckarf.
n
Unser Satz ist nur dann nicht trivial, wenn/(re) 25 - ist.
Bevor wir unseren Satz beweisen. wollen wir einige Bezeichnungen ein-
fiihren. Knotenpunkte werden mit xL. . . ., yt, ..., Kanten mit (x{, Xj) . . .
oder e{, . . . bezeichnet. Cmsei einKreis mit m Knotenpunkten, Cj^' sei eine
&-fache Pyramide, deren Basis ein Cm ist, das heiGt, xt, . . ., xm; y,, . . ., yk
sind die Knotenpunkte (xit xl + i), I ^igm-l, (xL, xm) und (xit y}),
363 ASSORTED GRAPH THEORY

14
Erdos, Saturierte planare Graphen
1 < i ^ ra; 1 5^ j :£ & sind die Kanten von C,'*'. C^' ist offenbar ein sa-
turierter planarer Graph mit m + 2 Knotenpunkten. Pm soil immer einen
saturierten planar-en Graphen mit m Knotenpunkten bezeichnen.
Anstatt Satz 1 wollen wir einen allgemcineren Satz beweisen:
71 ~ \ G fif?\
+ /(«) enthalt ein C(*} wit m > k y
4 / n
Satz 2. Jeder Gin:
Ab-
gesehen von dem Werte von. c'k ist der Satz scharf.
n'1
lemma 1. Jeder G In
+ 1 I hat eine Xante (xt, x-±) und w > c,n
■tceitere Knolenpunkte yi; ..., ym, so dap alle Kanten (a;,-, ijj), i — 1. 2;
j — 1,..., m zu G gehoren.
Das Lemma ist bekannt [1]. Die ^lenge der Knotenpunkte, die mit der
Kante e ein Dreieck in 67 bildcn, wollen wir mit S(e) bezeichnen. Unser
[n-
Lernma 1 besagt, daB in einem Gin;
\S(t)\ > c-, n existiert. '
+ 1 I immer eine Kante e mit
Wegen Lemma 1 enthalt unser Gin
f(n) I mindestens f(n)
n-
..4
Kanten eu ..., ef(n) mit S(et) > c> n. Bilden wir nun alle fc-Tupel von
Knotenpunkten von S(eL), i = 1, . . ., f(n). Offenbar gilt
... i(r)^'">r;])>(?)'/<.»(;).
Da die Anzahl allcr a us den Knotenpunkten von 67 gebildeten A>Tupel
] ist, folgt aus (1), daB mindestens ein /r-Tupel, z. B. {yx, .... yh), in
mindestens
* (¾ /(«>
der HSleA vorkommt. Es seien dieso Kanten (von 67)
., eu, u
(2) f(U)
Diese Kanten bestimmen einen Teilgraphen
G(n\ u) von G, und naeh einem Satz von Gallai und mir [2] enthalt aber
2u
G(n; u) einen Kreis mit mehr als - Knotenpunkten.
n
Dieser Kreis und y,, ..., yt bilden unser C,'*', damit sind die Satze 2
und 1 bewiesen.
Es ist leicht einzusehen, daB Satz 2 (mit Ausnahme des Wertos von c'k)
scharf ist. Um dies einzusehen, definieren wir einen Graphen G wie folgt:
Die Knotenpunkte sind x{, . . ., xt n,, y{, . . ., y.n ( ,,. Die Kanten sind
I 2 I
\Vi,yi"
?-i,yj),
1
j -\
n + 1
2 ~"~
, (yt , yit) ist da mi und nur dann
364 GRAPH THEORY

Erclos, Saturierte planare Graphen
15
Kante unscres Grapben, wenn fiir ein ganzes I
<^i\ < i-i <
n ' ' n
n-
ist. Offenbar hat unser Graph mindestcns -j- c;s/(n) Kanten, aber cr ont-
... /(¾)
bait kein C;,, mit wt > — .
re
Es ist etwas schwerer einzusehen, daB unser Graph auch kein P mit
G/, f(n) ..,
m > enthalt. Sei PUI in G enthalten. Zunachst zeigen wir, daB P
n
kcine zwei Knotenpmikte yu und yr enthalten kann, mit
lf(n) ^ At + i)f(n)
(2) - < w <. -- < l'.
« re
Der folgende cinfache Beweis stammt von T. Gallai. Jcdcr Knoten-
punkt von Pm muB in cinem Dreieck von Pm enthalten sein, es gibt aber
offenbar kein Dreieck, welehes die Knotenpunkte yu und yK enthalt. Be-
trachten wir eine durch Pm bestimmte Triangulation der Ebene. A'ehmeii
wir an, daB Pm zwei Knotenpunkte, die (2) befriedigen, enthalt. Dann kann
man die Dreiecke der Triangulation in zwei Klasscn einteilen. In der ersten
Klasso sind die Dreiecke, die mindestens einen Knotenpunkt
tf(n) (t^i)f(n)
Vu ■ <u< --- .
n n
enthalten. Diese Dreiecke enthalten dann mindestens zwei solche
Knotenpunkte. In der zweiten Klasse sind die Dreiecke, die keinen solchen
Knotenpunkt enthalten. Zwoi Dreiecke, die in verschiedenen Klassen sind, konnen
aber, wie leicht ersichtlich, keine gemeinsaine Kante enthalten. Es ist aber
urimittelbar klar, daB man die Dreiecke einer Triangulation nicht in zwei
solche Klassen einteilen kann. Daher kann Pm keine z.wei Knotenpunkte,
die (2) befriedigen, enthalten. Ohne Beschrankung der Allgemeinheit
konnen wir nun annehmon, daB die Knotenpunkte von Pm
f(n)
.r,, . . ., ru und 7/l, . . .,«/„, v SI
n
sind. Betracliten wir wieder eine durch Pm bestimmte Triangulation der
Ebene und die in Pm vorkommenden Kanten (y{, «/■), 1 Si i < j si c. Xach
dern EuLEBSchen Polyedcrsatz bestimmen diese Kanten hochstens 2 v — 4
Gebiete, und es ist leicht ersichtlich, daB in jedem dieser Gebiete hochstens
365 ASSORTED GRAPH THEORY

16
Erdos, Saturierte planare Graphen
ein xi liegen kann. Daher gilt u fg 2 v
also ist Satz 1 scharf.
Satz 3. Zujedem e > 0 existiert ein b
2f(n) 3/(re)
4 < — —. Daher ist m < --' - -,
n n
6(e) so, da fi jeder G ( n: — (1 + e)
fiir jedes m < bn m =!= 4, /n =+- ;"> -imd m ==- 7 em T3,,, enthalt.
Fiir gorades w ist der Beweis von Satz 3 sehr ahnlich dem Beweis von
Satz 1 und 2. Anstatt des Satzes von Gallai und mir muB man aber
folgenden Satz anwenden [4]: Jedes G(n; [t ns/2]) enthalt ein Cn fiir alle 2 sj I < c-, t'-.
Fiir ungerades m ist der Beweis von Satz 3 etwas kompliziorter. leh
zeige. daB jeder G
r;4
(1 + e) I fiir alle 6S2 m < bn ein C^'} niit den
Knotenpunkten xlt . . ., x2m; yi,y2,y-3, y^ und einen weiteren Knotenpunkt z
der niit yL, y2, y^ und mindestens drei der xit xt , xt , x,- . j", =: i.j = i, (mod 2)
verbunden ist. Der Beweis ist nicht ganz eirifach, und wir unterdriicken ihn.
YVie mir Herr Bollobas zcigte, enthalt dieser Graph ein P;niJ_:, mit den
folgenden Kanten: z ist niit xi , xu, a;,- , yt, y2, «/., verbunden, y,L ist mit alien
x'i s verbunden, y{ mit den x,j, i\ < j S= i-i> Vi mit den ^ , i2 5i j :S i:) und ?/■,
mit den xjy l. <-j <^ z,. unser P2m + :, enthalt noch das Glm(xx, ..., x2m).
Es bleibt noch der Fall m ===== 9. Wir erhalten offenbar ein drei-
kromatisches P9, indem wir zwei Oktaeder auf einer Seitenflache zusammen-
l n'
kleben. Es folgt unschwer a us [4], daB unscr G
(«;£(! + «))
dieses P<,
enthalt.
Es ist leicht ersichtlich, daB fiir m = 4, 5 und 7 kein drei-kromatischer
P,„ existiert, daher enthalt der dreikromatische TuRANsche Graph [7]
kein solches; es ist also leicht ersichtlich [6], daB jeder
gLi;
g(v:
3
«2-
"3.
+ 1) fiir alle 3
bn ein P„, enthalt.
Satz 4. Fiir jedes k und n > nn(k) enthalt jeder Gin:
n(\ + e)
fiir irgendein m ein Cfp.
Der Beweis ist recht kompliziert, und wir werden ihn bei einer anderen
Gelegenheit publizieren. Vielleicht gilt Satz 4 schon fiir alle
Gin:
enthalt kein P,„ mit m > 3.
G
+ /(*)
(n\
4
)■
+
~n - r
2
366 GRAPH THEORY

Erdos, Saturierte planare Graphen
17
x{, . . ., x.n + |., «/i • • ■ ■, 2/r„ ■ sind die Knotenpunkte. Die Kanten sind
(a-V, ■>/))
1 < i
■J ^
verbunden. Vielleicht aber enthalt jeder G I n :
, weiter ist a.-, auch mit jedem xt
n + 1"
:-'])
ein P„, mit
m > 3.
Zunachst wollen wir noch ohne Beweis folgenden Satz aussprechen.
Satz 5. Zu jedem f > 0 und m existiert ein 5 = d(e, m), so da/3 jeder
Gin; --(1 + e)) ein Cl°"] enthalt.
Der Satz ist scharf in dem Shine, dafi, wenn m gegen unendlich
1
strebt, dann fiir jedes e < d(e, m) gegen 0 strebt.
Es ist von Interesse, Satz 5 und Satz 2 zu vergleichen. Beide Satze
garantieren, da6 unser Gin;— (1 + e)l ein C'J*' enthalt, aber in Satz 2
kann m und in Satz 5 k von der GroBenordnung n werden, und beide Satze
sind in gewissem Shine scharf.
Der Beweis von Satz 5 kann mit den Methoden von [5] gefiihrt werden.
Literatur
[i]
[2]
[3J
[41
[0]
[7]
P. Erdos, On a theorem of Rademaciier-Turan. Illinois J. of Math, (i, 122—127 (1902)
(siehe Lemma 2, S. 124).
P. Erdos and T. Gallai, On maximal paths and circuits of graphs. Acta Math. Acad.
Sei. Hungar. 10, 337-350 (1959) (siehe S. 337).
P. Erdos, On the structure of linear graphs. Israel J. of Math. 1, 156—100 (1903)
(Satz 3).
P. Erdos and A. Stone, On the structure of linear graphs. Bull. Amer. Math. Soe. 52,
1087—1091 (1940), see also P. Erdos and M. Simonovits, A limit theorem in graph
theory. Studia Sci. Math. Hungar. 1, 51-57 (1906).
P. Erdos, On extremal problems of graphs and generalised graphs. Israel J. of Math. 2,
183-190 (1964).
M. Sijionovits, A method on solving extremal problems in graph theory, Stability
problems. Theory of Graphs, Proe. Coll. Held at Tihany Hungary 1900 Akad Kiado
and Academic Press 279-320.
P. Turan, Eine Extremalaufgabe aus der Graphentheorie (in ungariseher Spraehe).
Mat. es Fix. Lapok 4S, 430-452 (1941); see also P. Turan, On the theory of graphs.
Coll. Math 3, 19-30 (1954).
367 ASSORTED GRAPH THEORY

Part III
Combinatorial Analysis

Chapter 8
Ramsey's Theorem
Paul Erdos has worked on Ramsey's theorem throughout his career. His 1935
paper with Szekeres [18] was one of the first papers on the subject. In [124] he
defines f(k, I) as the least integer N so that a graph on ^V points must contain
a complete subgraph on k points or an independent set of/ points. The
calculation of the function / has involved many mathematicians over the last
twenty-five years. In [124] he gives what are still essentially the best
asymptotic bounds of f(k, k). The proof of the lower bound for f(k, k) is one of the
earliest uses of the "nonconstructive" method, a method that predominates in
many of his papers.
Paper [296] contains the following result: let k and n be positive integers.
Then there exists a graph which requires k colors but has no cycles of size less
than n. This is highly unintuitive since one might think that without small
cycles the chromatic number could not be forced up. A construction of this
graph was given by L. Lovasz, "Chromatic number of finite set systems,"
Acta. Math. Acad. Sci. Hungar. 79(1-2)(1968)59-67.
Paper [339] has a very good bound on /(3, k) derived by very complex
nonconstructive methods. The best upper bound is due to Graver and Yackel,
"Some graph theoretic results associated with Ramsey's theorem," J.
Combinatorial Theory, 4(1968)125-175.
Paper [407] concerns the number of monochromatic complete bipartite
graphs of a given size that must be found in a two coloring of a complete
graph. Question 2 posed in this paper is still open.
Papers [160] and [185] involve a "canonical" Ramsey's theorem.
It should be noted that many of the results on Ramsey's theorem,
including all of the asymptotic bounds, are given in ES.
Paper [iii] contains a good overview of many of the results and problems
on Ramsey's theorem. The reference to Szalai is incorrect and has caused
much confusion. In fact, any four coloring of Kf& produces a monochromatic
triangle. This result was shown by Jon Folkman and is being published
posthumously in the Journal of Combinatorial Theory in 1973 or 1974.
Papers in Chapter 8
[124] Some remarks on the theory of graphs
[160] (with R. Rado) A combinatorial theorem
371 RAMSEY'S THEOREM

[185] (with R. Rado) Combinatorial theorems on classifications of subsets of
a given set
[296] Graph theory and probability. I
[339] Graph theory and probability. II
[376] (with C. A. Rogers) The construction of certain graphs
[407] (with J. \V. Moon) On subgraphs of the complete bipartite graph
[iii] Applications of probabilistic methods to graph theory
[iv] (with R. J. McEliece and H. Taylor) Ramsey bounds for graph products
372 COMBINATORIAL ANALYSIS

SOME REMARKS ON THE THEORY OF GRAPHS
P. KKDOS
The present note consists of some remarks on graphs. A graph G
is a set of points some of w hich are connected by edges. We assume
here that no two points are connected by more than one edge. The
complementary graph G' of G has the same vertices as G and two
points are connected in G' if and only if they are not connected in G.
A special case of a theorem of Ramsey can be stated in graph
theoretic language as follows:
There exists a function /(k, I) of positive integers k, I with the
following property. Let there be given a graph G of n ^f(k, I) vertices.
Then either G contains a Complete graph of order k, or G' a complete
graph of order I. (A complete graph is a graph any two vertices of
which arc connected. The order of a complete graph is the number of
its vertices.)
It would be desirable to have a formula for f(k, I). This at present
we can not do. We have however the following estimates:
Theorem I. Let k ^3. Then
2*" < /(k, k) g C2t_2.t_, < 4*-».
The second inequality of Theorem I was proved by Szekeres,1 thus
we only consider the first one. Let N^2kn. Clearly the number of
different graphs of N vertices equals 2V(Ar_1)'2. (We consider the
vertices of the graph as distinguishable.) The number of different
graphs containing a given complete graph of order k is clearly
2Jvw-i)/i/2*(*-D/J. Thus the number of graphs of A^2*'2 vertices
containing a complete graph of order k is less than
2A'(n-d/i jyt 2N(N~l)>2 2Ar<Ar_1)'2
(1) CN * < <
2*<*^»;i kl 2*<*-'"* 2
since by a simple calculation for N^2kn and £3:3
2JV* < *!2*<*-'>'8.
But it follows immediately from (1) that there exists a graph such
that neither it nor its complementary graph contains a complete
subgraph of order k, which completes the proof of Theorem I.
The following formulation of Theorem I might be of some interest:
Received by the editors April 30, 1946, and, in revised form, October 20, 1946.
1 P. Erdteand G. Szekeres, Compositio Math. vol. 2 (1935) pp. 463-470.
292
373 RAMSEY'S THEOREM

THE THEORY OF GRAPHS
2"3
Define A(n) as the greatest integer such that given any graph G of n
vertices, either it or its complementary graph contains a complete
subgraph of order A (n). Then for A (n) £ 3
log n 2 log m
(2) —=—- <,!(«)< 5_-
2 log 2 log 2
The proof of (2) follows immediately from Theorem 1. (4A <">>«,
2*<»»*<n.)
The general theorem of Ramsey will now be stated.
Theorem (Ramsey). Given three positive integers i, ft, I, t gft, i£l,
there exists a function f(i, k, I) with the following property: If n ^/(t, k, I)
and if there is given a collection of combinations of order i of n elements,
such that every combination of order k contains at least one combination
of order i of the collection, then there exists a combination of order I all
of whose combinations of order i belong to the collection.
Several proofs of this theorem have been published.1 Szekeres's
proof gives the best known limits for/(», k, I). He proves1
(3) /(t. ft, /) ^ /(«' - 1. /(»'. k- 1,1), /(»-, ft. I - 1)) + 1;
also clearly/(1, ft, /)=ft+/-l;/(», i, I) = /;/(»', ft, »')=ft. By the same
method we used in the proof of Theorem 1 we obtain that for
sufficiently large A
(4) /(t, ft, ft) > c*"'*1,
A(i, n) < c,(log n)1"1'-1'.
(To see this put ft=Ci(log n)lHi~l) in (4) and observe/(»', ft, ft)>«,
for sufficiently large ct.) Here c and Ci depend only on n and t, and
A(i, n) is the greatest integer with the following property: Split the
combinations of order ft of n elements into two classes U\ and Uj in
an arbitrary way. Then there exist A (t, n) elements all whose
combinations of order ft are either in Vi or in Ut. The values given by
(4) are very much smaller than the values given by (3).
From (3) we obtain*
/(ft,/) =/(2, ft,/) SCt|,,>
Thus
«3, 0 ^ C,+1.2.
It is possible that
374 COMBINATORIAL ANALYSIS

2<M
p. erdOs
/(3. /) - O(0.
Our method used in the proof of Theorem 1 does not enable us to show-
that/^,/)//—».
Before concluding we prove the following theorem.
Thkorkm 11. Let there be given (*-l)(/-l) + l integers oi<a,
< • • • . Then either there exist k of them no one dividing the other or I
of them each a multiple of the previous one.
Construct a matrix aj" with the following properties: (1) no of'
is a multiple of ;iny ajr> with j£r; (2) every o|r+l) is a multiple of
some a,''1; (3) all the rt's occur among the a\ once and only once. If
any row contains k or more elements we have k o's, no one dividing
the other. If not, it clearly follows that the number of rows must be
at least I. Now we obtain from (2) that by considering any o{" we
obtain a sequence of I rt's, each being a multiple of the previous one,
which completes the proof. The (k — 1)(/-1) integers p'm, l^u^k — 1;
l£v£l— 1, pu primes, show that (k— 1)(/- 1) + 1 is best possible.*
By the same method we can prove the following theorem.
Theorem Ha. Let there be given a graph G of (£-1)(/-1) + 1
vertices. Then either G contains a complete graph of order k, or G' contains
a directed path of I vertices, for every orientation of the edges of G' in
which there are no directed closed paths.
Recently very much more general theorems have been proved by
Grtlnwald and Milgram.* They in fact proved (among others) that
the condition that G' contains no closed directed path is superfluous.
We suppress the proof of Theorem 11a since it is essentially the
same as that of Theorem II. (We only remark that a connected to b
by a line directed from a to b should be replaced by o divides b.)
Stanford University
* This proof is due to J. Brunings.
* Oral communication.
375 RAMSEY'S THEOREM

A COMBINATORIAL THEOREM
P. Erdos and R. RADof.
[Extracted from the Journal of the London Mathematical Society, Vol. 25. 1950.]
1. F. P. Ramsey (1) proved the following theorem. Let n be a positive
integer, and let A be an arbitrary distribution of all sets of n positive integers
into a finite number of classes. Then there exists an infinite set M of positive
integers which has the property that all sets of n numbers of M belong to the
same class of A. Apart from its intrinsic interest the theorem possesses
applications in widely different branches of mathematics. Thus in (1)
the theorem is used to deal with a special case of the " Entscheidungs-
problem" in formal logic. In (2) the theorem serves to establish the
existence of convex polygons having any number of vertices when these
vertices are to be selected from an arbitrary system of sufficiently many
points in a plane. In (3) it is a principal tool in finding all extensions of
the distributive law
{a-\-b){c-\-d) = ac-\-ad-\-bc-\-bd
to the case where the factors on the left-hand side are replaced by
convergent infinite series. Finally, Ramsey's theorem at once leads to
Schur's result (4), which asserts the existence of a number nk such that,
whenever the numbers 1,2, ..., nk are arbitrarily distributed over k classes,
at least one class contains three numbers x, y, z satisfying x+y = z. The
estimate of nk obtained in this way is, however, inferior to Schur's estimate
nk < e k{
The object of the present note is to prove a generalisation of Ramsey's
theorem in which the number of classes of A need not be finite. We consider
the term "distribution of the set Q." as synonymous with "binary,
reflexive, symmetrical, transitive relation in Q.". Let N= {1, 2, 3, ...},
and denote, for neN, by Q.n the set of all subsets {a1; a2, ..., an] of N,
where ax <a2 < ... <an. Let k, vlt v2, ..., vk be integers, O^Lk^n,
0 < vx < v2 < ... < vk ^n. Consider the following special distribution of
Q. , called the canonical distribution A'*' „ of Q„, Two elements
{a-L, ..., an), {.¾. ..., bn) of Qn are in the same class of A* ,, if, and only if,
a1<a2<... <an; b1<b2< ... <bn,
ari = b,x; a,t = b„t;...; a„k = b,k.
There are exactly 2" canonical distributions of Qn. We mention the
following two extreme cases of such distributions : (i) A'0), the distribution
t Received 26 August, 1949; read 17 November, 1949.
376 COMBINATORIAL ANALYSIS

260
P. Erdos and R. Rado
in which all elements of Q.n form one single class, (ii) A["'2i n, the
distribution in which every element of Q.n forms a class by itself. We shall prove
Theorem I. Let n be a positive integer. Let A be an arbitrary
distribution of all sets of n positive integers into classes. Then there is an infinite set
N* of positive integers and a canonical distribution A,, ,, such that, as far as
subsets of N* are concerned, the given distribution A coincides with the canonical
distribution A'*'
If, in particular, A has only a finite number of classes then the canonical
distribution of Theorem I, having itself only a finite number of classes,
must be A(0), so that Ramsey's theorem follows from Theorem I.
2. It may be worth while to state explicitly the special case n = 2 of
Theorem I.
Theorem II. Suppose that all pairs of positive integers (a, b), where
a <Cb, are arbitrarily distributed into classes. Then there is an increasing
sequence of integers xly x2, x3, ... such that one of the following four sets
of conditions holds, where it is assumed that a </3; y <§:
(i) All (xa, Xp) belong to the same class.
(ii) (xa, Xp) and (xy, xs) belong to the same class if, and only if, a = y.
(iii) (xa, Xp) and (xy, xs) belong to the same class if, and only if, /3 = S.
(iv) (xa, Xp) and (xy, x&) belong to the same class if, and only if, a = y;
p = s.
We shall deduce Theorem I from Ramsey's theorem. Our argument
does not make any use of Zermelo's axiom. Ramsey stated explicitly!
that his proof assumes Zermelo's axiom. It is, however, very easy to
modify his proof in such a way that this axiom is not required. In order
to establish Theorem I without the use of Zermelo's axiom, we give a brief
account, in §5, of such a modified proof of Ramsey's theorem.
3. We introduce some notations and definitions. The letter A denotes
distributions of objects into classes. The relation X=Y (.A) expresses
the fact that X and Y are objects distributed by A, and that X and Y
belong to the same class of A. Letters A, B, C, D denote typical finite
subsets of N. The number of elements of A is | A J. A relation J
(1) A1:A2:...:Am=B1:B2:...:Bm
t (1), Theorem A.
t Sefctheoretical operations are denoted by the common algebraic symbols, and
brackets { } are only used in order to define sets by means of a list of their elements.
377 RAMSEY'S THEOREM

A COMBINATORIAL THEOREM.
251
means that there exists a function/(x), defined for xzA1-\-...-{-Am and
having functional values in N, which has the properties:
if x<y, then f(x) <f(y),
Thus (1) simply means that, as far as the order relation in N is concerned,
the relative position of the sets A^ to each other is the same as that of the
B^. A relation
A1:A2=B1:Ba=C1:C2
is, by definition, equivalent to the simultaneous validity of the two relations
A.-.A^B^.B,, BX:B2=C^.C2.
4. Using the notation and definitions of §3, we can state Theorem I,
for a fixed n ^ 0, as follows-)".
Proposition Pn. Let A be a distribution of Qn. Let C0 be fixed,
j O0\ = n. Then there is an infinite subset N* of N and a subset C0* of CQ
such that the following condition holds: if
A + BCN*; \A\ = \B\ = n; A* : A = B* : B = C0* : CQ,
then A = B (A) if, and only if, A*=B*.
A corollary of the proposition Pn is the following test for a distribution
to be canonical.
Theorem III. A distribution A of 0.n is canonical if, and only if,
whenever
A = B(.A); A:B=C:D,
then C~D(A).
5. Our "choice-free" version of Ramsey's proof of his theorem runs
as follows. Let A be a distribution of Q.n into a finite number of classes.
We want to define an infinite subset M(A) of N such that, for some class k
of A, we have A e* whenever A c M(A), \A\ = n.
If 7i = 0, then we may put M(A) = N. Let n > 0, and use induction
with respect to n. If azN—M, where
M = {xlt x2, ...}cN; x1<x2<...,
f The case n = 0 is included merely in order to have an easy start of the induction
proof which ia to follow.
378 COMBINATORIAL ANALYSIS

252
P. Erdos and R. Rado
we define the distribution A{M, a) by putting
A = B{.A(M,a))
if, and only if,
A + B<zM, \A\ = \B\=n-l,
{a}+A=B{a}+B{.A).
By induction hypothesis, appliedf to the set M, in place of N, and the
distribution A(M, a), there is a well-defined infinite subset a(M, a) of M
and a class k[M, a) of A satisfying
{a}+A £K(M, a) whenever A<zo(M,a), \A\ — n—\.
Now we define, inductively, numbers ak and sets Mk not containing ak.
Puta1 = 1; M1 = N-{1). Let
a;+i ^6 the least number of a(Mu a;), and
put Ml+1 = a(Mlt «()~{«/+i} (1 — 0, 1, •••)• Let ^0 be the least number
such that K(Mk , ak ) = K(Mk, ak) for infinitely many k, and let ^0, fc1; ...
be all numbers k satisfying this last equation, k0 < k± < Then we may
put Jf (A) = {ak , ak , ...}. This proves Ramsey's theorem.
6. We now prove Pn. Clearly, P0 is true. For we may put N* = iV;
Cq* = CQ. Let n > 0, and use induction with respect to w. Let A be a
distribution of Q.n. Choose some fixed D0 satisfying \D0\ = 2n. Define,
for any A such that | A J = 2n, the set </>(A) of pairs of subsets of Da by putting
0(4)= 2 {(D',D'%
i'+i"C4
4': 4": 4 = D': D": Z>0
The set (f>(A) characterizes the effect of A on the subsets of A. We define
A* by putting
(2) A = B(.A*)
if, and only if, j A | = | B j = 2n; <f>(A) = <f,(B).
Since A* has only a finite number of classes it follows from Ramsey's
theorem that there is an infinite subset M of N such that (2) holds whenever
A + B<zM; \A\ = \B\ = 2n.
Without loss of generality we may assume that M = N. For all our
arguments are only based on order relations in N.
| The relation i><—*x„ sets up a one-one mapping of N oniW. By means of this mapping
there corresponds to every well-defined subset of N a well-defined subset of M, and vice
versa.
379 RAMSEY'S THEOREM

A COMBINATORIAL THEOREM. 253
Consider any sets A', B', C, D' satisfying
(3) A' = B'(A),
(4) A':B' = C':D'.
We want to deduce that
(5) C'^D'(A).
According to (3), (4), one can choose A and B satisfying
A' + B'czA; C'+D'<zB; \A\ = \B\ = 2n,
(6) A':B':A = C':D':B.
Then (2) holds and therefore, in view of (6), (3), and the definition of A*,
also (5). The fact that (3) and (4) imply (5) will briefly be described by
saying that A is invariant.
Case 1. Suppose that A = B (.A) only holds if A = B. Then the
conclusion of Pn is true if we put N* — N; C0* = C0.
Case 2. Suppose that there are sets A0, BQ satisfying 40 = 50(.A);
Aa =£ BQ. Put
41== 2 {2x}; Bx= 2 {2x}.
xtA0 xtB0
Then
■"■a '■ BQ = Ax : Blt
and therefore, since A is invariant,
(7) A^B^A).
As A0 ^= BQ, we can choose x0eB0—A0B0. Put
B2= (B1-{2x^) + {2x0+l}.
Then
■^■o'- B0 = A1: B2
and hence, again on account of the invariance of A
(8) A± = B2{A).
From (7) and (8),
(9) B1 = Ba[A).
There is C0' satisfying
(B0-{xQ}):BQ = CQ':C0.
Now consider any sets A3, Ai such that
1-^3 1 = I-^41 = n > Az^Ai; A': A3 — A' :A^= C0' :C0,
380 COMBINATORIAL ANALYSIS

254
P. Erdos and R. Rado
where A' is some suitable set. We shall show that
(10) A3 = At[A).
We may assume that
A' = A3-{x3] = Ai-{xi}; x3<xi.
Then B{: B.2 = A3:A^, and therefore, since (9) holds and A is invariant,
(10) follows. In other words, if sets A and A' satisfy A': A = C0': C0, then
the class of A which contains A only depends on A' and not on A~A'.
Hence every set A" satisfying
\A"\ = n-\; 4"c{2, 4, 6, ...} = JV",
say, determines a unique class K{A") of A, namely that class which contains
all A satisfying A" :A = CQ': C0. Such sets A always exist.
Define A" by putting
A" = B"(A")
if, and only if,
\A"\ = \B"\ = n-l; A" + B"<zN"; K(A") = K(B").
By induction hypothesis, the proposition Pn_x is true for A". Thus there is
an infinite subset N'" of N" and a subset C'a" of C0' such that the following
conditions are satisfied. Let
A" + B" C N'" ; A'" : A" = B'" : B" = C^" : C„\
Then A" = B" (.A") if, and only if, A'" = B'"'. In view of the definition
of A", this means that the conclusion of Pn holds for the given distribution
A if we put
N* = N'"; C0*=Co".
This proves Proposition Pn and hence Theorems I and II.
7. Wo now prove Theorem III. First of all, suppose that A is a
canonical distribution of £ln, say A = A[*'... „ . Then the relation A = B (.A)
means that
A = {alt ..., an}\ B= {blt ..., bn),
a1<...<an; b1<...<bn,
a„K = btt(\ ^k ^k).
If now C =--{cv ..., cn}; D = {du ..., dn],
Cj<... <cn; dx<...<dn,
381 RAMSEY'S THEOREM

A COMBINATORIAL THEOREM.
255
then the validity of A : B= C:D implies that cv = dv (1 ^C«r < k), i.e. that
C^D(.A). Hence A is invariant.
Vice versa, suppose that A is invariant. By Theorem I, we can find
an infinite subset N* of N, so that A is canonical in N*, say A = Ai4)___v , as
far as subsets of N* are concerned. But, since A is invariant in the whole
set N, this obviously implies that A = A!*' „ as far as all subsets of N are
concerned. This proves Theorem III.
References.
1. F. P. Ramsey, " On a problem in formal logic ", Proc. London Math. Soc. (2), 30 (1930),
264-286.
2. P. Erdos and G. Szekeres, " A combinatorial problem in geometry ", Composiiw Mathe-
matica, 2 (1935), 463-470.
3. R. Rado, " The distributive law for products of infinite series ", Quart. J. of Math.,
11 (1940J, 229-242.
4. I. Schur, " Ueber die Kongruenz x'" + y"' = z" (inodp) ", Jahresbericht der Deulschen
Mathemaliker-Vereinigung, 25 (1916), 114—117.
University of Illinois,
Urbana, Illinois, U.S.A.
King's College, London.
382 COMBINATORIAL ANALYSIS

COMBINATORIAL THEOREMS ON
CLASSIFICATIONS OF SUBSETS OF A GIVEN SET
By P. ERD(5S and R. RADO
[Received 5 Xovember 1951.—Read 15 Xovember 1951]
Introduction
Let 8 be a set, and denote by Cln(S) the set of all subsets of S which contain
exactly n elements. F. P. Ramseyf proved the following theorem:
Given any positive integers k, n, N, there is a positive integer M which has
the following property. If S = {1, 2,..., M}, and A is any distribution of
0,n(S) into k classes, there is always an element 8' of Q.y(8) such that the
(^) elements of Q.n(8') belong to the same class of A.
We denote by R(k,n,N) the least number M possessing this property,
and we call R Ramsey's function. Clearly
R(l,n,N) = .V; R(k,l,N) = k(N—1)+1,
so that only the case k ^ 2; N ^ n ^ 2 is of interest.
All known proofs (1), (2), (3), (4) of Ramsey's theorem give upper
estimates for R which are so large that they are hardly expressible explicitly
in terms of the fundamental algebraic operations. A modification of the
known methods of proof leads to a new upper estimate for R (Theorem 1)
which is in general much better than the known estimates and which is,
moreover, easily expressible in terms of (n— l)-fold exponentiation.
In (4), Ramsey's theorem, or rather its companion theoremj in which
both 8 and *S" are denumerably infinite, was generalized so as to cover
the case of arbitrary distributions into infinitely many classes. The main
step in the proof of this generalized Ramsey theorem consisted in showing
that every 'invariant' distribution of £ln(S) is 'canonical'. A distribution
A of D.n(8), where 8 is a set of real numbers, is called invariant if the following
condition holds. Suppose that A = {alv..,an}\ B = {61,...,ftn} are any
elements of Qn(S) belonging to the same class of A, and that/(¾) is any
function defined and increasing in the union A^B and having its values
in 8. Then always the sets {f(a1),...,f(an)} and {/(^),...,/(&„)} belong to
the same class of A. A distribution A of Qn(S) is called canonical if integers
k, vK can be found satisfying
0 < k < n; 1 < v1 < v2 < ... < vk < n,
such that the above sets A and B, under the assumption
ax < a2 < ... < an; bx < ... < bn,
t (1), Theorem B. + (1), Theorem A.
Proc. London Math. Soc. (3) 2 (1952)
383 RAMSEY'S THEOREM

418 P. ERDOS AND R. RADO
belong to the same class of A if and only if aVl = bVi; aVl = bV2\ ...; aVk = bV]c.
In this case we write A = A'*' „..
Clearly every canonical distribution is invariant. In (4), Theorem III,
it was proved that if S = {1,2,...}, then every invariant distribution is
canonical, so that the two classes of distributions coincide. It turns out
that for finite sets S this is no longer true. In Theorem 2 of the present
note necessary and sufficient conditions on n and N are established in
order that every invariant distribution of Qn({l, 2,..., N}) should be
canonical.
By means of Theorem 1 and Theorem 2 we obtain a finitist version
(Theorem 3) of the generalized Ramsey theorem of (4).
The next section of the paper is largely devoted to transfinite extensions
of Ramsey's Theorem A in the case k = n = 2. The general type of
problem arising in this field, of which we have only partial solutions, can
be characterized as follows. Suppose that all sets of two real numbers are
distributed into two classes Kx and K2. Let us say that an order type <j> is
realizable in the class K^ if there exists a set X of real numbers, of order type <j>
under the natural order according to magnitude, such that Q2(X) c K^.
Ramsey's Theorem A ensures that the first infinite ordinal number
oj = {1, 2,.,.} is always realizable in some class. A very interesting example
due to Sierpinski (last section, Example 4 a) shows that it can happen that
the only order types realizable in Kx are denumerable ordinals, and at the
same time the only types realizable in K2 are the converses of denumerable
ordinals, i.e. order types obtained from ordinal numbers by replacing every
relation x < y by the corresponding relation x > y. The most concrete
result obtained in this note (Theorem 7) implies that every order type aj + ra
is realizable in some class, for any finite m. In Theorems 4-8 various results
are established pointing in the direction of a fairly plausible conjecture:
in every distribution every denumerable ordinal is realized in some class. The
section concludes with two results (Theorems 9 and 10) concerning
distributions of Q„(£) for arbitrary finite n and having any number of classes,
finite or infinite.
In the final section of the paper a number of examples are given which
show that in certain directions Ramsey's theorem cannot be generalized.
Some of these examples give rise to unsolved problems, the most interesting
of which seems to be the following one, discussed in §§ 2 and 3 of the last
section. Given an infinite set S, is it possible to divide all finite subsets of S
into two classes in such a way that every infinite subset of S contains two finite
subsets of the same number of elements but belonging to different classes ?
The answer is in the affirmative if the cardinal of S is at most that of the
continuum, and there are, in fact, many different methods of effecting
384 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 419
the required classification. Nothing, however, is known for sets whose
cardinal exceeds that of the continuum.
The last example of the paper is not concerned with Ramsey's theorem
but with the following theorem due to van der Waerden (5). Given positive
integers k and I, there is a positive integer m such that, if the set {1, 2,..., m) is
divided into k classes, at least one class contains l-\-1 numbers which form an
arithmetic progression. The least number m possessing this property is
denoted by W(k,l) (van der Waerden's function). Our final example yields
what seems to be the first non-trivial, no doubt extremely weak, lower
estimate of W, namely W(k, I) > ckH^. An upper estimate of W, at any
rate one which is easily expressible explicitly in terms of the fundamental
algebraic operations, seems to be beyond the reach of methods available
at present.
Notation and definitions
Brackets {...} are used exclusively in order to define sets by means of a
list of the elements they contain. Thus order and multiplicities are
irrelevant, so that {1,1,2} = {2, 1}. If A and B are sets, then A + B, AB,
and A—AB denote their union, intersection, and difference respectively,
and \A\ denotes the cardinal of A. Set inclusion, in the wide sense, is denoted
by A c B. The symbol {a1,a2,.-.,an}^ denotes the set {a1,....an} and also
expresses the fact that ar ^ as (1 ^ r < s ^ n), and similarly {^,..., &„}<
denotes the set {6^..., &n} and at the same time expresses the fact that
b-L <C ... <C bn. Throughout, even in arguments involving transfinite
ordinals, the symbol xv x2,... denotes a sequence of type oj, where a> stands
for the first infinite ordinal.
For typographical reasons we frequently replace symbols like
I a„ by ILSI™^
and similarly when in place of ]£ we have YJ_, min, or max. As stated in
the introduction, we put £ln(S) = ^ S' c S; \S'\ = n[]{S'}.
The letter A is used to denote partitions or distributions of sets S into
classes, i.e. equivalence relations on S. The fact that A is a distribution
of S is also expressed by saying that A(x) is defined for x e S. We use the
notation and the calculus of partitions developed in (6), which will now be
briefly described.
A relation x = y (. A) expresses the fact that x, y e 8, and that x and y
belong to the same class of A. The number of non-empty classes of A is
denoted by | Aj. The following two methods are employed for generating
new distributions from given ones.
385 RAMSEY'S THEOREM

420 P. ERDOS AND R. RADO
(i) If A1,..., Am are denned in S, then the equation
A*(x) = n 1 < fi < nOV1) (xeS)
defines that distribution A* of S for which the relation
x = y(.A*)
is equivalent to the system of relations
x = y(.^) (l</i<m).
(ii) If A(y) is denned for y e T, and if /(¾) is a function on S into T, then
the equation A'(z) = A(/(z)) (x e S)
defines that distribution A' of S for which the relation
x1 = x2 (.A')
is equivalent to the relation
f(Xl)=f(x2)(.A).
Estimate of Ramsey's function R(k, n, N)
We define a binary operation * by putting, for positive numbers a and b,
a%b = ab.
Furthermore, we put, for n ^ 3,
<z1*a2* a3 *...*«„ = «!* (a2* (a3 *(...* («„-!* «„)...))).
Then, if 1 <; m < w,
<z1*a2* ...*am*(am+1*...*«„) = o1*o2*...*on,
where the symbol <z1*a2* •••*am has the value a1 if m = 1.
Theorem 1. Ramsey's function R, defined in the introduction, satisfies
the inequality
R{k,n,N) < k*(kn-1)*(kn-2)*...*(k2)*[k(N — n)+l] (1)
(k > 2; A7 ^ w > 2).
ira particular, R(k, 2, AT) < feW-a+i.
Ramsey, in his paper (1), states that his method yields the estimate
R{k,2,N) < (...((A7!)!)!...)! (*—1 symbols '!'),
but he emphasizes that he believes the right-hand side to be far too large.
The result obtained in (2) is
R{k,2,N) < (fc«-v-i>+2)/(fc—1).
For k = n — 2, the result of (3) is slightly better than (1). We find
from (3): R(2, 2,N) < (25j2) ~ 77-* 22^2^^ (N -> oo);
while Theorem 1 gives R(2,2,N) < 2
2iV-3
386 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 421
But for k = 2; n = 3 the comparison is heavily in favour of the new value.
We find
from (3): an estimate considerably weaker than
B(2,3,N) < 2* 2*...* 2 (2N—2 'factors')
and from Theorem 1: R{2, 3,N) < 2i2N'\
In the special case k = n = 2 Theorem 1 asserts that every graph of order
22N~3 contains either N independent nodes or a complete subgraph of order N.
A transfinite analogue of this result is proved in (8).
Proof of Theorem 1. Let k ^ 2; N ^ n > 2. We first dispose of a trivial
case. If N = n, then R(k,n,N) = n, and the theorem asserts that
n < k*(kn-1)*(kn-2)*...*(ki)*[k.0+l]. (2)
If n = 2, then (2) asserts that 2 ^ k, which is true. Now suppose that
(2) holds for all k and some n = n0 > 2. Then, when n = n0 is replaced
by n0-\-l, the right-hand side of (2) increases by at least a unit, so that
(2) holds for n — n0-\-l. Hence the conclusion holds for N = n, and we
may assume that N > n.
Let ibea finite set. The construction we shall describe will be possible
provided that \A | is sufficiently large. A sufficient condition on \A | will be
determined after the construction has been defined. Throughout this proof
the letter B denotes subsets of A such that \B\ = n~~2.
We are given a distribution A of Qn(A), such that [ A| ^ k. We choose
{a1,a2,...,an_1}lt cA and put
K-i(x) = ^({oi.-^n-i^}) {xeA~~{a1,...,an_1}).
Then |Are_1| ^ k, and there is Anc A~~{a1,...,an_1] such that |Are_1| = 1
{nAnA \An\ > (\A\~n+l)k-K
We choose an e An and put
K(*) = n5c{«iv-,a„-i}DA(JB + {aw,x}) (z e An—{an}).
Then (Aj < k*(%l\), and there is An+1c An—{an} such that [A„| = 1
^^^ \An+1\>{\An\-l)k-(nn--\).
Generally, let m > n, and suppose that elements o1,..., am_i and sets
An, An+1,..., Am have been defined, and that Am ^ 0. Then we choose
am e Am and put
Am(*) = H B c{aly...,am_^U^{BJr{am,x}) (x e Am—{am}).
Then |Am| < ^*(Jfr21)> anci there is Am+1 cAm—{am} such that |Am| = 1
[nAm+1' \Am+1\>(\AJ-~l)k-(Z-i).
Now put I = 1 + R(k,n— 1,N— 1).
f i.e. all elements of An belong to the same class of An_i.
387 RAMSEY'S THEOREM

422 P. ERD6S AND R. RADO
Then I > N > n. If \A\ is sufficiently large, then Am + 0 (n < m < I),
so that «!,..., «; exist. Let
^'({Pi>->Pn-i}) = A({opi,...,opn_1,o,}) (1 < Pi < ... < ?„_! < £)•
Then |A'| ^ k, and by definition of? there is I)c{l 1-1} such that
|A'j = 1 infV^D), l-DI = N—l. Finally, we put
A' = fa}+ !PeDn{ap} = {%,-,%}-,
say. Then [A[ == 1 in Qn(A'). (3)
For let C = {a ,..., apn}cA'; 1 < p1 < ... < pn < ?. Then, since opn, ateApii,
we have <z„ = a, (. A. ), and hence
pn l v Pn—\'J
Also, since {api,...,a„ J c<4', we have
{fflpi>->aP-i) = {aAi'->aA»-i} ('A')'
i.e. {%, ^.1-^ = ^---^-1^} (-A)- (5)
By (4) and (5), C = C0 (.A), where C0 = {a^1,...,o^n_1,a.} is independent of
the choice of C Hence (3) follows.
We shall now obtain a value for \A | which will ensure that the
construction can be carried through. For such a value we shall then have
E(k,n,N) <\A\.
Put <n = ^(1^1-^+1),
tm+i = k-(n-t)(-l+tm) (n^m<l).
All we require is that tt > 0. Now, if fc-U-^) = km,
t, = */-2(-l+*V8(-(-l + ^-l(-l + «„))-))
= «-'/-2 kl-2'H-a •'• "i-2 ^(-3 '•• "-n-l + ""f-2 •" "-'n-1 '«•
Hence a sufficient condition on [ A \ is
^-2--^-2(1^1-^+1) > kl_2+kl_2kl_3 + ...+kl_2...kn_1,
i.e. |^4| —W+l > fc(»-2)+- + (^-2) + ^(+- 2)+- + (^-2) + ...+^-2).
A possible value is
\A\ = w+fc(»-i)+A.-(»-i)+... + fc(»zi),
so that
1-2 , ■.
R(k,n,N) ^n + Y k*l
Zw \w—1
A-»-l v
1-2 ' 1-2
+ w + J &*A*(rc—1) < rc + 2 [&*(A+l)*(rc— 1) — &#A#(w —1)]
X = n — 1 A = ?t — 1
= W + fc*(£— 1)*(W—1)—fc*(»l—l)*(7l—1) < &# (?— l)*(7l— 1)
= fc*iJ(fc,»-l,xV— 1)*(W— 1),
R(k,n,N)*n < (/kn)*iJ(fc,7i—l,iV—l)*(rc—1).
388 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 423
After n—1 applications of this last inequality we get
B{k,n,N)*n < (1cn)*(kn-1)*...*(k2)*R(k,l,N—n + l)*l
= (Jfc»)*(*»-1)*...*(*2)*[i(JV-n) + l],
which is the desired result.
Invariant and canonical distributions
The terms invariant distribution and canonical distribution are defined
in the introduction.
Theorem 2. Let n and N be integers, 1 ^. n ^. N. and lei A be an invariant
distribution of£ln(S), where S = {1, 2,..., N}. Then A is canonical, provided
that at least one of the three conditions (i) n = 1, (ii) n < £J\T, (iii) n = N
holds. If, on the other hand, the numbers n and N do not satisfy any of the
conditions (i), (ii), (iii), then there exists an invariant distribution of D.n(S)
which is not canonical.
Proof. We recall the following definitions used in (4). If A , B are
sets of positive integers (1 ^ /j, <; m), then the relation
A1:A2:...:Am = B1:B2:...:Bm
means that there exists a function f(x), defined and increasing on
A1JrA2^r ...JrAm, which maps, for all /j,, A on B^. The relation
A: A' = B:B' = C:C
means that simultaneously
A: A" = B:B'; B:B' = C:C.
It is worth noting that the relations
A: A' = B-.B'; A': A" = B': B"
do not imply that A: A" = B:B". This is shown by the example in which
A, A', A"; B, B', B" are
{1,3}, {2,4}, {3,5}; {1,4}, {2,5}, {3,6}
respectively.
We begin by proving the last part of Theorem 2. If n and N do not
satisfy any of the conditions (i)-(iii) then
2 5^ n < .V < 2n. (6)
Put S = {1,...,N};A0 = {1,2,...,7i}; B0 = {Ar~~n+l,X—n + 2,...,N}, and
define a distribution A' of Cln(S) as follows. If A, BeD.n(S), then
A ~ B (. A') if and only if either A = B or A = A0; B = B0 or A = B0;
B = A0. Then A'is invariant. For let A, B, C, D eQn(S); A = B (. A'),
A:B=C:D. (7)
We have to deduce that C = D (.A'). If A = B, then, by (7), C = D,
389 RAMSEY'S THEOREM

424 P. ERDOS AND R. RADO
and hence C eee D (.A'). If A 7= B, then we may assume that A — A0;
B = B0. Then, in view of (6), the relation (7) implies that C = A0; D — B0.
For, clearly, the order relations between the elements of A0 and B0 are
such that there exists only one pair of elements of Qn(S) for which they all
hold. Hence in any case C = D (.A').
We now show that A' is not canonical. We have 1 < iV—»+1,
{1,2,...,¾} 5EE {N-n+\,N-n+2,...,N} (.A').
Hence, if A' were canonical, say A' == A^ ^, then the only possibility
would be k = 0, i.e. [A'| = 1. But this is a contradiction against
{1,2,...,¾} == {1,2,...,71-1,71+1} (.A').
Thus the last part of Theorem 2 is proved.
We now suppose that n and ^satisfy at least oneof the conditions (i)-(iii),
and that A is an invariant distribution of Qn(S), where S = {1,..., N}.
We shall prove that A is canonical. If in what follows we assume N = 00 so
that (ii) holds, then we obtain a new proof of Theorem I of (4) which is
simpler than the original proof.
All congruences are understood (.A). First of all, let n = 1. If there is a
set {^o'2/o}< c & such that {x0} ==■ {y0}, then we have, for {#,?/}< c S,
K>} -{Vo} = {*}■■&}
and hence, by the invariance of A, {x} ==■ {y}. Hence, if n = 1, then either
A = A(0) or A = A'1', so that A is canonical.
Next, let n = N. Then A is canonical for the trivial reason that
\nn(S)\ = i.
Thus we need only to consider the case 2 < n < ^N. Put
Hv = {1,2,...,^-1,^+1,...,71+1} (1 < v < 71+1).
Let I = {v1,i>2,...,vk}< be the set of all v such that
1 < v < n; #, == Hv+1.
Then 0 < k < n. We shall prove that A = A^L4.
Define n operators Tv by putting, for 1 < v ^ n,
TV{A,B) = ({«!,..., <zI,_1,min(aI„6I,),aI,+1,...,«„}, {blt..., min^, bv),...,bn}),
whenever A = {a1,...,ay^< c S; B = {b1,...,bn}< c S. (8)
Thus, in particular,
TV{HV,HV^) = (HV+1,HV+1) (1 < v < 7i).
(a) Suppose that (8) holds, and av = bv {v e 7).
We have to prove that A == 5. Put
p(4,B) = 2o^J,ni.
390 COMBINATORIAL ANALYSIS

CLASSIFICATIOXS OF SUBSETS 425
We may assume that p(A, B) > 0, and we may use induction with respect
to p(A,B). Let v0 = mina,, =£ bv\Jv. Then v0e I. We may assume that
«,„ < &*„• Then
vo"
TVt(A,B) = (A,B'),
where B:B' = Hv :HV ...
Also, since v0 e I, HVo = Hv +1. Hence, since A is invariant, B = B'.
But p(A,B') <p(A,B), and therefore, using the induction hypothesis,
A = B' = B.
(/3) Suppose that (8) holds, and A = B.
We have to prove that av = bv (v e I).
Suppose that, on the contrary, there exists v e 7 such that av, 7= bv>.
Let v0 be the smallest of such indices v'. Put
This means that, in particular, A = C, B = D, if v0 = 1. Then, using
the definition of I, we deduce that C = A = B = D. Let
c == {<+•••, c»}<; .D = K,. ..,<}<•
Then c„ = <2„ (v < v0); c„o =£ <2Vo,
and we may assume that c„o < dVo. Since JV > 2w, it is possible to find sets
E and F such that
E = {e1,...,en}< c^-{JV); i1 = {f1,...Jn}< c S~{N],
C:D = #: F.
Then e, =/, (v < v0); eVo < /„o.
Also, by the invariance of A, E = F. Put
77' = ^,...,6^,6^ + 1,6^2+1,...,6^+1),
■* == {/lv-;/v0-l;/v0+ l;/y0+l"T 1; • • •;/n "T V'
Then 77' and F' are obtained from E and 71 respectively by an application
of the mapping
x->x (x < eVo); z->x+l (x > eVo),
and therefore we have E:F = E':F'. Hence, by the invariance of A,
E' ~ F'.
Similarly, if
E" = {6,,...,6^^,6^+1,6^^+1,...,6^+1),
then E:F=E":F', since E" and F' are the images of E and F respectively
under the mapping
x->x {x < eVo); x->z+l (x > eVo).
Hence 77" = J1', and so, finally, E' = F' = E". But
E': E" = HVo+1: HVo,
391 RAMSEY'S THEOREM

426
P. ERDOS AND R. RADO
and therefore, by the invariance of A, HVo+1 = HVo, i.e. v0 e I, which is the
desired contradiction. This proves Theorem 2.
An estimate of the generalized Ramsey function R*(n, N)
Theorem 3. Given any integers n and N such that 1 <; n <; N, there exists
a positive integer M* having the following property. If 8 = (1, 2,..., M*},
and if A is any distribution of 0,n(S) into any number of classes, then there is
S* c S, where \S* | = N, such that A is invariant in £ln(S*). If, in addition
N > 2n, then A is canonical in Qn(S*). If R*(n, N) is the least possible value
of M*. then i/2»\r/2»\ i
K2n,N + n~~l)
(9)
S*(n, N) < R(
where R is Ramsey's function.
Proof. Let M* be the number on the right-hand side of (9), and
S = {1,..., M*}.
Consider any distribution A of £ln(S). We define A'(X) for X e£l2n(S) as
follows. We put E = F (. A') if and only if E, F e Q2„(£), and the following
condition holds. Whenever
A,Be£ln(E); C,De£ln{F)
A-.B-.E = C:D:F
then the two relations
A = B(.A); C = D(.A)
are either both true or both false.
For fixed E the number of unordered pairs A, B is n
Hence [A'j < 2™', and therefore, by definition of M*
8' = {Oi,02»-";«V+n-l}< C $
such that [A'[ = linQ2n(fi"). Then the set 8* = {av...,aN} has the required
property. For suppose that A, B, C, D eQ.n{8*),
A = B(.A); A:B=G:D.
Then \A + B\ = \C+D\ = m, say, where n < m < 2
D, and so C = D (. A). If m > n, then we put
(10)
(11)
_ l/2rc'
"~ 2\n
there is
(»)
If
(12)
n, then
C
E = A + B+{c
F
;A-+l'°A'+2
^+211-^.
}.
*~> +-^ + (^+11 al\7+2>-"' aX+2n-my
Then E, F e£l2n(S'), and hence, by choice of S', E = F (.A'). Now,
(10) holds and the first relation (11) is true. Therefore, by definition of
A', both relations (11) are true.
Hence, in any case, (12) implies (11), and so A is invariant. The rest of
Theorem 3 follows from Theorem 2.
392 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 427
Some more definitions. Transfinite extensions of Ramsey's
theorem
We consider a fixed, non-empty, set S of real numbers. The letters
A, B, X, Y denote subsets of S. and always \A\, \B\ > X0. The order type
of X under the natural order by magnitude is denoted by X. The letters
a, /3, /jl, v, denote ordinal numbers, and always
0<|«|,|j8|<X0; 0< \p\,\v\ <2*S
and the letters m, n denote positive finite ordinal numbers. By </> is denoted
any order type of cardinal \<j>\ <I X0, and <^>* denotes the converse, of </>,
i.e. the order type obtained from <f> by replacing every relation x < y by
the corresponding relation x > y. We recall that, according to the definition
of multiplication of order types, (</>1</>2)* = 4>*4>t- As usual, oj denotes the
least denumerable ordinal number.
We shall be concerned with a fixed distribution A of Qn(S) into
nonempty classes Kx. In the special case n = [ A [ = 2 we denote the classes
of A, in some fixed order, by K1: K2, and we define certain sets and
numbers as follows.
FX{A) ^^XcA; a2(X) c XAD {X},
F'x(A) = nBc^nFx(B),
fx(A) = mm^eFx(A)[]^
f'x(A) = min Be An fx(B).
Thus FX(A) is the set of all order types <j> which are 'realizable' by some
suitable subset of A all of whose subsets of two elements belong to Kx,
and F'X(A) is the set of all those <j> which are even realizable in every non-
denumerable subset of A. Also, fx(A) is the least ordinal number which is
not realizable in A, and f'x{A) is the least ordinal number which is not
realizable in every non-denumerable subset of A.f
It follows from the definitions that, if A c B, then
leF'x(B)cF'x(A)cFx(A)cFx(B),
2 < f'x(B) < f'x(A) < fx(A) < fx(B).
It is well known that corresponding to every <j> there is a set X such that
X = </>. Furthermore, if </>0 denotes the order type of the set of all rational
numbers, then </>0 e FX{A) implies that every </> e FX(A).
Theorem 4. If S is the set of all rational numbers, and n = \ A| = 2, then
either (i) oj e F^S), or (ii) oj* e F^S), or (iii) every cf>eF2{S).
f Ordinal numbers realizable in a given ordered set were studied by J. C. Shepherd-
son, Proc. London Math. Soc. (3), 1 (1951), 291-307.
393 RAMSEY'S THEOREM

428 P. ERDflS AND R. RADO
A common hypothesis of Theorems 5-8 is \S\ > X0; n = |A| = 2.
Theorem 5. Either (i) oj 6 2^(^), or (ii) eren/ oceF2{S).
Theorem 6. If /[(S) is not a limit number, then there is A and fi > 0
■™chthat f1(A)=f'1(A)^f'1(S),
f2(A) = f'2(A) = ^.
If <f>veF'2{A) (*<«), <Ac» (2^)*e^2(^). 7/ ^6^(^), <Ac» erery
</>a* e J"2(^4). Finally, every a* e F'2(A).
Theorem 7. Either (i) eren/ aj+ra e F^S), or (ii) e^en/ aj.ra e F2(S).
Theorem 8. Either (i) oj+oj* e i*i($), or (ii) every a. e F2(S), or (Hi) every
<x*eF2(S).
Theorem 9. If \S\ > X0, and n and A are arbitrary, but such that, for
everyaeS, | J {a,x2)...,x„}< eZAD{A}|< X0,t
UK Vi»«}<e^DWI < N0.
<Aew <Aere is S' c S such that \S'\ = X0, and |A[ = H» Qre(5").
Theorem 10. If \ S\ > X0, and w and A are arbitrary, but such that, for
every {a1,...,a„_1}< c £,
[ yZ,{a1,...,an_1,x}< eZAD{A}[ < N0>
<Aen there is a set S' = {xltx2,...}< c $ sitcA <Aa< iw £\(*S") A is canonical and,
in fact, A = A^'..^, where 0 < k < n—1; 1 < v1 < v2 < ... < vfc < n—1.
The following theorem of Dushnik and Miller (7) belongs to the group of
Theorems 4-10.
If \S\ > X0, then either (i) u> e F^S), or (ii) oj* e-?!(£), or (iii) #,(£)
contains some non-denumerable order type.
Proofs of Theorems 4-10
Proof of Theorem 4. The letter I denotes open intervals. A set X is called
i-dense if there exists some I such that X is dense in I.
Lemma 1. If neither X nor Y is i-dense then X-\-Y is not i-dense.
For, given any I, there is 71 c I such that XIX = 0, and then there is
72 c 7j such that YIS = 0. Then (X + F)72 = 0.
We now prove the theorem. We introduce the notation
Lx(a) = 'Z{x,a}<eKxU{x}; *a(«) = I {«,*}< eJTADH (13)
J7A(o) = 7,A(a) + 2?A(a).
f Here, as in similar cases later on, the summation indices or symbols are all
symbols which are not stated to have fixed values. Thus, in the present case,
summation extends over varying x2,..., xn, A.
394 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 429
A set 8' c 8 is said to be of type 1 if there is a0 e S' such that S'U^ttg) is
»-dense, and of type 2 if there is no such a0.
Case 1. Suppose that every »-dense set 8' c 8 is of type 1. Then there is
a0 e S0 = 8 such that S± = S0 U^a^) is i-dense. Generally, if 8m has already
been defined for some m, and if 8m is i-dense, then there is am e 8m such that
Sm+1 = 8m U^dm) is i-dense. Then S0d Sxd ..., and if 0 < r < s, then
ase Ssc Sr+1 = Sr U^a,.); {<zr, as} e K^.
Since every sequence contains a monotonic subsequence—this is, in fact,
a simple case of Ramsey's theorem—there is a set {b^b^...} c{a1,a2,..?j
such that either 61 < b2 < ... or b1 > 62 > ••• • Hence either (i) or (ii) of
Theorem 4 holds.
Case 2. There is a set S' c S which is i-dense and of type 2.
Then we can choose 7 such that S' is dense in I. We choose
Iv c I (v = 1,2,..,) such that the Iv are dense in 7, which means that if
I' c I then Iv c 7' for some v. Then we can find xx e Ix S'. Suppose that for
some m the numbers already been defined, and that, if m > 1,
xv+1eIv+18'U2{x1)U2{x2)... U2{xv) (0 < v < m).
Then none of the sets S'TJx{xv) (1 ^ v ^ ra) is »-dense, and therefore, by
Lemma 1, the set
{x1,...,xm}+ 2 1 < " < mQS'U^x,,)
is not i-dense. Hence it is possible to choose
xm+l e-Sn+l^ ^2(^1)^2(^2)---^2(^)-
This process defines the sequence xlt x2,... which is dense in 7 and satisfies
{xr,xs}eK2 (1 < r < s).
Thus (iii) of Theorem 4 holds, and the proof is completed.
Proof of Theorem 5. If a < 6, then we denote by (a, b) the open interval
with ends a and 6. A set A is called a/wW set if for a<J<c;iei always
|(<z,6)^|>X0; \(b,c)A\>X0.
Lemma 2. Given any A, there is a full set B c A.
Proof of Lemma 2. Put, for § > 0,
A_{8) = ^aeA; \(a-8,a)A\ < X0D{«};
^+(8) = 2aeA; \(a,a+8)A\ < X0D{«}.
Let a e -4_(§). We shall prove that a is not a point of condensation of
A_(8). If (a,a+8)A_(8) = 0, then
\(a—8,a+8)A48)\ < |(a—8,aL4| < X0>
395 RAMSEY'S THEOREM

430 P. ERDOS AND R. RADO
and our assertion is proved. If there is 6 e (a, aJr8)A_(8), then
a—8 < 6—8 < a < 6,
|(6—8,6)^i_(8)| < \(a — B,a)A| + |(6—8,6)^i| < X0,
and the same conclusion follows. Therefore no point of -4_(8) is a point of
condensation of A_(8), and so |-4_(8)| < X0. Similarly, |^4+(8)| < X0.
Then the set , ,., ,.
^—> I \m) +\m
is non-denumerable and full, and the lemma follows.
We now prove Theorem 5. Let us suppose that oj e F^S). Then there is
A such that 1^(0)1^ X0 (aeA), (14)
where R^a) is defined by (13). For otherwise we could define sequences
Am, am as follows. Put A0 = S. Then, since (14) is false for A = A0,
there is a0 e A0 such that |^41| > X0, where A1 = ^0-R1(a0). Then there is
a1 e A1 such that \A2\ > X0. where A2 = Ax R^a^. Generally,
ameAm = Am_1R1{am_1) (m = 1,2,...).
If we put X = {a0, a1;...}, then X = oj; £22(^) c ^i> which is a contradiction
against cueF^S). Hence (14) holds for some suitable A. By Lemma 2
we can find a full set B c A.
Now consider any ordinal a ^ oj. We can choose a set T = {^,^,...}^
of real numbers such that T = a. Then we can successively find intervals
71, 72.... such that
4 = (6«,.cm); JrJs = o (r^r
{6^,...,6^ = ¾^^} J-;*) (15)
|7m7?| >X0 (m= 1,2,...) J
For if 71;..., 7m_j have already been chosen, then we can choose deB
such that -U,.*o
lV--> K-i>d} = ft---,^-i=W ^Ty
and then put Im = (d—e, ^+e) for a sufficiently small e. Then
{61;62,...} = a,
and therefore there is a one-to-one mapping of the set of all v in the range
0 ^ v < a onto the set of intervals 71, 72,... in such a way that, ifthe interval
associated with v is Jv = (b'v,c'v), then
b'n < V < K < c'v
whenever 0 ^ /j, < v < a. Now, using (14) and the definition of Jv, we can
find, by transfinite construction, numbers xve,JvB such that
{x^,xv} eK2 (0 < /x, < v < a).
396 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OV SUBSETS 431
Then, putting X = 2 {xv}> we have X = a] Q2(X) c K2, and therefore
a e F2(S). This proves Theorem 5.
Proof of Theorem 6. There is Axc S such that f'^S) = f^A-A, and
.4 c ^41 such that/:^) = f2(A). Then
A(^) <,A(A) =/i(^) ^/;(.4) < /,(,1),
A(^) =/2(^1)^/2(^)^/^),
and therefore /^)=/1(4)=/^)-, /2(^)=/2(^)- ^ow, by
hypothesis, /i(£) = v0+ 1. Then
^(a)! <X0 (aEi), (16)
For otherwise we could find a0 e A such that ' ALy(aQ) > X0. Then, since
vo < f'i(S) = /i(-4). there is X0 c 4.2^(¾) such that A"0 = v0: Q2(X()) c &i-
Then, putting X0+{a0} = X1;
X, = „0 +1 = AM); X, c 4; Q2( X±) c ^,
which is a contradiction against the definition of fi(A).
Let a be arbitrary, and suppose that </>r e 2¾^.) (" < <*)• In order to
prove that (J </>*)* e 2^(.4). let us consider any set A2 c 4. By Lemma 2
there is a full set B c 42. Then we can choose T, tm, Im, bm, cm, Jv, b'v. c'v
exactly as in the proof of Theorem 5. so that (15) holds, but with a* in place
of a. Then, using (16) and the definition of ,/„, we can find, by traiisfinite
construction, sets XvcJvB such that Xv = </>„; Q2(X)c/\2. where
X = J " < « D Xv; X = ( 2 #:)*. Hence
(Y<j>*)*eF2(B)cF2(A2).
Since A2 is arbitrary, this implies that (2 </>*)* G -^2(-^)- -^ H1 particular,
(j)v = cf) for all v < a, then
^*=(W-(I^re^(4 (17)
Finally, since 1 e F'2(A), we may put <f> = I and find a* e 2^(4).
There only remains to prove the statement about /j,. It is sufficient to
show that/2(4) is of the form oj^, since then, in view of f'2(A) ^ 2, we have
/j, > 0.
f'2(A), being a non-zero ordinal number, can be written in the form
f'2(A) = co^^+l).
Then oj^/jl-^ < f'2{A), and therefore, by applying (17) to </> = oj^; a = 2.,
W find w'Vi+Z'i) < /2(^) = ^(^+1),
i.e. ^ = 0. This completes the proof of Theorem 6.
Proof of Theorem 7. We assume that
to+m^F^S); w.m2eF2(S),
where mx and m2 are given numbers, and we shall deduce a contradiction.
397 RAMSEY'S THEOREM

432 P. ERDOS AND R. RADO
We have, for any Alt ajra2 e F2(A1), and therefore, by Theorem 5, o> e F^Aj).
As Ax is arbitrary, o> e F'^S), and so
Hence it follows that f'^S) is not a limit number, and by Theorem 6 there is
A and /j, > 0 such that
o^ = f2(A) < /2(£) < o>m2.
Then jU = 1, o> e ^(-^)- anc^ a second application of Theorem 5 shows that
co-\-m1 e FX(A) which is the desired contradiction. This proves Theorem 7.
Proof of Theorem 8. We require the following lemma.
Lemma 3. If the hypothesis of Theorem, 8 holds, and
\B1(a)\ < N0 (aeS), (18)
then every a e F'2(S).
Proof of the lemma. Let /3 > 1, and assume that a e F'2(S) for every
a < /3. We have ]£ 0 < a < j8[]{a} = {a1; a2,...}. Let A be arbitrary.
By Lemma 2 there is a full set B c A. Then there are numbers bm, cme B
such that 61 < Cj < 62 < c2 < .... Put 7m = (bm,cm). Since 0¾ e F'2(S),
there is i?j c i?71 such that B1 = 0^; Q2(-^i) c -^2- Generally, if/ij,..., -Bm_1
have already been found for some m, then we can find Bm such that
Bm c BIm-BIm 2 lf£<m □ ir^a),
Bm = ocm; 0.2{BJcK2 (m=l,2,...).
Here we make use of (18). If B' = ]£ i?m, then
£'c^L; 5' = 2«m>/3; H2(5')cZ,
Hence/3 e i^(^)- Since ^ is arbitrary, this implies /3 e F'2(S), and thelemm£.
follows by transfinite induction.
Corollary. By applying Lemma 3 to the set S1 = 2 x e ^0(—#} we
find that i/ I L^b) \ < N0 (6 e £), <Aen even/ a* e F'2(S).
WTe now turn to the proof of Theorem 8. Let us assume that both (ii)
and (iii) of Theorem 8 are false. Then, by Lemma 3 and its corollary, given
any T c S, | T\ > X0, there are numbers a, b e T such that | TR^a)] > X0;
I TL1(6) I > X0. By means of repeated applications of this result we find
numbers am, bm and sets A0, Am, Bm such that A0 = S,
b.eB,; A1 = B1L1{b1),
and, generally, . . r _ J F (a \
am fc ^-m-l' -°m — ■a-m-\ -al\um)>
bmeBm\ Am= BmLl{bm)-
398 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 433
Then AmcBmc Am_^. If 0 < r < s, then
os e As_-l c Arc Brc E^a?); {ar, as}< e Klt
bseBsc Br+1 cArc L^b,); {6S, br} <eK1,
as e As^ cArc L^b?); {as, br] <eK1,
6S_! e £g_! c Src E^); {ar, bs_1}< e iq.
Hence, if X = ~^,{am,bm}, then X = oj + oj*; Q2(-^) c ^i> and therefore
o> + o>* 6 2^(^). This proves Theorem 8.
Proof of Theorem 9. Let S(m) be the set of all a e S such that
\2,{a,x2,...,xn}<eKxU{fy\ < m>
l2K,...,^-i,«}<eirAnWI <m-
Then | 2 $(m)| = | $1 > X0, and therefore | $(771,,)1 > X0 for some suitable
m0. An application of Lemma 2 shows that S(m0) contains an increasing
sequence, and by applying to this sequence the generalized Ramsey
theorem of (4) we find a set S' = {a1,a2,..?j< c S(m0) such that in
Qn(5") the distribution A is canonical, say A = A*.^, where 0 ^ k ^ n;
1 < v1 < v% < ... < vk < n. If k > 0 and vfc > 1, then
(01,0,.+2,-,0,.+^=^(0^0,,+2,-,0^} (-A) (0 < r < s < m0),
which is a contradiction of the definition of S(m0). If k > 0 and vfc = 1,
then & = 1,
K,omo+2,-,omo+„} =£ {os,omo+2,...,amo+„} (.A) (1 < r < s < m0+l),
which is again a contradiction of the definition of S(m0). Hence k = 0, and
Theorem 9 follows.
Eemark. The hypothesis \S\ > X0 cannot be replaced by |£| = X0,
as is shown by the distribution Aj1J in Qn(S), where.S = (1, 2,...}. In this
case, for a £ S, , -^ r •> 77 n oi i i
life Vi.a}<e^nWl = max(0,a—n+1).
Proof of Theorem 10. We choose a full set ^ c £, and we put
Ax = A2 = ... = An_1 = A.
Choose {o1(—, oM_1}< c .4. Then there is a full set An c ^4„_x such that
{a1,...,an_1,x}< e K(l, 2,...,n—I) (x e An),
where, generally, K(plt..., p„_i) denotes one of the K^. We choose an e An.
Generally, if m ^ n, and if the ar and Ar have been chosen for
1 ^ r ^ m—1,
399 RAMSEY'S THEOREM

434 P. ERDOS AND R. RADO
we can find a full set Am c Am__1 such that
{api>->apn-i>X}< eK(Pl>->Pn-l)
for 1 < p1 < ... < pn^ < m; x e Am.
We now choose am e Am. This defines a sequence <zm such that
{oPi»Opa»-,OpJ< e% p„-i) (1 < Pi < P2 < - < pj- (19)
By the generalized Ramsey theorem there is a set
8' = {xux2,...}< cfa^a2,...}
such that in Q„(*S") the distribution A is canonical, say A = A^..^. Then,
by (19), k < n; 1 < v1 < v2 < ... < vfc < w, and Theorem 10 is proved.
Some counter-examples for extensions of Ramsey's theorem
1. Ramsey's theorem makes an assertion about distributions of Qn(S)
when n < X0, )A) < X0. The question arises how much of the theorem
remains true when n = X0. The following example shows that there is
very little scope for a reasonable extension in such a direction, even for
(A| = 2 and arbitrarily large cardinals \S\. In this final section the
conventions about the use of the letters S, A, B are no longer valid.
Example 1. Let \S\ ^ X0- Then there exists a distribution of the set T
of all infinite subsets of S into two classes such that, given any 8' e T, there are
infinite subsets A, B of S' satisfying A ^ B (.A).
Proof. Let X < Y be a well-ordering relation of the set T. Let K^ be
the set of all X e T such that X' < X for at least one X' c X, and put
K2 = T—Kx. Then the distribution A whose classes are K^ and K2 has the
required property. For let S' e T, and let A be the first infinite subset of S'.
Then, whenever A' c A and A' e T, we have A' c A c S' and hence, by
definition of A, A ^ A'. Therefore A e K2. We can write
A = {a^a^a.^,...)^ + 0,
where, for all m, ame C. Put Am = {a2, o4,...,o2m, o1,a3, o5,...} + C. Then
there exists vaxn.Am = Amo, and we have
^iiii <- m0+l> -"-ma ^- mi+11
Then B = Amo+1 e Kly and we have A ^ B(. A). This proves the assertion.
Clearly, in Example 1 we may replace the set T of all infinite subsets of
S by the set of all subsets of S having a given fixed infinite cardinal not
exceeding \S\.
2. Let | $| = X0, and let A be a distribution of the set of all finite subsets
of S. Suppose that jAj < X0. Then, by means of N successive applications
of Ramsey's theorem it is easy to find an infinite set S'N c S such that, for
every n ^ N, we have ) A | = 1 in Qn(S'N). The question arises whether the
400 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 435
set S'N can be chosen so that it is independent of N, say S'N = *S" for all N,
so that now simultaneously for all n we have | A| = 1 in£2„(*S"). The
following example shows that such a set S' need not exist.
Example 2. If \S\ = X0 then there exists a distribution A of the set
T = ^Qn(S), where |A| = 2, such that the following condition holds. If
n
S' 6^^0(^) then, for every sufficiently large n, there are sets A, B eD.n(S')
such that A =£ B (.A).
Proof. Let S = {1,2, 3,...}. Denote by Kx the set of all A e T such that
j A I ^ me A for at least one m, and put K2 = T—Kx. Let A be the
distribution whose classes are Kx and K2. Now suppose that S' = {a^a^,..?^ c S.
Then, for n ^ ax, we have
{0^02,...,0^} # {an+1,an+2,...,a2n} (.A).
3. The last example quite naturally leads to the question -whether a
distribution exists which has a property similar to that described in
Example 2, but with respect to a set S which is not denumerable. This
question has only been decided for \S\ ^ 2^°, and the general case seems
to be well worth studying. In the case | S | ^ 2Xo several examples of
distributions are known which have the required property. In order to throw
more light on the problem we give three such examples, in the hope that
perhaps one of the methods used might turn out to be capable of an
extension to cardinals exceeding that of the continuum.
Example 3 a. If \S\ = 2X°, then there exists a distribution A of the set
T = *£,Qn(S), where |A| = 2, such that the following condition holds. If
n
S' e £^0(^)1 then, for infinitely many n, there are sets A, B e £ln{S') such that
Proof. Let S be the set of all sequences 0 = (a(1), o(2),...), where o(r) e {0, 1}.
Let i?[ be the set of all sets {alt o2,..., om}_^ c S such that
at = (o;(1),op>,...); o<™> = aim) = ... = a™,
and put K2 = T—Kx. Now consider any set *S" = {61,62,...}^ c S, where
bl = (ftp'.&P',...) {I = 1,2,...). Then there are infinitely many r > 1 such
that integers s, t can be found satisfying 1 < s < t; b^ 7^ bfK For every
such choice of r, s, t we have
A = {bs,bbbl+1,...,bl+r_2}eK2.
Furthermore, among the 2r— 1 numbers 6^, b(2\..., b2^r)_1 there are r which
are equal to each other. Hence there is a set B = |6Sl,6S2,..., 6Sr} e Kx for
suitable s such that 1 ^ s1 < s2 < ... < sr ^. 2r—1.
The following example is due to N. G. de Bruijn.
401 RAMSEY'S THEOREM

436 P. ERDOS AND R. RADO
Example 3 b. Let \S\ = 2X°. Then there exists a distribution A of the
set T = 2^»(^)> w^6re jAj = 2, such that the following condition holds.
n
If \a,b}^ c 8, then there is a positive integer n = n(a,b) such that
{a}+C # {6} + C (.A) for all 0 6^.,(5-(0,6}).
If {a1} o2,...}# c 5, ^en ^e numbers n{as,at) are unbounded for 1 ^ s <c t.
Proof. Define 8 as in Example 3 a, but modify the definition of Kx as
follows. Let Z, be the set of all {0,,...,0^ # c £ such that a^'+a^' + .-.+a^'
is even, and put K2 = T—Kx. If we now define n(a,b) to be the least
number m such that <z(m) ^ 6(m), then the desired conditions hold.
Example 3 c. Let \S\ = 2X°. Then there exists a distribution A. of the set
T = 2^»(^)> where jA| = 2, smc/j £/mz£ the following condition holds. If
S' eQ$0(S), then, given any sufficiently large integer m, there are sets
A, Be£lm{8') such that A =£ B (.A).
Proof. Let 8 be the set of all real numbers x in the range 0 < x < 1.
Denote by .K, theset of all sets {x1,x2,..., xm}< c 5 such that m(xm~x1) ^ 1,
and put K2 = T—K^. Now let 8' = {0,,02,...}^ c 8. As is well known,
there is an increasing sequence of positive integers rlt r2,... such that the
sequence bm = aTm is monotonic. Then, for m > j62—61j_1, we have
A = {b1,b2,...,bm}eK2.
On the other hand, since, for fixed m, br+m—br+1 -»- 0 as r -> 00, we have
B — {br+1, br+2,...,br+m) e Kx if r is sufficiently large.
4. Our next example, due to Sierpinski, shows that Theorems 4-8
cannot be strengthened very much.
Example 4 a. Let 8 be a set of real numbers. Then there is a distribution of
Q2(S) into two classes Klt K2 such that, in the notation defined on p. 427,
cc cc
Proof. Let a <^ 6 be a well-ordering relation in 8. Define if, to be the
set of all {a, b}< c S such that a <^b, and put i?2 = Q2(^)~^i- If> now,
i^2(^) c -^1 then, in .4, the relations x <C y and x <^. y are equivalent, and
therefore 4 is an ordinal number. Similarly, if Q2(B) c -^2» then (£)* is an
ordinal number. In either case, the ordinal number is at most denumerable.
If, in particular, | $ | = X0, then the same construction, which now no
longer requires Zermelo's axiom, leads to a distribution of an even more
special character.
Example 4 b. Let S be a denumerable set of real numbers. Then there is a
distribution of £l2(S) into two classes Klt K2 such that
F±(S) c {o>, 1,2,...}, F2(S) c {o>*, 1,2,...}.
402 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 437
The classes K^ can be defined without the use of Zermelo's axiom. If, in
addition, S is well ordered according to magnitude, then the stronger result
F2(S)c{l,2,...} holds.
Proof. Let S = [xly x2,...}^, and take as Kx the set of all {xr, xs}< such that
r < s, while K2 = Q2($)—.¾. If we now assume that co+l e F^S), then
there is A c S such that A = o>+l; £l2(A) c Kx. Then there is xmo eA
such that xm < xma; xme A for infinitely many m. Then, by definition of
Kl7 m < m0 for infinitely many m, which is impossible. Similarly, if
1+oj* eF2(S), then B = l+o>*; Q2(.B) c if2 for some Be S, and there is
xmi e .B such that xm > xmi; xm e B for infinitely many m. Then, by
definition of K2, m < mx for infinitely many m, which is impossible. This
proves our assertions.
5. The construction in Example 4 a required the axiom of choice. The
following example shows that a weaker result can be obtained without
the use of that axiom.
Example 5. Let 8 be the set of all real numbers. Then, without the use of
Zermelo's axiom, a distribution of 0.2(S) into two classes Ku K2 can be defined
which has the property that neither F^S) nor f2(S) contains any order type
which is dense in an interval.
Proof. Let K\ be the set of all sets {a, b}< c S such that
22r+A <^ &_0 < 22r+A+l
for some suitable integer r, positive, negative, or zero, where A = 1 or A = 2.
Now suppose that £12{A) c K^, and that A is dense in the open interval
(a0, b0). Then we can find real numbers a, b and an integer r such that
«0 < a < b < b0, b—a = 22r+x,
and hence numbers a', b' e A such that
a < a' < b' < 6; 2^+^1 < b'—a' < 22r+\
Then {a',b'}e K\, and we have obtained a contradiction.
6. The next example shows that Ramsey's theorem becomes false in
the case of denumerably many classes, even if we replace the hypothesis
|S| = X0by |S| = 2«°.
Example 6. If \S\ = 2X°, then there exists a distribution A of £12{S) into
X0 classes such that, given any set 8' 6^0(^)1 there is a sequence of sets
Am eD.2{8') satisfying Ar =£ As (.A) for 1 ^ r < s.
Proof. Let S be the set of all real numbers x in 0 < x < 1, and denote,
for A = 0, 1, 2,..., by K^ the set of all sets {a, b}< c S such that
A < (b—a)-1 < A+l.
Then the distribution whose classes are the K^ has the desired property.
For if S' eQ.$.a(S) then there are sets Am = {am, bm}< c S' such that
bm~am -+ ° and Am e K\J Am^ooasm^oo.
403 RAMSEY'S THEOREM

438 P. ERDOS AND R. RADO
The last example is a best-possible one, in view of the following theorem,
which is a special case of (8), Theorem 1.
If\®\ > 2^°; ^2(^) = K1+K2+..., then there is always a set S' c S such
that \S'\ > X0; Q2(S') c K% for some A'.
7. Our next example shows that in Theorem 10 we cannot replace the
set 8 of real numbers by any arbitrary non-denumerable abstract set in
which an order has been defined. Some properties of the order type of the
real continuum are essential for the truth of Theorem 10.
Example 7. There exists a non-denumerable ordered set S and a
distribution A of £l2(S) into classes K\ such that, for every a e S,
\^{a,x}<eKxnm^K
but, at the same time, {x, y) =£ [x, 2} (. A)
whenever {x,y, z}< c S.
Proof. Let Wj be the first non-denumerable ordinal number, and
consider an abstract set S, ordered according to the order type oj*. For Xe S,
let Kx be the set of all sets {a, A} < c S, for varying a. Then the assertion holds.
8. In conclusion we prove a theorem which belongs to the present
chapter in so far as it asserts the existence of a distribution having a special
property, although the property in question is not concerned with Ramsey's
theorem but with the following theorem of van der Waerden (5). Given
any positive integers k and I, there is a positive integer m which has the
following property. If A is any distribution of the set {l, 2,...,m}, and j Aj $^ k,
there are positive integers a and d satisfying
a = a+d = a+2d = ... se a+ld (.A). (20)
We define van der Waerden's function W(k, I) as being the least value of such
a number m. The existing proofs of van der Waerden's theorem lead to
upper estimates of W which are far beyond the range of explicit expressions
in terms of common algebraic operations. The following example of a
distribution gives what seems to be the first non-trivial lower estimate of
W, namely W(k, I) > {2lkl)K
Example 8. Let k and I be integers not less than 2, and let m0 be the largest
integer such that ml ^. ilk1. Then there exists a distribution A of
S0 = {1,2,...,m0],
where |A| $^ k, such that (20) does not hold for any positive integers a, d.
Proof. We assume that, given any distribution A (|A| ^ k) of the set
S = {1,..., m}, there are positive numbers a, d satisfying (20), and we shall
deduce that m2 > ilk1.
404 COMBINATORIAL ANALYSIS

CLASSIFICATIONS OF SUBSETS 439
We have m ^ /+1. For any integer d in the range 1 <; d < (m—1)//,
the number of increasing arithmetic progressions of/+1 terms with common
difference d and terms in 8 is m—Id. Hence the total number of such
progressions, for varying d, is r
M = ^(m-ld).
a---i
where r is the integer satisfying r < (m—1)// < r-\-l. The number of
functions / on S into the set {1, 2,..., k} is km, and the number of those /
which take a given value k0 at the /+1 places corresponding to the terms of a
fixed progression is &™~'_1. Hence, in view of our initial assumption.
kMkm-l~x > km,
IJ <:\ M = hr(2m~l~lr) < rJ!--L[2m-(m-l)] < ■—,
which is the desired conclusion.
REFERENCES
1. F. P. Ramsey. Proc. London Math. Soc. (2), 30 (1930). 2G4-8C.
2. Th. Skolem, Fundamenta Math. 20 (1933), 257, Satz 3.
3. P. Eedos and G. Szekeres, Compositio Math. 2 (1935), 464.
4. and R. Rado. Journal London Math. Soc. 25 (1950), 251.
5. B. L. van der W.ierden, Nieuw Archief voor Wiskunde, 15 (1927), 212-1 li.
6. R. Rado, Proc. London Math. Soc. (2), 48 (1943), 124, § 1.
7. B. Dushnik and K. W. Miller, American J. of Math. 63 (1941), G06. Theorem
5.22.
8. P. Eedos, Bevista, Universidad Nacionat de Tucnindn. Serie A. 3 (1942). 363-7.
University College, London, W.C. 1,
King's College. London, W.C. 2.
405 RAMSEY'S THEOREM

GRAPH THEORY AND PROBABILITY
l\ erdOs
A well-known theorem of Ramsay (8; 9) states that to every n there exists
a smallest integer g(») so that every graph of g(») vertices contains either
a set of n independent points or a complete graph of order n, but there exists
a graph of g(») — 1 vertices which does not contain a complete subgraph
of n vertices and also does not contain a set of n independent points. (A graph
is called complete if every two of its vertices are connected by an edge; a set
of points is called independent if no two of its points are connected by an
edge.) The determination of g(«) seems a very difficult problem; the best
inequalities for #(«) are (3)
a) 2*- < g(W)< (2; : *).
It is not even known that g(n)1/n tends to a limit. The lower bound in (1)
has been obtained by combinatorial and probabilistic arguments without an
explicit construction.
In our paper (5) with Szekeres f(k, I) is defined as the least integer so that
every graph having f(k, I) vertices contains either a complete graph of order
iora set of I independent points (fik, k) — g(k)). Szekeres proved
(2) /(*,/)<(*+[~2).
Thus for
* = 3,/(3, *)<(* "J 0-
I recently proved by an explicit construction that /(3, I) > /1+e' (4). By
probabilistic arguments I can prove that for k > 3
(3) /(*, I) > I
C^T2)''
which shows that (2) is not very far from being best possible.
Define now h(k, I) as the least integer so that every graph of h(k, I) vertices
contains either a closed circuit of k or fewer lines, or that the graph contains
a set of / independent points. Clearly h(3, I) = /(3, I).
By probabilistic arguments we are going to prove that for fixed k and
sufficiently large I
(4) *(*, I) > /1+1/2t.
Further we shall prove that
Received December 13, 1957.
34
406 COMBINATORIAL ANALYSIS

GRAPH THEORY AND PROBABILITY 35
(5) h(2k +1,/) < c3 /1+1", A(2¾ + 2, J) < c3 ll+m.
A graph is called r chromatic if its vertices can be coloured by r colours
,o that no two vertices of the same colour are connected; also its vertices cannot
be coloured in this way by r — 1 colours. Tutte (1,2) first showed that for
every r there exists an r chromatic graph which contains no triangle and
Kelly (6) showed that for every r there exists an r chromatic graph which
contains no /fe-gon for k < 5. (Tutte's result was rediscovered several times,
for instance, by Mycielski (7). It was asked if such graphs exist for every k.)
Now (4) clearly shows that this holds for every k and in fact that there exists
a graph of n vertices of chromatic number > n' which contains no closed
circuit of fewer than k edges.
Now we prove (4). Let n be a large number,
0 <t <{
R
is arbitrary. Put m = [n1+t] ([x] denotes the integral part of x, that is, tht
greatest integer not exceeding x), p = [n1^] where 0 < ij < e/2 is arbitrary.
Let @(7l) be the complete graph of n vertices x\, x-i x„ and @(p) any of
its complete subgraphs having p vertices. Clearly we can choose &m in Q)
ways. Let
«in), 1 < a <
be an arbitrary subgraph of ®(n) having m edges (the number of possible
choices of a is clearly as indicated).
First of all we show that for almost all a ®aw has the property that it
has more than n common edges with every (y(P). Almost all here means: for
all as except for
Let the vertices of §YV) be X\, x2, . . . , xv. The number of graphs
(^^containing not more than n of the edges {xu Xj), 1 < i < j < p equals by a
simple combinatorial reasoning
t(®(0-(J))<8.+.,(®)((d-®)
^«:<»(--1)'<(©H-s>-
\ m I
it)
m I
%
407 RAMSEY'S THEOREM

36
P. ERDOS
Now the number of possible choices for ®(p) is
<P".
(;)<»■
\p/
Thus the number of as for which there exists a ®w so that O*'"' /°\ Qf>atn)
has not more than w'edges is less than (rj < e/2)
V2/]p3"exp(-M1+-2') = i W
Km/ \ m J
as stated.
Unfortunately almost all of these graphs Qftain) contain closed circuits of
length not exceeding k (in fact almost all of them contain triangles). But we
shall now prove that almost all Q&aln) contain fewer than n/k closed circuits of
length not exceeding k.
The number of graphs ®„c") which contain a given closed circuit (xuX2),
(x2, x3), ..., (xi, Xi) clearly equals
The circuit is determined by its vertices and their order—thus there are
n(n — \) . . . {n — /+1) such circuits. Therefore the expected number of
closed circuits of length not exceeding k equals
&tM{th^M$
since t < l/k. Therefore, by a simple and well-known argument, the number
of the as for which Q/iam contains n/k or more closed paths of length not
exceeding k is
as stated.
Thus we see that for almost all a Qbam) has the following properties: in every
®(P) it has more than n edges and the number of its closed circuits having k
or fewer edges is less than n/k. Omit from ®„(l,) all the edges contained in a
closed circuit of k or fewer edges. By what has just been said we omit fewer
than n edges. Thus we obtain a new graph (^J^ which by construction
does not contain a closed circuit of k or fewer edges. Also clearlv ©„'<"» C\ (&iv)
408 COMBINATORIAL ANALYSIS

GRAPH THIiORY AND 1'KOU.VHILITY
11
is not empty for every ®(P). Thus the maximum number of independent
points in (Ma/(n) is less than p = [nl~v], or
h(k, [n'"]) > n
which proves (4).
By more complicated arguments one can improve (4) considerably, thus
for k = 3 I can show that for every e > 0 and sufficiently large /
/(3, /) = h(3, I) > I'2",
which by (2) is very close to the right order of magnitude.
At the moment I am unable to replace the above "existence proof" by a
direct construction.
By using a little more care I can prove by the above method the following
result: there exists a (sufficiently small) constant c4 so that for every k and /
(6) h(k, /) > a /1+sV
(If k > c log I (6) is trivial since h(k, I) ^. I.)
From (6) it is easy to deduce that to every r there exists a cb so that for
n > nB(r, c5) there exists an r chromatic graph of n vertices which does not
contain a closed circuit of fewer than [c6 log n] edges. I am not sure if this
result is best possible.
We do not give the details of the proof of (3) since it is simpler than that
of (4). For k = 3 (3) follows from (4). If k > 3, put
2 2_
m = cs[n *-]]
and denote by (^>a(n> the "random" graph of m edges. By a. simple computation
it follows that for sufficiently small cfi, Qdain) does not contain a complete graph
of order k for more than
\ m /
values of a, and that for more than this number of values of a 0V' does not
contain a set of C7M2/*_1 log n independent points (c7 = c7(c6) is sufficiently
large). Thus
f(k, cintk~^ log n) > n,
which implies (3) by a simple computation.
Now we prove (5). It will clearly suffice to prove the first inequality of (5).
We use induction on /. Let there be given a graph @ having h(2k +1,/)-1
vertices which does not contain a closed circuit of 2k + 1 or fewer edges and
for which the maximum number of independent points is less than /. If every
point of © has order at least [/"*] + 2 (the order of a vertex is the number
of edges emanating from it) then, starting from an arbitrary point, we reach
in k steps at least / points, which must be all distinct since otherwise (M would
409 RAMSEY'S THEOREM

38
P. ERDOS
have to contain a closed circuit of at most 2k edges. The endpoints thus
obtained must be independent, for if two were connected by an edge ® would
contain a closed circuit of 2k + 1 edges. Thus (M would have a set of at least
/ independent points, which is false.
Thus ® must have a vertex x, of order at most [/"*] + 1. Omit the vertex
xi and all the vertices connected with it. Thus we obtain the graph OV and xi
is not connected with any point of ®', thus the maximum number of
independent points of ®' is / — 1, or GV has at most h(2k + 1,/-1)-1
vertices, hence
h(2k + 1, /) < h(2k + 1,/-1) + [l1'"] + 2
which proves (5).
References
1. Blanche Descartes .4 three colour problem, Eureka (April, 1947). (Solution March, 1948.)
2. Solution to Advanced Problem nn. 4526, Amer. Math. Monthly, 61 (1954), 352.
3. P. Erdos, Some remarks on the theory of graphs. B.A.M.S. 5$, (1947), 292-4.
4. Remarks on a theorem of Ramsey, Bull. Research Council of Israel, Section K, 7
(1957).
5. P. Erdos and G. Szekeres, .4 combinatorial problem in geometry, Compositio Math. 2 (1935)
463-70.
6. J. B. Kelly and L. M . Kelly, Paths and circuits in critical graphs, Amer. J. Math., 76 (1954),
786-92.
7. J. Mycielski, Sur le colorage des graphs, Colloquium Math. S (1955), 161-2.
8. F. P. Ramsay, Collected papers, 82-111.
9. T. Skolem, Ein kombinatonscher Sat: mit Anwendung auf einlogisches Entscheidungs problem.
Fund. Math. 20 (1933), 254-61.
University o/ Toronto
and
Technion, Haifa
410 COMBINATORIAL ANALYSIS

GRAPH THEORY AND PROBABILITY. II
p. erdOs
Define f(k, I) as the least integer so that every graph having f(k, I) vertices
contains either a complete graph of order k or a set of I independent vertices
(a complete graph of order k is a graph of k vertices every two of which are
connected by an edge, a set of I vertices is called independent if no two are
connected by an edge).
Throughout this paper cu c2, . . . will denote positive absolute constants. It
is known (1, 2) that
(1) /1+cl</(3,/)<
(T).
and in a previous paper (3) I stated that I can prove that for every e > 0
and I > /(e),/(3,1) > P~l. In the present paper I am going to prove that
(2) w>l>>o$?-
The proof of /(3,I) > l1+Cl was by an explicit construction. I can only
prove (2) by a probabilistic argument, and I cannot explicitly construct a
graph which satisfies it. The method used in the proof of (2) will be a
combination of that used in (3) with that in my recent paper (4) with Renyi. It
is possible that (2) can be strengthened to/(3,I) > c3l2, but it seems impossible
to improve (2) by the methods of this paper
Theorem. Let A be a fixed, sufficiently large number. Then for every n > n»
there is a graph & having n vertices, which contains no triangle and which does
not contain a set of [A n^ log n] = x independent vertices.
Clearly our theorem implies (2).
To prove the theorem put y = [n3/2/Al/2]. Denote by ®(n) the complete
graph of n vertices and by @(I) any of its complete subgraphs having x
vertices. Clearly we can choose @(I) in ( 1 ways. Let
(3) ®*\ I <a<l\2/j = t
be an arbitrary subgraph of &m having y edges (we use the notations of
(3)). Now we need
Received June 24, 1960.
346
411 RAMSEY'S THEOREM

GRAPH THEORY AND PROBABILITY. II
347
Lemma 1. Almost all Qt>am have the property that for every &(x) there is an
edge ea.x contained in both ($,,^ and ®(I), which is not contained in any triangle
whose edges are in 0>)am and whose third vertex is not in @(T).
"Almost all" here means for all but o(t) graphs &ain>. We could prove
Lemma 1 even if we would omit the words "and whose third vertex is not
in @(I)," but the proof would become very much more complicated, and
Lemma 1 suffices for the proof of our theorem.
The proof of Lemma 1 will be difficult and we postpone it. Assume that
the Lemma has already been proved, then it is easy to prove our theorem.
Let ®ain) be one of the graphs which satisfy Lemma 1. We construct a
subgraph @a(n) as follows: Let ei(a\ e2(a\ • • • , e»(a) be an arbitrary enumeration
of the edges of ©„<">. We put e^ C ®c,l"> and we have e/°» C OV"' (1 < k < 3.)
if and only if ekta) does not form a triangle with the edges eTla\ 1 < t < k
which we had already put in &ain). &am has n vertices, contains no triangle,
and does not contain a set of x independent vertices. The first two statements
are obvious; now we prove the third one. It will suffice to show that for
every @(I) @(I) /~\ @a(n) is not empty. Consider the edge eaa = eT (see Lemma 1),
if it is contained in ®a(n) our statement is proved, if not there must exist a
triangle eu ejt eT (i < t, j < t), whose edges are all in &>ainK But by Lemma 1
the third vertex of this triangle must be also in ®(I), thus ef C W(T), e3 C @(3">,
or e* and e_, are both in @(I) C\ ($^. This completes the proof of our third
statement, and thus if we put &aw = ® the proof of our theorem is complete.
If we had proved Lemma 1 in the stronger form without the words "and
whose third vertex is not in ©(J°," we could have defined (¾^5 as the union
of those edges of ©a(n) which are not contained in any triangle of &a("\
To complete our proof we now have to prove Lemma 1. First we need
borne lemmas. Denote by £<,(@(I)) the number of edges in CV° coiuiecting
the vertices in ®(I) with the vertices not in &{I).
Lkmma 2. For almost all &am we have
(4) max£.(®(1)) < [n13] = m,
where the maximum is taken over all the ( 1 possible choices of @(T).
We could easily prove the lemma with (1 + 0(1))2.4½. but (4) will suffice
lor our purpose.
The number 9?(m) of as for which (4) is not satisfied is not greater than
\ y — m I \ y — m I
To prove (5) observe that there are f J choices for WX), and the number
of edges in Wn) connecting the vertices of (9(I) with those not in (9(I) is
■v(n — x). Thus (5) follows by a simple combinatorial argument.
412 COMBINATORIAL ANALYSIS

348 1\ ERDOS
In estimating binomial coefficients we will make use of the following simple
mequc
(6)
(7)
and
ilities
(:)<£<(f)"
\2/ n - 1 1
(8) —j- = -5 > 5 for « > 3.
n In 3
From (5), (6), (7), and (8) we have (by substituting the values of x, y, and
m)
*<«/'< »'(v)"(i)"<»'fe)"-°<'>'
which proves the lezimia.
Lemma 3. for almost all @a(n) /Ae degree of every vertex of ($am is less than
By a theorem of Renyi and myself (4) it follows that p can be replaced by
(1 + 0(1))2^,
but the weaker result will suffice here.
The number of a's for which the condition of Lemma 3 is not satisfied is,
bv a simple combinatorial argument, less than
„(»r)(0-;) <"(;)(!;)■
(since the number of &ain) for which a given vertex has degree > p is
\ y -
-M
y - p I
and there are n possible choices for this vertex). From (6), (7), and (8), wt
have
■©(®-;)/'<-fe/<-(¾-•<"
which proves the lemma.
413 RAMSEY'S THEOREM

GRAPH THEORY AND PROBABILITY. II
349
Put
(9)
and
(10)
z, = [2'.4"Jlogn],
0,1,...,
.Wi
4"(*+l)2J lor0<i<\\ogn
-" J for - log n < i.
We shall say that ®aCB) has property Pt if there exists a @(I) and an t > 0 so
that there are at least wt vertices not contained in ($(Z), each of which is
connected in (Sa(B) with at least zt vertices of Q)(I).
Licmma 4. T'Ae number of graphs (y„CB) which have property Pt for some i
is o(t).
Since by Lemma 3 we can assume that the degree of every vertex of GV'
is less than p, we can assume that for sufficiently large A
(11) 2'A2" log n < p = 10(~) , or2'<
.4 log n '
Thus there are less than log n choices of i, and it will suffice to show that
for every i satisfying (11) the number of a's for which ®a(B) satisfies P( is
o{t/\ogn). Denote by 9^ the number of a's for which Qda(n) satisfies P<. A
simple combinatorial argument shows that
\ y — wxZi I
I 7t\ ( Tl — X\
To see (12) observe that there are ( J ways of choosing @(I); I J ways
of choosing the wt vertices not in ®(I), which are connected with at least zt
vertices of W(z); ( 1 ways of choosing the vertices in 0)(J°, with which the
w, vertices not in 6i(I) are connected in (Sa(B). For the remaining y — WiZi
edges of 6V' there are clearly
\ y - WiXt
choices; thus (12) 'is proved. From (12), (6), (7), and (8) we have, by
xy < .1* n- log n,
(13)
Ki
<
\ y — WiZi I
<
nX+*JlOAhogn\h
414 COMBINATORIAL ANALYSIS

350
P. ERDOS
Now 22i > n since z, > [A-'3 log n). Thus 2WiZi > nw\ hence from (13), by
substituting zt = [2V12/3 log n], we have for sufficiently large /1
(14) -<n\YA^) <n\2^) •
Assume first 0 < i < \ log n. Then from (9) and (10) we have
(15) w*( > ^jp^Y > »*■
From (14) and (15) we have (expw = e")
(16) y < M* exp(-n*log 2) = oW .
Assume next I > j log m. From (9), (10), and (11) we have, by i < log n
for sufficiently latge A,
(17) W.-2, > ^Tlf— > A ' T? log M.
Thus from (14) and (17), by 2,+ 1 > nuu),
(18) y < wIexp(-/l3/V(logM)7lO) = o(£)
for sufficiently large .4. Equations (1G) and (18) complete the proof of
Lemma 4.
Lemma 5. Almost all 6J„W have the property thai for every @(I) there are more
than i\r,) edges of ©(I) which do not occur in any triangle, the other two sides
of which are in &Jn) and whose third vertex is not in &ZK
We could prove Lemma 5 even if we omit the words "and whose third
vertex is not in @(I)," but the proof would be more complicated and Lemma 5
in its present form suffices for our purpose.
Denote by U\{a\ «2(a), . • • , un-z{a) the number of edges in ($a(n) which
connect the n — x vertices of @(rl) not in ®(I) with the vertices of @(I). The
number of edges of @(I) which are contained in triangles the other two sides
of which are in @a(B) and whose third vertex is not in ©(I) is clearly at most
n-i /„,<«)\
sG"')-
Thus to prove Lemma 5 it will suffice to show that for almost all a we have
for every choice of @(I)
415 RAMSEY'S THEOREM

GRAPH THEORY AND PROBABILITY. II
351
By Lemma 4 we can assume that ®a(7l) does not satisfy Pt for all i > 0. But
then the number of indices j for which Uj(a) > s, is not greater than wt for
all i > 0, or by (9) and (10) and w0 = n
w £ (f) < ? •&•) < r ^¾^ +
v n2uAil3(\ogn)2
2 4 2
where in 21 . 0 < i < - log n; and in ^ . 7 log m < i < log m
by (11). Thus, finally, from (20),
S (f) < T ^^ »)' + ^4"»(log ^)2 < | (2)
for sufficiently large A, and this proves the lemma.
Now we can prove Lemma 1. It suffices to consider those (Sa(B) which
satisfy Lemmas 2 and 5 (since the number of the other graphs is o{t)). Let
©(I) be a fixed graph having x vertices. We are going to estimate the number
of graphs ®a(rl' which satisfy Lemmas 2 and 4 and which fail to satisfy Lemma 1
with respect to @(I) (that is which do not contain an edge ea.z C @(x) ^ &a{"\
where ea_x is not contained in any triangle whose other two sides are in @a(">
and whose third vertex is not in ®(I)). Let us assume that we have already
chosen the u edges e^1', e^z\ ..., eu(z) (u = uz) which connect (in ©„<"') the
vertices of @(I) with the vertices not in Wz). Since Lemma 2 holds we have
u < m4'3. The number of the ®„(,l> for which e^, ejz\ ..., e„(I) are all the
edges which connect the vertices of @(I) with those not in ®(I) clearly equals
'©-<-
(21) I \2/ ,*v" "*' l — m^!" ^z)\
y — it
since we have at our disposal (9) — x(n — x) edges and have to choose
y — u of them. But by Lemma 5 there are at least U J edges of ®(I) which
do not form a triangle with any two of the e,'s 1 < i < w, and if we put
any of these edges in ®a(7l) Lemma 1 will be satisfied. Hence the number
9t''(ei(I), ..., eu{z)) of graphs, which do not satisfy Lemma 1 with respect to
&(1) and for which the edges connecting the vertices of ®(I) with those not
in ©<*> are ex{z) eu(z\ satisfies (u < n4li < y/2 for n > n0(A))
(22) W(e[z) elz>) <
MI) - x(n - x) - \[£) j.
\ y — u I
416 COMBINATORIAL ANALYSIS

352
P. ERDOS
Thus from (21), (22), and (7), we have
(2) -*(»-*)-! (2)
(23) "W^ ^)
Since (23) holds for all choices of ex(x), ..., e„(I) which satisfy Lemmas 2
and 4, we obtain that the number of ®a(B) which satisfy Lemmas 2 and 4
but do not satisfy Lemma 1 with respect to ®(I) is less than
(24) 'exp("4$)-
Since these are ( J choices for @(I) we obtain from (24) and Lemmas 2
and 4 that the number of graphs ®a(B) which do not satisfy Lemma 1 is less
than ((») < „.
l\x) exp\ £*V + °W <l exp(x log n) CXp\ 4¾) + °^
= *exp((l + 0(l))i4n*(log M)2)exp[-(1 + o(l))i4,/2n*(log w)2/4] + o(<)
= o(0,
which completes the proof of Lemma 1. Thus our theorem is proved.
The difficulty of trying to improve our theorem by the methods used in
this paper is due to my belief that there exists a constant c3 = Ci(A) su
that almost all graphs &aw contain an independent set of [c3n312 log n]
vertices. I am unable at present to prove or disprove this conjecture.
References
1. P. Erdos and G Szckeres, On a combinatorial problem in geometry, Compositio Math.. 3
(1935), 463-470.
2. P. Erdos, Remarks on a theorem of Ramsey, Bull. Research Council of Israel, Section F, 1
(1957).
3. P. Erdos, Graph theory and probability. Can. J. Math., 11 (1959), 34-38.
4. P. Erdos and A. R6nyi, On the evolution of random graphs, Publ. Inst. Hung. AcacJ, §ei.> ?
(1960), 17-61.
Australian National University, Canberra
417 RAMSEY'S THEOREM

THE CONSTRUCTION OF CERTAIN GRAPHS
V. ERDOS and C. A. ROGERS
1. Introduction. A graph G is called complete if any two of its vertices
are connected by an edge; a set of vertices of G are said to be independent if
no two of them are connected by an edge. It follows from a well-known
theorem of Ramsay (1) that for each pair of positive integers k, I there is an
integer f{k, /), which we take to be minimal, such that every graph with
f(k, I) vertices either contains a complete graph of k vertices or a set of /
independent points. Szekeres (2) proved that
/<*.o < (* 2-i-; *)•
and Erdos (3; 4) that
f(k,k) >2*'2,
/(3,1) > /1+",
for ;i positive constant c3.
Clearly
f(k, /) > /(3, /) > l]+c\
for k > 4. Our object is to prove a stronger result. We say that a set 5 of
points of a graph G is ^-independent, if there m no complete subgraph of G
having m vertices in 5. Let h(k, I) be the mini nal integer such that every
graph of h(k, /) vertices contains either a coniplete graph of k vertices or a
set of I points which are (k — l)-independent. Then clearly
h(k, I) </(&, /)
for all k, I. However we can still prove that
h(k,l) > ll+c\
for k > 3. This problem is due to A. Hajnal (oral communication).
Our construction is geometric, and is based on a lemma (§2) of some
geometric interest.
2. Regular simplices on the surface of a sphere. We define the relative
surface area of a set 5 on the surface of a sphere in w-dimensional euclidean
space to be the surface area of 5 divided by the surface area of the sphere.
We prove
Received October 26, 1961.
702
418 COMBINATORIAL ANALYSIS

THE CONSTRUCTION OF CERTAIN GRAPHS
703
Lemma. Suppose n and k are positive integers (k < n) and that f satisfies
o < r < V2,
k[i - m2\nn < l.
Then, if S is a set on the surface of the unit sphere 2 in n-dimensional space of
relative surface area
v> ii - m2\ni\
there is a regular k-simplex, with its vertices each on 2 within a distance* f of S,
and with its centre at the centre of 2.
Remark. This lemma shows that in a space of many dimensions even a set
of rather small relative surface area on the unit sphere will always contain a
^-simplex, which is very nearly a regular ^-simplex of unit circum-radius.
Proof. Let C be the minor spherical cap cut from 2 by a plane passing at a
distance $f from its centre. Since f < y/1, it is clear that the union of the
segments joining the centre O to the points of C is contained in the sphere
with radius
U - Qf)2!"2
with its centre at the centre of the base of the cap C. Consequently the relative
surface area of C is at most
U - (*r)a)"'2 < v,
and so is less than the relative surface area of S. Let Q and S( be the sets of
points on 2 within the distance f of the points of C and 5 respectively. Then
by a well-known result of Schmidt (5) the relative surface area of S{ will be
at least that of Cf. But Cf is the major cap cut from 2 by a plane passing at the
distance \'c from 0, and so has relative surface area at least
i - li - (ir)T'2> i - (i/k).
So the relative surface area of the set T of points of 2 not in Sf is less than l/k.
Consider the space ©t of all ordered sets X = {xi, x2, . . . , xk\ of k points
of 2 forming a regular ^-simplex of circum-radius 1 with the metric
d(X,Y) = j/jZl*,-y,l2}.
It is possible to introduce a measure on the Borel sets of 2¾ giving the whole
space unit measure and such that, for i = 1, 2, . . . , k, the measure of the set
Xt of points X = [xt, X2, ■ ■ ■ , Xk\ with xt € T is equal to the relative surface
area of T and so is less than l/k. Hence we can choose a point X of 3¾ not in
j
U Zu
i=i
*A11 our distances are measured in the K-dimensional space, not on the surface of the sphere.
419 RAMSEY'S THEOREM

704
P. ERDOS AND C. A. ROGERS
The points x,, x± xk form a regular ^-simplex of circum-radius 1 in S(
and so within distance f of S. This proves the lennna.
3. Theorem. Let k > 3 fre aw integer. If ct is a positive constant less than
log i/{i - (fo*)ai
2 log 4/¾
1/,= 1/^ = *(* - l),,2(fe- 2y/2l{2(k - 1)*)>»+ {2fe(fe -2))"*],
and I is a sufficiently large integer, there is a graph G, with less than
/] + c*
vertices, which contains no complete k-gon, but such that each subgraph with I
vertices contains a complete (k — Y)-gon.
Remark. We can take ck ~ 1/(512&4 log k) as k —>•<».
Proof. Let 77 be the greatest integer less than l]+ct. Let e be a small positive
constant and let n be the nearest integer to
(l + e)logi?/log
■vV(i - (iv)2)S
We take the vertices of our graph to be a set X of II points on the surface of
the sphere 2 in euclidean w-dimensioiial space with centre at the origin 0 and
with unit radius, and we join each pair whose distance apart exceeds
V[2k/(k- 1)).
Since the unit sphere contains no simplex with k vertices with all its edges
exceeding this length our graph contains no complete &-gon. But if (k — 1)
points of A' have mutual distances apart exceeding
V{2(k - 1)/{k - 2)j - „fc = V{2k/(k - 1)|,
they will form a complete (k — l)-gon in the graph. Thus to prove the theorem
it will suffice to prove that the points of X can be chosen, so that from any
set of I points of -Y a subset of (k — 1) points may be chosen with their mutual
distances apart exceeding
V{2(k - 1)/{k - 2)| - r,.
With each point x of 2 and each £ with 0 < J < 1 we associate the spherical
cap C(x, £) of all points of 2 within a distance £ of x. Now the union of the
segments joining 0 to the points of C(x, £) contains a cone with 0 as vertex
of height
i - he,
420 COMBINATORIAL ANALYSIS

THE CONSTRI'CTION OF CERTAIN GRAPHS
705
with a (n — l)-diineiisional sphere of radius
as its base. But the unit sphere is itself contained in a cylinder of height 2
with a (n — 1 )-diniensional unit sphere as its base. 1 Ience the relative surface
area of C{x, £) is at least
2«(1 ~ l?mi ~ ^V'8]"-1 > him ~ iWr-
Since 0 < 7] < 1 we can choose £ with 0 < £ < 3-57 so that the relative surface
area V of C(.v, £) is exactly
^ = -^11- (iv)Ys]H-
Let 5 be the union of all the caps C(x, £) with x in A. Let h be the integer
nearest to IT. Since
log (h + 1)- log j(//+ 1)L|
= e log // - log // + w log [(4/,) { 1 - (hr\-'} + O(log «)
= 2e log // + O(log log //),
we have
h + 1 > (//+ 1)1',
provided / is sufficiently large. A simple probability argument, which we have
recently used elsewhere (6), shows that, if the II points of the set A are
distributed independently uniformly over 2, then the expectation of the relative
surface area of the set F„ of points of i which lie in h or more of the caps
C(.v, £) with .v in A'
is at most
__jn T/*n _ n«-» (/7. + 1)(1- in
h\{II -A)! l ; [h + 1) - (/-/ + 1) F ■
So we may suppose that the points of A' are chosen so that the relative surface
area Vh of the set Fn satisfies
V„ <
HI
V"{\ - 1
„H-H (¾ + 1)(1 — V)
N ow
and
" " h\(H - h)l ' ^ '' (/; + 1) - (//+ 1)L"
h = IT + 0(1),
V
l/2l»
-in
Mi- (biY)'-]
= exp
n log —-/T j-yj- + C)(log n)
vV (1 — iiv) ) -J
exp[- (1 + e)log// + <9(log log//)]
Kiog//)0'1')//-1-'.
421 RAMSEY'S THEOREM

706
P. ERDOS AND C. A. ROGERS
So, using Stirling's formula and making some elementary reductions, we have
log Vh - log \V
^i L Hi rrh-ln g_» (h+ 1)(1 - V)
< l0gL2 h\{H-k)\ V {1'V) (A+i)-(ff+i)7.
= -2eH'logH+0(H'\og\ogH).
Thus Vh < \ V, when / is sufficiently large.
Let L be a subset of ;V with / elements. Let C'(x, £) be the part of C(x, £)
not lying in Fh. The relative surface area of C'ix, £) is at least
V- Vh > i V.
The points of the union SL of the sets C'(x, £) with x in Z, belong to at most
A — 1 of the sets C'{x, £). So the relative surface area VL of Si is at least
\Vl/(h- 1).
Hence
log VL - log[l - (h)2}'"2
>\og[hVl/{h- 1)| - log[l - (h)2]"'2
= log/ - 6logi7- (1 +e) \ogH + *m log 1/{1
1 ,. . , 1
= (1 + ch)
.1 +c,
-(l + f)
1+ [log 1/11 - (¾) |]/[2 1og4A]
(h)\ +0(]og\ogH)
log I
+ 0(\og]ogl).
c, <
log 1/U - (fo)l
2 log 4/,
provided c is chosen to be sufficiently small, we have
vL > [l - (^rr2,
for all sufficiently large I.
Since
(k- i)U - (^)T'°- < i,
for all sufficiently large I, we can now apply the lemma, with f = }rj, to the
set SL. Thus we can choose a regular (& — 1) simplex with each of its vertices
on 2 within a distance \-q of Si and with its centre at the centre of 2. So we
can choose k — 1 points xu xi xf_i of L, each point within a distance
\-t) of a different vertex of a regular (k — l)-simplex of circum-radius 1 and
edge-length
V{2(k- 1)/(k - 2)}.
422 COMBINATORIAL ANALYSIS

THE CONSTRUCTION OF CERTAIN GRAPHS 707
Since all the edges of the simplex, xJt x2 x^-i exceed
V{2(k - 1)/{k - 2)| - r, = V{2k/(k - 1)|,
the subgraph of G with vertices xlt .v2 x,;_i is a complete (k — l)-gon,
as required. This completes the proof.
References
1. P. P. Ramsay, On a problem of formal logic, Proc. London Math. Soc. (2), 30 (1930), 264-286.
2. P. Erdos and G. Szekere.s, A combinatorial problem in geometry, Compositio Math., 2 (1935),
463-470.
3. P. Erdos, Some remarks on the theory of graphs, Bull. Amer. Math. Soc., 53 (1047), 292-21)4.
4. — Remarks on a theorem of Ramsay, Bull. Research Council Israel, 7 (1957), 21-24.
5. E. Schmidt, Die Brunn-Minkowski Ungle.ichung und ihr Spicgelbild sowie die isoperinietrische
Eigenschaft der Kugel in der euklidischcn und vichteuklidischen Geometric I, Math. Xach.
Berlin, 1 (1948), 81-157.
6. P. Erdos and C. A. Rogers, Covering space -with convex bodies, Acta Arithmetic;! 7 (1962),
281-285.
University College, London
and
University of Toronto
423 RAMSEY'S THEOREM

ON SUBGRAPHS OF THE
COMPLETE BIPARTITE GRAPH
P. Erdos and J. W. Moon
(received March 22, 1963)
G(n) denotes a graph of n vertices and G(n) denotes
its complementary graph. In a complete graph every two
distinct vertices are joined by an edge. Let C (G(n)) denote
the number of complete subgraphs of k vertices contained in
G(n). Recently it was proved [1] that for every k
(1) min(Ck(G(n))+ Ck(G(n))) <-^~
J2I
2[
<
where the minimum is over all graphs G(n). It seems likely
that (1) is not far from being best possible and that
C (G(n)) + Cn(G(n))
,.,. .. . k k £■
(L) 11m mm =
-' |k\
a m
That this is true for k = 3 follows from the results of
Goodman [2], Sauve [5], and Lorden [3]. We are unable to
prove (2) for k > 3 but we can prove an analogous result for
bipartite graphs.
The bipartite graph B(m,n) consists of the vertices
x, x and y , y and some of the edges (x , y ).
1 m 1 n 1 j
B(m,n) has the same vertices, and the edge (x., y.) is in
B(m,n) if it is not in B(m,n). If B(m,n) contains mn
Canad. Math. Bull, vol.7, no. 1, January 1964
35
424 COMBINATORIAL ANALYSIS

edges then it will be referred to as the complete (m,n) graph.
Let C (B(m,n)) denote the number of complete (k,i )
k, a
graphs contained in B(m,n). We shall prove the following
THEOREM. For fixed positive integers k and i
Ck j[ (B(m,n)) + Ck ^ (B(m,n)) 2
lim min '■ '
/mi I n i ki
m-oo 1..(. 2
Q-CI
where the minimum is to be taken over all graphs B(m,n) as
m and n tend to infinity independently.
We shall prove this theorem in two steps. First we
observe that
2f
ulil)
(3) min(Cki (B(m,n))+Cki(B(m,n)))< .
The proof of this is quite similar to that of (1) but for the sake
of completeness we shall outline it.
There are j if 1 complete (k,i ) graphs contained in
the complete (m,n) graph. The probability that any one of
these is entirely contained in either B(m,n) or B(m, n) is
ki
clearly 2/2 , assuming all possibilities equally likely.
Hence, the expected value of Y = C (B(m,n)) + C (B(m,n))
ki k, i
equals the right-hand side of (3). Since Y is greater than the
right-hand side when B(m, n) is the complete (m,n) graph,
the strict inequality (3) now follows immediately.
The rest of our proof makes use of the following simple
LEMMA. Let there be given integrable functions f.(x)
and g.(x), i = l,2,...,m, such that f.(x) + g.(x) = 1 and
0<f.(x)<l for all 0<x<l. Let
36
425 RAMSEY'S THEOREM

0 v 1
(x) . . . f. (x) + S g. (x)
\ 11
V*')
dx ,
/here the sums are over the I ] unordered k-tuples of distinct
functions. Then, for every e > 0,
1
T > ^-'^T (T) ' if m>m
— 1 k > k' o
Proof. Let
1
r
o
H = kT ■[
S f.(x) ) +( S g (x)
.i=l
■ i=l
dx
Usiag the Cauchy-Schwarz inequality and crude estimates, we
find that
wIt) <h<t + f (m -^^
k k-1
But m - m =0(m ), for fixed k, as m -► oo . The
(k)
lemma now follows directly.
We now show that for every < > 0
(4) C (B(m,n)) + C (B(m,n))> (1 - « ) -jj
rain-
if m > m and n > n .
o o
j-1 j
If —— < x <— set f (x) = 1 or 0 according as the edge
n —n i
(x., y.) is in B(m, n) or B(m, n), respectively, for i =1,2. m
and j=l,2,...,n. Set f.(0)=0 and g.(x) = 1 - f.(x) for all i.
Label the k-tuples of the x points from 1 to j J . Let t
and h denote the number of points y for which the edge
v
37
426 COMBINATORIAL ANALYSIS

(x. ,y) is in B(m, n) or B(m,n), respectively, for each point
x. in the v k-tuple of points. It is not difficult to see that
ra
S (t + h ) = nT.
v v
v = l
Applying Jensen' s inequality, we have that
C (B(m,n)) + C (B(m,n)) = S
K'* '* v = l
ra'C/M/)!
>- €) , lk.
Using the lemma, a simple calculation shows that this last
quantity is greater than or equal to
<«->:= 00 ■
for sufficiently large m and n. This completes the proof
of (4) and the theorem now follows by combining (3) and (4).
It seems very unlikely that one can replace (4) by an exact
lower bound in general. Lower bounds are given in [4] for the
case that m =n and k= I -2. but it isn't known if even these
are exact when n > 10.
In closing, we remark that the above theorem can easily
be extended to the case where the edges of the complete (m,n)
graph are split into an arbitrary number of classes instead of
just two, as supposed here.
38
427 RAMSEY'S THEOREM

REFERENCES
1. P. Erd<3s, On the number of complete subgraphs
contained in certain graphs, Publ. Math. Inst. Hung.
Acad. Sci. 7 (1962), 459-464.
2. A. W. Goodman, On sets of acquaintances and strangers
at any party, Amer. Math. Monthly, 66 (1959) 778-783.
3. G. Lorden, Blue-empty chromatic graphs, Amer. Math.
Monthly, 69 (1962) 114-120.
4. J. W. Moon and L. Moser, On chromatic bipartite graphs,
Math. Mag. 35 (1962) 225-227.
5. L. Sa uve, On chromatic graphs, Amer. Math. Monthly,
68 (1961) 107-111.
University College, London
39
428 COMBINATORIAL ANALYSIS

Applications of Probabilistic Methods
to Graph Theory
Paul Erdos
Both in his lectures and his written work, Erdos takes great pleasure in offering
a monetary reward to anyone who can settle a stated conjecture; such an offer
is made in this lecture. The reader is assured that this offer is bona fide and that
Erdos has occasionally had the pleasure of paying. Whenever Erdos spoke {or
"preached" as he put it), our seminar audience was always a bit larger than
average.
The area of probability in graph theory arose from a theorem of Ramsey,
which may be simply explained by the following celebrated problem: Prove
that among any six people at a gathering, there will always be three mutual
acquaintances or three mutual nonacquaintances.
Here is the basic idea of a probabilistic argument. To prove that there
exists a graph with a specific property, one derives an estimate on the number
of graphs which do not have the property. If it can be shown that this number is
definitely less than the total number of graphs with a given number n of points,
then there must exist a graph with the property in question. But this does not
give any clue on how to construct such a graph. For example, it had been
shown by probabilistic methods that there always exists a graph with an
arbitrarily large chromatic number as well as an arbitrarily large girth, but no
one had any idea of how to find such graphs. In September 1966, a high school
student from Budapest named L. Lovdn developed a method for constructing them,
having nothing to do with the probabilistic proof.
F.H.
The application of probabilistic methods to graph theory stems from a
well-known theorem of Ramsey [16], an English philosopher and
mathematician, whose brother is the Archbishop of Canterbury. A special case of
Ramsey's result may be stated in the language of set theory: For any infinite
set S and any partition of S x S into two subsets 7\ and T2, there exists an
infinite subset A of S such that A x A is contained in Tl or in T2- Clearly the
same result holds if the partitioning is into any finite number of subsets.
This result can be stated in graphical language as follows: Every infinite
graph contains an infinite complete subgraph or an infinite independent set
60
429 RAMSEY'S THEOREM

APPLICATIONS OF PROBABILISTIC METHODS TO GRAPH THEORY 61
of points. In other words, for any infinite graph G, G or its complement G
contains an infinite complete subgraph.
The finite form of Ramsey's theorem is more involved. Let/(/-, s) be the
smallest integer such that every graph G with/(/-, s) points contains at least r
mutually adjacent points or s independent points. Symbolically, G contains
the complete graph Kr or G contains Ks. Of course/(/-, s) = f(s, /-) . An upper
bound for the value of/(/-, s) was found by Erdos and Szekeres [11], and there
has been no serious improvement of this result:
The proof of (1) follows readily from the following recursive inequality
by double induction on /- and s:
f(r,s) </(/-- l,s)+f(r,s- 1). (2)
To prove (2), let G be any graph with/(/- - 1, s) + f(r, s — I) points, and
let v be any point of G. Let A be the set of points adjacent with v and B the
remaining points. Then A has at least/(/- - 1, s) points or B has at least
/(/-, J — 1) points, since the sum is the total number of points in G. Assume
that A has/(/- — 1, j) points. Then it follows that the subgraph induced by A
contains Kr_ l or Ks. If A contains Kr_l, then the subgraph induced by v and
A contains Kr; otherwise, G has s independent points. If A does not have
/(/- — 1, s) points, then B has/(/-, s — 1) points, and similar arguments show
that G contains Kr or has s independent points, proving inequality (2) and
hence the theorem.
The special case of (1) when /- = s is
/(/-,/-) <(2;;ij). (3)
This has been improved by Frasnay [12] to the inequality
*■* 4:(2::?)-
A question on a Putnam examination a few years ago asked the contestants
to prove that among any six people at a gathering, there will always be 3 who
know each other or 3 who do not know each other. In other words, for any
graph G of six points, either G or G contains a triangle. Thus this question asks
for the proof that/(3, 3) = 6.
It is also known that/(4, 4) = 18, and it is easily shown that/(/-, 2) = /-.
Gleason (unpublished) has recently extended the known values of/(/-, s), but
the determination of exact values in general remains a difficult unsolved
problem, even for /(/-, /-). The conjecture has been made that (/(/-, /-))1/r
approaches a limit as /- ->■ oo. The following bounds on (/(/-, /-))1/r are known
V2 < (/(/-, /-))1/r < 4 . (5)
430 COMBINATORIAL ANALYSIS

62 APPLICATIONS OF PROBABILISTIC METHODS TO GRAPH THEORY
The upper bound in (5) follows immediately from (3) since
f::5)-'-
The proof by Erdos [3], [6], and [7] of the lower bound uses probabilistic
methods as follows. From n given points, fix any r of them and suppose that
either all lines or no lines joining them appear, that is, they induce Kr or Kr.
Since there are two choices for each other line of a graph with n points, namely
(")~(r)
"to be or not to be," there are exactly 2V v possibilities for the occurrence
of these other lines. There are | I ways of choosing /- points from n given
points, and there are 2
satisfying the inequality
(")
points, and there are 2 labeled graphs with n points. Therefore, for every n
(ffr) -2-2^-^ <T'>
(6)
there is a graph with n points containing no Kr or Kr . Note that the factor 2
in the left side of (6) refers to the choice of Kr or Kr. But it is easy to verify
that (6) holds whenever n > 2r/2. Hence/(/-, /-) > 2r/2, completing the proof
of (5). Note that no explicit construction is known for such a graph. Only its
existence has been proved by this probabilistic (or one might say
computational) argument.
An interesting special case is the study of/(3, s). When /- = 3 is
substituted into inequality (1), we get
f(3,s)<(S+2l'y (7)
This inequality has not been much improved. But it has been observed by
several mathematicians that you can subtract cs, a constant times s, from the
right side of (7). Incidentally, a trivial lower bound is given by
/(3, s) > 3s. (8)
By elementary but complicated computations, Erdos [4] has proved the strict
inequality in the following result.
„2
"" 2<f(3,s)<(S+2l))-cs. (9)
(log*)2
Conjecture. As s -»■ oo, /(3, s)/s2 approaches a constant. (I offer fifty
dollars to anyone who can prove or disprove this conjecture.)
A well-known problem (see Dirac [2]) asked whether there exists a graph
G with an arbitrarily high chromatic number containing no triangle. Writing
under the pseudonym of Blanche Descartes [l], Tutte proved this result by
providing an explicit construction of a graph with an arbitrary chromatic
number containing no triangle. As is often the case, this theorem was later
rediscovered independently; see Zykov [17] and Mycielski [15].
431 RAMSEY'S THEOREM

APPLICATIONS OF PROBABILISTIC METHODS TO GRAPH THEORY 63
Tutte's result was extended by Kelly and Kelly [14], who proved by an
explicit construction that for any positive integer /-, there exists a graph G that
contains no triangle, quadrilateral, or pentagon and that has a chromatic
number greater than /-. They conjectured that, for any two positive integers /-
and s, there exists a graph G, whose chromatic number is at least /-, which
contains no polygon with fewer than s sides. This conjecture was proved by
Erdos [5], using a probabilistic argument, the outline of which follows.
Consider a graph G with a large number n of points and let c be a large
constant. Consider further that G is a random graph with n points and en lines.
In a series of papers on the evolution of random graphs, Erdos and Renyi [10]
studied the probable structure of G and its dependence on the value of c. They
showed that for sufficiently large c almost all graphs contain triangles,
quadrilaterals, and so on. A simple computation shows that the expected
number of small cycles is very small. Destroy them by deleting one line from
each small cycle. But another simple computation shows that most of the
remaining graphs still have an arbitrarily large chromatic number.
Erdos and Hajnal [8] gave a simple construction of an infinite graph
having infinite chromatic number and containing no triangle. Let N(2) be the
set of all unordered pairs of distinct positive integers {!,]} with i < j. Now
construct a graph G with point set JV<2) in which, for every two integers i and k,
the pairs {i,j} and {j, k] are adjacent. Clearly, G has no triangle. To prove that
the chromatic number of this infinite graph G is infinite, assume that the
chromatic number x(G) = /- < oo. Then the points of G can be split into /-
classes of independent points. Applying Ramsey's theorem to the complete
graph H whose points are the integers, we find that at least one of these classes
contains an infinite complete subgraph H' of H. Therefore, it also contains
two lines {i,j} and {j, k], which shows that G can not have chromatic number /-.
A classical problem is to find the maximum number n of points so that the
lines of the complete graph Kn can be colored with /- colors, in such a way that
there is no triangle which is unicolored (all one color). The solution, that n = 5
when /- = 2, is (in disguise) the Putnam examination question mentioned
above. Recently, Greenwood and Gleason [13] showed that for 17 points,
every coloring with three colors contains a unicolored triangle, but for 16
points, not every coloring contains such. For 66 points and four colors, there
always exists a unicolored triangle, but until very recently, the situation for 65
points was unsolved. However, a Hungarian sociologist, Szalai, showed by an
explicit construction that one can color the lines of K6S by using four colors
so that there is no unicolored triangle. Incidentally, in this process Szalai
rediscovered Ramsey's theorem.
Erdos and Rado [9] proved the following related theorem concerning
infinite graphs without the continuum hypothesis but by using the axiom of
choice. For every infinite cardinal number m, there exists a graph G with m
points containing no triangle and having chromatic number m.
Erdos conjectured that there exists a graph G containing no quadrilateral
432 COMBINATORIAL ANALYSIS

64 APPLICATIONS OF PROBABILISTIC METHODS TO GRAPH THEORY
and whose chromatic number is nondenumerable. He offered ten pounds to
anyone who could prove or disprove it, and Hajnal has just settled this
conjecture negatively. Actually, the proof is not difficult.
We conclude by noting that probabilistic methods do not usually give the
best possible results, but they can be used in many different situations and
enable one to attack problems which could not even be started otherwise.
References
[1] Blanche Descartes, A three colour problem. Eureka 9(April 1947) 21. Solution.
Eureka 10(March 1948) 24. (See also the solution to Advanced problem 1526.
Amer. Math. Monthly 61(1954) 352.
[2] G. A. Dirac, The structure of ^-chromatic graphs. Fund. Math. 40(1953) 42-55.
[3] P. Erdos, Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53(1947)
292-294.
[4] , Graph theory and probability I. Canad. J. Math. 11(1959) 34-38.
[5] , Graph theory and probability II. Canad. J. Math. 13(1961) 346-352.
[6] , On the number of complete subgraphs contained in certain graphs.
Publ. Math. Inst. Hung. Acad. Sci. 7(1962) 459-464.
[7] , On a problem in graph theory. Math. Gaz. 47(1963) 220-223.
[8] and A. Hajnal, Some remarks on set theory IX. Mich. Math. J. 11(1964)
107-127.
[9] and R. Rado, A construction of graphs without triangles having pre-
assigned order and chromatic number. J. London Math. Soc. 35(1960) 445-448.
[10] and A. Renyi, On the evolution of random graphs. Publ. Math. Inst.
Hung. Acad. Sci. 5(1960) 17-67.
[11] and G. Szekeres, A combinatorial problem in geometry. Compositio
Math. 2(1935) 463-470.
[12] C. Frasnay, Sur des fonctions d'entiers se rapportant au theoreme de Ramsey.
Comptes Rendus Acad. Sci. Paris. 256(1963) 2507-2510.
[13] R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic
graphs. Canad. J. Math. 7(1955) 1-7.
[14] J. B. Kelly and L. M. Kelly, Paths and circuits in critical graphs. Amer. J. Math.
76(1954) 786-792.
[15] J. Mycielski, Sur le coloriage des graphes. Colloq. Math. 3(1955) 161-162.
[16] F. P. Ramsey, On a problem of formal logic. Proc. London Math. Soc. Ser. 2,
30(1929) 264-286.
[17] A. A. Zykov, On some properties of linear complexes. Math. Sbornik. (66)
24(1949) 163-188. Amer. Math. Soc. Translation No. 79 (1952).
433 RAMSEY'S THEOREM

PACIFIC JOURNAL OF MATHEMATICS
Vol. 37, No. I, 1971
RAMSEY BOUNDS FOR GRAPH PRODUCTS
Paul Erdo's, Robert J. McEliece and Herbert Taylor
Here we show that Ramsey numbers M(klt- ■ ■ ,kn) give
sharp upper bounds for the independence numbers of product
graphs, in terms of the independence numbers of the factors.
The Ramsey number M(ku ••-,&„) is the smallest integer m with
the property that no matter how the ( „) edges of the complete
graph on m nodes are partitioned into n colors, there will be at least
one index i for which a complete subgraph on kt nodes has all of its
edges in the ith color. Ramsey's Theorem tells that these numbers
exist but only a few exact values are known.
The complement graph G has the same nodes as G and the
complementary set of edges.
The independence number a(G) of a graph G, is the largest
number of nodes in any complete subgraph of G.
The product Gt x • • • x Gn of graphs Gu ■ ■ ■, Gn is the graph
whose nodes are all the ordered w-tuples (au ---,¾) in which a; is a
node of G; for each i from 1 to b, and whose edges are as follows.
A set of two nodes {(au • • -, an), (by, ••-,&„)} will be an edge of
Gi x • • • x Gn if and only if the nodes are distinct and for each i from
1 to n, a,; = 6; or {ait &;} is an edge of G;.
Theorem 1. For arbitrary graphs Gu ••-, Gn
a(G, x • • • x Gn) < MiaiG,) + 1, • • •, a(Gn) + 1) .
Proof. We have a complete subgraph of Gt x • • • x Gn on
a(GL x • • • x Gn) nodes. Its edges can be n colored by the following
rule: give {(au ••-, an), (xu •••, xn)} color i if i is the first index for
which {a;, x;} is an edge of G;.
With this coloration any case where all the edges on k nodes
have color i requires a complete k subgraph of G; and so requires
k < a(Gt) + 1. With the definition of the Ramsey number this ensures
that
a(G, x • • • x Gn) < MiaiG,) +1,---, a(Gn) + 1) .
Theorem 2. If ky, • • -, kn are given, there exist graphs Gu • • •, G„
such that for each index i from 1 to n, a(G;) = ki and
a(G, x • • • x Gn) = M(kL + 1, • • -, kn + 1) - 1 .
45
434 COMBINATORIAL ANALYSIS

46 PAUL ERDOS, ROBERT MCELIECE AND HERBERT TAYLOR
Proof. From the definition of the Ramsey number there must
exist an n color partition of the edges of the complete graph on
Mik,, + 1, • • •, k„ + 1) — 1 = m modes such that for every i from 1 to
n the largest complete subgraph in the ith color is on &; nodes. For
each i let Gt be the graph on the same m nodes having all the edges
not of color i. Thus for each i, a(G;) = &;. These Gt make the
diagonal a complete m subgraph of Gt x • • • x Gn, and so
a(Gi x • • • x Gn) ^ m .
Applying Theorem 1 we have
a(G, x • • • x Gn) = M{k, + 1, •••,&„ + 1) - 1
Theorem 3. If n and k are given, there exists a graph G such
that a(G) = k and putting k; — k for every i,
a(Gn) = M{ky + 1, • • •, kn + 1) - 1 .
Proof. With m = M^ + 1, • • •, kn + 1) - 1 and every A.-; = k,
refer to the graphs Glt '",Gn as specified for Theorem 2. Now
construct G as follows. Let the nodes of G be all the ordered pairs
(a, i) such that 1 < i < n and a is a node of G;. Let {(a, i), (b, j)} be
an edge of G if and only if i ^= j or {a, b} is an edge of G;.
Thus constructed a(G) = k because each a(Gj) = k. Gn will have
a subgraph isomorphic to G: x • • • x G„ and consequently
a(G") ^ a(G, x • • • x Gn) = m .
So again with Theorem 1 we have
a(Gn) = m = M{k, + 1, • • •, kn + 1) - 1 .
A question remains whether for every k, n with
k2 < n < M(k + 1, k + 1)
there exists G such that a(G) = k and a(G2) = w. It is known that
M(4, 4) = 18, and for each n between 9 and 17 we have found a graph
G such that a(G) = 3 and a(G2) = n. However it is only known that
37 < M(5, 5) < 58 and for example we have no proof that there exists
a graph G such that a(G) = 4 and a(G2) = M(5, 5) - 2.
Received May 25, 1970. The work of the latter two authors represents one phase
of research carried out at the Jet Propulsion Laboratory, California Institute of
Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and
Space Administration.
Jet Propulsion Laboratory
435 RAMSEY'S THEOREM

Chapter 9
Property B
A family of sets is said not to have property B if no matter how the elements
are two colored at least one of the sets will have all its elements the same color.
These papers fix bounds for the minimal sized family that does not have
property B under varying conditions. These conditions include size of the sets
(they are usually required to have the same cardinality) and the size of the
union.
Results on property B may be used to find lower bounds for Ramsey
numbers and Van der Waerden numbers (see ES).
A "property B game" is considered in a paper by P. Erdos and J. Sel-
fridge, to appear in the Journal of Combinatorial Theory in 1973 or 1974.
Further results on property B are found by H. L. Abbott and D. Hanson,
"On a combinatorial problem of Erdos," Canad. Math. Bull. 72(1969)
823-829.
Erdos and Hajnal have shown in [461] (not included in this volume) the
existence of a family of n-sets not having property B such that no two sets
intersect in more than one element. Their proof was nonconstructive; a
constructive proof is given by H. L. Abbott, "An application of Ramsey's
theorem to a problem of Erdos and Hajnal," Canad. Math. Bull. 5(1965)
515-518.
Papers in Chapter 9
[390] On a combinatorial problem. I
[430] On a combinatorial problem. II
[527] On a combinatorial problem. Ill
437 PROPERTY B

ON A COMBINATORIAL PROBLEM
P. erdOs
Let g be a family of sets. Qr is said by E. W. Miller [3] to possess
property B if there exists a set B such that
F n B 4= 0 for every F e g ,
F $ B for every f eg.
Miller used the letter B in honour of Felix Bernstein, who in the early
years of this century proved that the perfect sets have property B and
using this "constructed" a totally imperfect set of power continuum
(that is, a set of power continuum which does not contain a perfect set).
I put constructed in quotation mark, since he used the axiom of choice
(in fact, without the axiom of choice the existence of a totally imperfect
set has never been proved).
Several other well known theorems can be formulated in terms of
property B. For example, a well known theorem of van der Waerden
states that if we split the integers into two classes, then at least one
class contains for every k an arithmetic progression of 1c terms. This
theorem can be formulated as follows: The family of all arithmetic
progressions of 1c terms does not have property B.
Hajnal and I [2] recently published a paper on the property B and
its generalizations. One of the unsolved problems we state asks: What
is the smallest integer m{p) for which there exists a family $ of finite
sets Alt . . .,Am(p), each having p elements, which does not possess
property B ?
For p=l there is no problem since m(p)=l. Trivially ra(2) = 3 and
by trial and error we showed m(3) = 7. m(3)^7 is shown by the set of
Steiner triplets (1,2,3), (1,4,5), (1,6,7), (2,4,7), (2,5,6), (3,4,6), (3,5,7).
It is easy to see that every set which has a non-empty intersection with
each of these sets must contain at least one of them. By a somewhat
longer trial and error method we showed m(3) > 6. Thus ra(3) = 7. The
value of m(p) is not known for p> 3 and it does not seem easy to
determine m(p) even for £>=4. We further observed that m(p)S I 1 by
[5]
439 PROPERTY B

6
P. ERDOS
defining the family g as the set of all subsets taken p at a time of a set
of 2p—l elements.
We shall now show that for all p=?2:
(1) m(p) > 2P-1 ,
and for every e > 0 if p > p0(e):
(2) m(p) > (l-e)2*>log2 .
Ak will denote the number of elements of Ak, and At\Aj will denote
the set of those elements of Ai which are not contained in Aj. Instead
of (1) and (2) we shall prove the following
Theorem 1. Let {A^, l^i^k be afamily $ of finite sets, Ai = xi^2. If
(3) Z—^-
or
(4) Si1-**)-1*
holds, then $ has property B.
(1) clearly follows from (3) and (2) from (4). In fact (4) clearly implies
(3), and we include (3) only because its proof is very simple.
I do not know the order of magnitude of m(p) and cannot even prove
that
(5) Y\mm(p)l/P
p—>oo
exists. Quite possibly the limit in (5) is 2.
Put \J'[=lAi = T, T = n. If g is a family of sets, Jy will denote the
number of sets in the family. Denote by $T the family of sets S for which
(6) S <= T, Ai n S * 0, At £ 8, 1 ^ i ^ k .
We have to show that if (3) holds then ^T > 0 (since this implies that
the family of sets At, l^i^k satisfying (3) has property B). Denote
by ^- the family of sets S satisfying
(7) S <= T, Ai c S or At n S = 0 .
Clearly an S<=T is in the family %T if it is in none of the families ^4,
l^i^k (that is, it satisfies (6) if it does not satisfy (7) for any i', l^i^ k).
By a simple sieve process we thus have
&
(8) 3^2--2^+1-
440 COMBINATORIAL ANALYSIS

ON A COMBINATORIAL PROBLEM
(
The proof of (8) is indeed easy. 2n is the number of all subsets of T,
and to obtain f^T we have to subtract away all the sets of $it 1 5S i <. fc.
But the sets which contain A^i)A2 have been subtracted away twice
and there is at least one such set (namely T), which explains the sum-
mand + 1 on the right hand side of (8). We evidently have
(9) & = 2-^+1 ,
since clearly there are 2n~ai sets S<=T satisfying Ai <^S and 2n~ai sets
satisfying AinS = 0. From (8) and (9) we have %T S 1 if (3) is satisfied.
This proves the first statement of Theorem 1.
To prove the second statement we need the following
Lemma. Let T<=TU Tl = m>n. The number of subsets S<^Tl which do
not contain any of the sets At, l^iiSk is greater than or equal to
(10) 2^/7(1-1),
with equality if and only if the sets Ai are pairwise disjoint.
We use the set T±^>T only to make our induction proof easier. Denote
by f(At, . . .,A}-; Tx) the number of subsets S of Tx not containing any
of the sets At, l^i^j, and by f(Alt .. . ,A}-; Ai+1,Tt) the number of
sets S^T-l which contain Aj+1, but do not contain any of the sets At,
l<i<j.
If the sets Ai are pairwise disjoint, we evidently have
(11) f(Alt...,Ak;Tx)= 2»-»/7(2«<-l) = 2-//(l-l),
since we obtain the sets S<= T±, At^S, 1 5Si <;k by taking the unions of
all the proper subsets of the sets Ai with any subset of T±\T. Thus
there is equality in (10).
Assume next that the sets Ai are not pairwise disjoint, say A^A^ + 0.
If k = 2, a simple argument shows that
/(A^A^Tj) = 2m - 2m-ai - 2m-"2 + 2m~n > 2m(l -) (l-—),
where n = Ali)A2 <ot1 + otz. Thus for k = 2 (10) holds with the sign of
inequality. Assume next that if we have any k — 1 (k S 3) sets which are
not pairwise disjoint, then (10) holds with the sign of inequality. We
shall show that the same is true for k sets Ax, . . .,Ak, Axr\A%^0.
By a simple argument we have
(12) f(Av .. .,Ak; Tt) = /(,1,,. . .,Ak_i; ^)-/(^, . ...Ak-i\ A^i) •
441 PROPERTY B

8 P. ERDOS
By our induction hypothesis we have
(13) f(Al,...,Ak_1;Tl) > 2-/7(1-^).
Further clearly
(14) f(Ax,...,Ak_x;Ak,Tx) = f(A,\Ak,.. .,Ak^\Ak;T^\Ak) .
To every subset S' of T-^A^ which does not contain any of the sets
A^Afi., l^i^k— 1, we make correspond 2"k subsets 8 of Tl which do
not contain any of the sets At, \r^i<k—\. It suffices to consider the
sets
(15) S'vS", S" <= Ak.
Clearly if two subsets $5/ and S2' of Tj\.4fc are distinct, all the sets
(15) are distinct. Thus we have
f(Au.. .,Ak ,; Tx)
(16) f(Ax\Ak, . ..,Ak^\Ak; T^A,) Z J[ ^k~1' ^.
From (12), (13), (14), and (16) we obtain
f(A1,...,Ak;T1)^f(A1,...,Ak_1;T1)^l-^ > 2-/7(1-!),
which proves the Lemma.
The Lemma in fact follows immediately from the following special
case of a theorem of Chung [1]: Let Et, l<i<k be k events of
probability /?4, Ei denoting the event (of probability 1 — /?4) that Et does not
happen. Assume that for every {, 2 <;«<;&:
(17) P(E, u .. . U Et^ I Et) > P{EX u . .. U Et_x) ,
where P(E \ F) denotes the conditional probability of E happening if
we know that F has happened. (17) implies
k
(18) P(Ein...nEk) > /7(1 -pt),
with equality only if there is equality in (17) for every i, 2^i<.k. We
obtain our Lemma by defining the event Et as the event that S^T-^
contains At.
To complete the proof of Theorem 1 we have to show that if Ait
l^i^k satisfies (4), then ^T>0 (see the proof of (3)). Clearly
2n—f(A1,...,Ak;T) equals the number of subsets S of T for which
Ax c S holds for some i, 1 :£ i S k, and it also equals the number of subsets
442 COMBINATORIAL ANALYSIS

ON A COMBINATORIAL PROBLEM
9
S<=T for which At<=T\S for some i, 1 ^i^k. Denote by L the number
of subsets S<=T for which At <=S and Ai2<=T\S for some ix and i2.
A simple argument shows that
(19) gr= 2--2(2--/(^,...,^7)) + ^.
If the sets At are not pairwise disjoint, then (4), (19) and our Lemma
implies %T > 0. If the sets At are pairwise disjoint (in fact if A1nA2 = 0),
then L>0 since A1<=Al, A2<=T — AV Thus in any case (4) implies
|jr > 0 and hence Theorem 1 is proved.
By slightly more complicated arguments we could prove the following
Theorem 2. Let A±,A2,... be a finite or infinite sequence of finite sets
satisfying
X > 2 and /7(1 ] >-,
i \ 2"*/ _ 2
and Ax' ,A%,... a finite or infinite sequence of infinite sets. Then the
family {.4^11(.4/} has property B.
Now one can ask the following problem which I cannot answer: Let
{Af} be a finite or infinite family of finite sets which does not have the
property B and for which Ai>p>2 for all i. What is the upper bound
C(p> of /7i(l-2-a<) and the lower bound Cp of 2^2-°'? Very likely
C^ = H and C2=i Probably
lim C(p) = 0, lim Cp = oo .
p—>oo p—>oo
If f(Al,...,Ak;T)>2n-1, our proof immediately shows that the
family {At}, \<i^k has property B, but if f(Alt . . .,Ak; 7) = 2--1. the
family {At}, l^iSk does not have to have property B, for instance if
it Consists of the subsets taken p at a time of a set of 2p — 1 elements.
A family of sets ^f is said to have property B(s) if there exists a set
B such that FnB + 0 and FnB<s for every i1 of g.
Hajnal and I asked [2] what is the smallest integer m(p,s) for which
there exists a family g of sets At, l^iSm(p,s) not having property
B(s) and satisfying At = p, 1 <i^m{p,s). Clearly m(p,p) = m(p), and we
remarked that m(p,s)^
Using the methods of this note we can show that positive absolute
constants c± and c2 exist so that
(1+(^)8 < m{p,s) < (l+c2)s.
r.)
443 PROPERTY B

10
P. ERDOs
REFERENCES
[1] K. L. Chung: On tnutually favorable events. Annals of Math. Stat. 13 (1942), pp.
338-349.
[2] P. ErdOs and A. Hajnal: On a property of families of sets. Acta Math. Acad. Hung.
Sci. 12 (1961), pp. 87-123; see in particular problem 12 on p. 119.
[3] E. W. Miller: On a property of families of sets. Comptes Rendus Varsovic 30 (1937),
pp. 31-38.
444 COMBINATORIAL ANALYSIS

ON A COMBINATORIAL PROBLEM. IP
By
P. ERDOS (Budapest), member of the Academy
Let M be a set and F a family of its subsets. F is said by £. W. Miller [5]
to possess property B if there exists a subset K of M so that no set of the family F
is contained either in K or in K (K is the complement of K in M).
,Hajnal and I [2] recently published a paper on the property B and its
generalisations. One of the unsolved problems we state asks: What is the smallest integer
m(n) for which there exists a family F of sets Ax, ..., Am(n) each having n elements
which does not possess property Bl Throughout this paper A: will denote sets
having n elements.
We observed w(n)^| J, m(l)=l, m(2) = 3, w(3) = 7. As far as 1 know
the value of w(4) is not yet known.
Recently I [3] showed that w(n)>2"_1 for all n and thatfor n-^n0(e) m(n)>
>(1 -e)2"log2. W. M. Schmidt [6] proved w(«)>2"(l + ^n~x)~x and up to date
this is the best lower bound known for m(n).
Recently Abbott and Moser [1] proved that
(1) m(ab)^m(a)m(b)a.
From (1) they deduced that for n>n0, m(n) <(]/7-!-e)" and that lim m(ny " exists.
Their method is constructive. By non-constructive methods 1 now prove
Theorem 1. w(n)-=/722" + 1.
Theorem 1 thus implies lim m(nY" =2. Theorem 1 and the result of Schmidt
n= »
gives
(2) 2--(1+4^(-1)-1 <m(n)<n22"+1.
It would be interesting to improve the bounds for m(n). A reasonable guess
seems to be that m(n) is of the order n 2".
A family of sets Fis said to have property B(s) if there exists a set S which has
a non-empty intersection with each set of the family, but the cardinal number of
the intersection is ^s. Hajnal and I asked what is the smallest integer m(n,s)
for which there exist sets {A;}, l = i^m(n, s) which does not possess property
B(s) ? Clearly m(n,n) = m («).
* This paper was written while the author was visiting at the university of Alberta in Edmonton.
445 PROPERTY B

446 P- ERDOS
Mr. H. L. Abbott pointed it out to me that m{2k, 2) = 3, m{2k +1,2) = 4.
Now we prove Theorem 1. We shall construct our n22" + 1 sets of n elements
not having property B as subsets of a set M of 2n2 elements. Suppose I have chosen
already k of the sets (&<n22" + 1) Alt ..., Akand suppose that there are uk pairs of
subsets {Ki, AT,-}, 1 ^i^uk of M so that no set Ah 1 ^i^k is contained either in
A^ or in K. If wfc = 0 our Theorem is proved. Assume henceforth wt>0. We shall
prove that we can find a set Ak+l so that
1
~ 2"
(3) w, + 1^wk|l
(For each /', 1 ^i^uk, consider all subsets of n elements of K: and K,.) For
fixed /' the number of these subsets is clearly {\B\ denotes the number of elements
Thus the total number of subsets of n elements under consideration (1 5 i = uk)
, (n2\
is at least 2uk I I.
(2n2\
The total number of subsets of M taken n at a time is I I. Hence at least
one of these sets, say Ak + l, occurs either in K-, or in K; for at least
in
2w,.
n
n~ l
4) -77^- = 2m* /7 («2-0 /7 (2«2-/)| =
i'=0 Vi= 0
(r)
. i
M-1
A Vt' i__J_L3
2"-1 ,« 2n2-/ 2"
values of /'. Hence from (4) uk+l-=-Uk\\ — — I and (3) is proved.
Clearly u0 = 22nl~l (since M has 22"2 subsets). Hence from (3)
1
(5) Wr-22"2-i(l-2„
Hence from (5) if r = n22" + 1, w,-=1, thus wr = 0 and our sets Ah I 5=/s=n22"+1
do not have property B and the proof of Theorem 1 is complete.
2
elements we could show by slightly more careful
By taking M to have
n
~2
calculation that for every e>0 and »>/70(X)
(6) m(n)<(l+e)elog2n22"-2.
It seems unlikely that (6) can be improved to any great without some new idea.
446 COMBINATORIAL ANALYSIS

ON A COMBINATORIAL PROBLEM. II
447
By methods used in a paper of Renyi and myself [4] I can prove the following
Theorem 2. Let M be a set of N elements. Put
(7) k=cN»'n[i-ii1-.
where C is a sufficiently large absolute constant. Then for all but
O
choices of k subsets Ait l^i^k of M, the A's will not have property B.
1 can show that the order of magnitude in (7) cannot be improved, but 1 can
not determine the correct value of C.
Let M be a set of N elements. Denote by mN(n) the smallest integer for which
there exist subsets /1,-,1 ^i^mN(n) of M which do not have property B. The problem
makes sense only for/V S 2» —1 and clearly/??2,,-i(«) —I n )• Tor IVs2«-l,
mN(n) is a non-increasing function of N and for sufficiently large N, mN(n) =m(n).
Let N0 be the smallest integer for which wVo(«) =m(n), probably N0 = Cn2. It seems
to me that perhaps the order of magnitude of rns(n) is
«-i i j
N2nffAl~N-i
This would in particular imply that if /V<c,w, ms(n) > (2 + t2)". 1 have been
unable to throw any light on any of these questions.
MATHEMATICAL INSTITUTE,
EOTVOS LOR AND UNIVFRSITV,
BUDAPEST
( Received 6 January 1964)
References
[1] A. L, Abbott and L. Moser, On a Combinatorial Problem of Erdos and Hajnal, Canad. Math.
Bull. 7 (1964).
[21 P. Erdos and A. Hajnal, On a property of families of Sets, Acta Math. Acad. Sci. Bung.. 12
(1961), pp. 87 — 123; see in particular problem 12 on p. 119,
[31 P. Erdos, On a Combinatorial Problem, Nordisk Mat. Tidskrift, 11 (1963), pp, 5-10,
L4] P. Erdos and A, Renyi, On the Evolution of Random Graph, Pub/. Math. Inst. Hung. Acad.,
5 (i960), pp, 17-67,
[5] E.W. Miller, On a property of families of sets, Conrp. Rend. Varsovie, (1937), pp, 31—38,
[6[ W, M, Schmidt, Ein Kombinatorisches Problem von P. Erdos und A, Hajnal, Acta Math.
Acad. Sci. Hung.. 15 (1964), pp. 373-374.
447 PROPERTY B

ON A COMBINATORIAL PROBLEM III
P. Erdos
(received October 16, 196S)
A family of sets { Aj is said by Miller [3] to have property B if
there exists a set S which meets all the sets A and contains none of
a
them. Property B has been extensively studied in several recent
papers (see the references in [2] and the last chapter of P. Erdos and
A. Hajnal, On chromatic number of graphs and set systems, Acta. Math.
Acad. Sci, Hung. 17 (1966) 61-99). Hajnal and I define m(n) as the
smallest integer for which there is a family of m(n) sets A , |A | = n,
1 < k < m(n), which do not have property B [1]. Trivially
2n-l
m(n) < ( ) (take all subsets taken n at a time of a set of 2n-l
— n
elements), m(2) = 3, m(3) = 7, m(4) is not known. It is known [2],
[4] that for n > n(€ )
(1) 2n(l + | f1 < m(n) < (1 + 6 ) e Iog2 n22n"2
m (n) is the smallest integer for which there are m (n) sets
Nv ' B Nv '
Ak> IV
n, 1 < k < m (n) which are all subsets of a set S,
— — N
N
and which do not have property B. I conjectured in [2] that for N < en,
m (n) > (2 + c ) . In this note we prove this conjecture and get fairly
good upper and lower bounds for m (n). In fact we prove that if
N = (c + o(l))n
(2)
Iim m (n) /n - 2 (c-2) * (c"2) (c-l)1"0 *c for c > 2 and
n=oo N c2
Iim m (n)
n=°o w
1/n
4 if N = (2 + o(l))n.
Canad. Math. Bull: vol. 12, no. 4, 1969
413
448 COMBINATORIAL ANALYSIS

THEOREM 1.
1 n_1
(3) m (n) > m2N(n) > 2n" II (1 + ^-) .
i=0
Let |S| = 2N and |A I = n, 1 < k< m (n) where {A } is
a family of subsets of S which does not have property B. Clearly S
1 2N (t)
can be split in —( ) ways as the union of two disjoint sets S and
S, 1 < t < -( ) for every t, |S(t) | = '|S^' | = N. By assumption
the family {A } , 1 < k < m (n) does not have property B. Thus for
1 2N (t)
every t, 1 < t < — ( ), at least one of the sets S , i = 1 or 2,
— — 2 N l
contains one of our A' s. A fixed A can clearly be contained for only
k k
( ) values of t in one of the sets S. , i = 1 or 2 (i.e. there are
N-n i
( ) subsets of S having N elements which contains a given A ).
Thus clearly
, , 1 2N. ,2N-n 1 n_1 2N-i ,n-l n_1 IA i .
m2N(n) ± 2( N)/( N-n> = 2 " ITT = 2 ." (1 + 2^
i=0 i=0
Thus since m (n) < m (n) is obvious, Theorem 1 is proved.
k+1 — k
THEOREM 2.
<*> m2N+1(n) < m2N(n) < [N2n "n (l-^)"1]
i =0
= N2n £0li+*hi> =f(N->
The proof of Theorem 2 follows very closely the proof in [2],
Let |S | = 2N. We shall construct our f(N, n) sets A 1 < k < f(N,n),
ACS, | -A. | = n, not having property B by induction. Suppose I
have already chosen t of the sets A., 1 < j < H < f(N,n) and suppose
that there are u pairs of subsets ofS{K,K},l<i<u so that
I i i - - I
no set A 1 < j < p is contained either in K. or in K..
J - J — i i
414
449 PROPERTY B

If u = 0 Theorem 2 is proved. Assume henceforth u > 0. We
shall prove that we can find a set A so that
i+1
(5) Vi-V1" " (1"*k,/2a~1)'
1 = 0
For each i, 1 < i < u , consider all subsets of n elements of
_ ~- I
K and K . For fixed i the number of these subsets is clearly
l l
<|Ki') + (|Ki') > 2(N) (|K. | + |K.| = |S I = 2N).
n n — n l l
Thus the total number of subsets under consideration
(1 < i < u I is at least 2u ( ). The total number of subsets of S
-- I £^
2N
taken n at a time is ( ). Hence at least one of those sets say A
n _ £+1
occurs either in K or in K. for at least
l i
-JSJ- - 2.4 n 0,.,,,^ ,„.,„ . _ ^ (1. _|_,
values of i, which proves (5).
2N-1 2N
Clearly u = 2 (since S has 2 subsets). Hence from (5)
n-1
11 (1 ' 2NTT>
r
Thus by (6) if r = f(N,n), u < 1 and our sets A., 1 < j < f(N,n),
do not have property B, which completes the proof of Theorem 2.
(2) follows easily from Theorems 1 and 2 by Stirling' s formula.
415
450 COMBINATORIAL ANALYSIS

For large values of N instead of m (n) it seems more
appropriate to consider m' (n) where m' (n) is the smallest integer
for which there is a family {A } 1 < k < m' (n) not having property
B and satisfying A C S, |S | =N and the further property that the
set of A' s contained in any proper subset of S has property B.
For n = 2, m' . (n) = 2N+1, and, for even N,m' (n) is not defined;
2N+1 N
this is just a restatement of the fact that the only critical three chromati
graphs are the odd circuits.
2n-l.
ot
<sn- 1
It is easy to see that m (n) = m (n) =( )-1 can n
2n-l 2n n
compute rn (n) and in fact do not know the value of m (4).
2n+l 9
It would be interesting to find an asymptotic formula for m (n)
and m' (n), but I have not been able to do so. The upper and lower
bounds for m (n) given by Theorems 1 and 2 differ by 2N; I could not
even decrease this to o(N).
I wish to thank the referee for some very useful remarks.
REFERENCES
n
1. P. Erdcs and A. Hajnal, On a property of families of sets.
Acta. Math. Acad. Sci. Hung. 12 {1961) S7-123 (see in
particular problem 12 on p. 179)
it
2. P. Erdos, On a combinatorial problem II. Acta. Math. Acad.
Sci. Hung. 15 (1964) 445-447.
3. E. W. Miller, On a property of families of sets. Comp. Rend.
Varsovie (1937) 31-3S.
it
4. W. M. Schmidt, Ein Kombinatorische s Problem von P. Erdos
un
d A. Hajnal. Acta. Math. Acad. Sci. Hung. 15 (1964) 373-374.
McGill University
416
451 PROPERTY B

Chapter 10
Systems of Sets
Paper [564] is a survey article giving many interesting results and conjectures.
It includes references to some of the earlier papers in this section.
The conjecture in [108] was settled by D. J. Kleitrnan in "On a lemma of
Littlewood and OfTord on the distributions of linear combinations of vectors,"
Advances in Math., 5(1970)155-157.
Further results will appear in a forthcoming paper by Erdos, Rado, and
Milner in the Australian Journal of Mathematics. Further results also appear
in "Intersection theorems for systems of sets," H. L. Abbott and D. Hanson,
J. Combinatorial Theory, 72(1972)381-389.
Papers in Chapter 10
[108] On a lemma of Littlewood and OfTord
[312] (with R. Rado) Intersection theorems for systems of sets
[357] (with Chao Ko and R. Rado) Intersection theorems for systems of
finite sets
[539] On a lemma of Hajnal-Folkman
[542] (with J. Komlos) On a problem of Moser
[564] (with D. J. Kleitrnan) Extremal problems among subsets of a set
[569] (with D. J. Kleitrnan) On collections of subsets containing no 4-member
Boolean algebra
453 SYSTEMS OF SETS

ON A LEMMA OF LITTLEWOOD AND OFFORD
P. ERDOS
Recently Littlewood and Offord1 proved the following lemma:
Let X\, Xi, • • • , xn be complex numbers with |x,| ^1. Consider the
sums 23t«i€txt, where the e* are ±1. Then the number of the sums
St«ie*x* which fall into a circle of radius r is not greater than
cr2"(log »)»~1/2.
In the present paper we are going to improve this to
cr2"n~112.
The case Xi = 1 shows that the result is best possible as far as the order
is concerned.
First we prove the following theorem.
Theorem 1. Let be n real numbers, | x,-| ^ 1. Then the
number of sums / !?_r tkXh which fall in the interior of an arbitrary
interval I of length 2 does not exceed Cn,m where m~ [ra/2]. ([x] denotes
the integral part of x.)
Remark. Choose Xi=\, n even. Then the interval ( —1, +1)
contains Cn.m sums ^2t~iekXk, which shows that our theorem is best
possible.
We clearly can assume that all the x, are not less than 1. To every
sum ^St-i6*** we associate a subset of the integers from 1 to n as
follows: k belongs to the subset if and only if e4=+l. If two sums
2t-ie*** anc' St=ie*'x* are both in I, neither of the corresponding
subsets can contain the other, for otherwise their difference would
clearly be not less than 2. Now a theorem of Sperner2 states that in
any collection of subsets of n elements such that of every pair of
subsets neither contains the other, the number of sets is not greater than
Cn.m, and this completes the proof.
An analogous theorem probably holds if the x,- are complex
numbers, or perhaps even vectors in Hilbert space (possibly even in a
Banach space). Thus we can formulate the following conjecture.
Conjecture. Let xx, xi. ■ • • , xn be n vectors in Hilbert space'
\\xi\\ jSl. Then the number of sums ^t.i**** which fall in the interior
of an arbitrary sphere of radius 1 does not exceed C„,m.
Received by the editors March 28, 1945.
1 Rec. Math. (Mat. Sbornik) N.S. vol. 12 (1943) pp. 277-285.
' Math. Zeit. vol. 27 (1928) pp. 544-548.
898
455 SYSTEMS OF SETS

A LEMMA OF LITTLEWOOD AND OFFORD
899
At present we can not prove this, in fact we can not even prove that
the number of sums falling in the interior of any sphere of radius 1
iso(2").
From Theorem 1 we immediately obtain the following corollary.
Corollary. Let r be any integer. Then the number of sums 222,,16¾^¾
which fall in the interior of any interval of length 2r is less than rCn,m.
Theorem 2. Let the xt be complex numbers, \xt\ 2:1. Then the
number of sums /l^itkXk which fall in the interior of an arbitrary circle of
radius r (r integer) is less than
crCn,m < cxr2nn~w.
We can clearly assume that at least half of the Xt have real parts
not less than 1/2. Let us denote them by xx, Xi, • • • , xt, t^n/2. In
the sums ^J.i6*** we ^x e<+i> •••»«»• Thus we get 2' sums. Since
we fixed et+i, • • • , en,jL*-ie*** nas to fall in the interior of a circle
of radius r. But then zl'^x ekR(xk) has to fall in the interior of an
interval of length 2r (R(x) denotes the real part of x). But by the
corollary the number of these sums is less than
crCt.it/2i < cxr2'/t1'2.
Thus the total number of sums which fall in the interior of a circle
of radius r is less than
c2r2n/nl'\
which completes the proof.
Our corollary to Theorem 1 is not best possible. We prove:
Theorem 3. Let r be any integer, the x,- real, | x,| 2:1. Then the
number of sums y^J"„,ekXk which fall into the interior of any interval of length
2r is not greater than the sum of the r greatest binomial coefficients
{belonging to n).
Clearly by choosing x, = 1 we see that this theorem is best possible.
The same argument as used in Theorem 1 shows that Theorem 3
will be an immediate consequence of the following theorem.
Theorem 4. Let Ax, Ai, • • • , Au be subsets of n elements such that
no two subsets Ai and Aj satisfy Ai^)Aj and Ai—Aj contains more
than r — 1 elements. Then u is not greater than the sum of the r largest
binomial coefficients.
Let us assume for sake of simplicity that n = 2m is even and
r = 2/+1 is odd. Then we have to prove that
456 COMBINATORIAL ANALYSIS

900
P. ERDOS
[December
+ i
W ^ 2^ C2m,n + i-
i-—;'
Our proof will be very similar to that of Sperner.2 Let AUA2, • • -,AU
be a set of subsets which have the required property and for which
u is maximal. It will suffice to show that in every A the number of
elements is between n—j and n+j. Suppose this were not so, then
by replacing if need be each A by its complement we can assume that
there exist A's having less than n —j elements. Consider the A's with
fewest elements; let the number of their elements be n—j — y and let
there be x .,4's with this property. Denote these A's by Ah A 2, • • • Ax.
To each A i, i = 1, 2, • • • , x, add in all possible ways r elements from
the n-\-j-\-y elements not contained in A. We clearly can do this in
Cn+j+yiT ways. Thus we obtain the sets _BX, _B2, • • • , each having
n+j—y+i. elements. Clearly each set can occur at most CB+/-i/+i,r
times among the B's. Thus the number of different B's is not less than
Xl^n+ i+v >r\^n+ j—u+l,r )-i > X.
Hence if we replace Au Ait ■ ■ • , Ax by the B's and leave the other
A's unchanged we get a system of sets which clearly satisfies our
conditions (the B's contain n+j — y+i elements and all the A's now
contain more than n—j—y elements, thus B— A can not contain more
than r — 1 elements and also B($_A) and has more than u elements,
this contradiction completes our proof.
By more complicated arguments we can prove the following
theorem.
Theorem 5. Let Ax, Ai, ■ • • , Au be subsets of n elements such that
there does not exist a sequence of r + 1 A's each containing the previous
one. Then u is not greater than the sum of the r largest binomial
coefficients.
As in Theorem 4 assume that n = 2m, r = 2/+1, and that there are
x A's with fewest elements, and the number of their elements is
n—j—y. We now define a graph as follows: The vertices of our graph
are the subsets containing z elements, n—j—y ?£z?£;n+j+y. Two
vertices are connected i( and only if one vertex represents a set containing
z elements, the other a set containing z+1 elements, and the latter
set contains the former. Next we prove the following lemma.
Lemma. There exist dn,n-j-v disjoint paths connecting the vertices
containing n—j—y elements to the vertices containing n-{-j-\-y elements.
Our lemma will be an easy consequence of the following theorem
457 SYSTEMS OF SETS

'9451
A LEMMA OF LITTLEWOOD AND OFFORD
901
of Menger:8 Let G be any graph, V\ and Vi two disjoint sets of its
vertices. Assume that the minimum number of points needed for the
separation of V\ and Vi is w. Then there exist w disjoint paths connecting V\
and Vi. (A set of points w is said to separate V\ and Vi, if any path
connecting V\ with Vi passes through a point of w.)
Hence the proof of our lemma will be completed if we can show
that the vertices V\ containing n —j —y elements can not be separated
from the vertices Vi containing n-\-j-\-y elements by less than
C*2n,n-j-» vertices. A simple computation shows that V\ and Vi are
connected by
Cin^j-yin + j + y){n + j + y - 1) • • • (n - j - y + 1)
paths. Let z be any vertex containing n+i elements, —j—y^i^j+y.
A simple calculation shows the the number of paths connecting V\
and Vi which go through z equals
(n+i)(n+i— 1) • • • (n— j— y+l)(n — i)(n — i— 1) • • • (n—j — y+l)
^(n+j+y)(n+j+y-l) ■ ■ • (n-j-y+1).
Thus we immediately obtain that V\ and Vi can not be separated by
less than Ctn.n-i-y vertices, and this completes the proof of our lemma.
Let now Ax{x\ A2U), ■ ■ ■ , Ax(l) be the ,4's containing n—j—y
elements. By our lemma there exist sets Ai{l), i = \, 2,---, x;
/=1, 2, • • • , 2j + 2y+l, such that ^.-W+^+u has n+j+y elements
and i4i('>Ci4i(,+1) and all the A's are different. Clearly not all
the sets A^l), /=1, 2,---, 2j+2y+l, can occur among the
Ai, Ai, • ■ ■ , Au. Let i4,(s' be the first A which does not occur there.
Evidently s ^r. Omit Aia) and replace it by A,-(*>. Then we get a new
system of sets having also u elements which clearly satisfies our
conditions, and where the sets containing fewest elements have more
than n—j — y elements and the sets containing most elements have
not more than n+j+y elements. By repeating the same process we
eventually get a system of A's for which the number of elements is
between n—j and n+j. This shows that
+ i
U ^ 22 Cin,n+i,
•— J
which completes the proof.
One more remark about our conjecture: Perhaps it would be easier
to prove it in the following stronger form: Let | x,| = 1, then the num-
8 See, for example, D. Konig, Theorie der endlichen und unendlichen Graphen, p. 244.
458 COMBINATORIAL ANALYSIS

902
P. ERDOS
ber of sums 23J,!«*»;* which fall in the interior of a circle of radius 1
plus one half the number of sums falling on the circumference of the
circle is not greater than C„,m. If the Xi are real it is quite easy to
prove this.
We state one more conjecture.
(1). Let \xt\ =1. Then the number of sums Xjb-i«*** with
|^Ci-i«***| =1 is greater than c2nn~l, c an absolute constant.
University Of Michigan
459 SYSTEMS OF SETS

85
INTERSECTION THEOREMS FOR SYSTEMS OF SETS
P. Erdos and R. Rado|
A version of Dirichlet's box argument asserts that given a positive
integer a and any a2-f-l objects there are always a-\-1 distinct
indices v (0 ^ v ^ a2) such that the corresponding a-f-1 objects xv are either
all equal to each other or mutually different from each other. This
proposition can be restated as follows. Let N be an index set of more than a2
elements and let, for each element v of N, Xv be a one-element set. Then there is
a subset N' of N having more than a elements, such that all intersections X^ Xv
corresponding to distinct elements li, v of N' have the same value. In this note
we investigate extensions of this principle to cases when the sets X„ are of
any prescribed cardinal 6. Both a and 6 are given cardinals, finite or infinite.
In the case of finite a and 6 we obtain estimates for the number which
corresponds to a2 in Dirichlet's case, and we show that when at least one
of a and b is infinite then ab+1 is the best possible value of that number.
We introduce some definitions}. A system Sx : Yv (vzN) of sets Yv,
where v ranges over the index set N, is said to contain the system
S0 : X^ (fx e M) if, for every li0 of M, the set X occurs in Sx at least as often
as in S0, i.e. if
\{v: vbN; 7,, = ^(^1^: psM; X, = *„}|.
If Sj contains S0 and, at the same time, S0 contains Sj, then we do not
distinguish between the systems S0 and Hv
The system S0 is called a (a, b)-system if it consists of a (not necessarily
distinct) sets of cardinal b, i.e., if |M\ = a and \Xfi\=b for lieM. The
system S0 is called a ^-system if it has the property that the intersections
of any two of its sets§ have the same value, i.e. if for
Po> Pv ft, ft6^; ft^ft! ft ^ft
we always have XM X^ = X X^. More specifically, S0 is a A(a)-system
with kernel K if \M\ =a and X^X = K whenever li0, ll-^zM ; fx0 ^liv
In the special case when | M | = 1, say M = {ft0}, we stipulate that K <= X^ ,
and the empty system 20, for which M = 0, is considered as a A(0)-system
with any arbitrary, set K as kernel. Expressions such as
(> a, <6)-system, A(>a)-system
have their obvious meaning. Trivially, every (> a, 0)-system is a
A-system, and the box principle stated above asserts that every
f Received 5 December, 1958; read 18 December, 1958.
X The cardinal of the set A is denoted by \A |, and set union by A -\-B or 2(f e N) Av and
set intersection by AB or n{v e N) Av. AcB denotes inclusion, in the wide sense. We use
the obliteration operator ^ whose effect consists in removing from a well-ordered series the
term above which it is placed. Unless the contrary is stated all sets are allowed to be empty.
§ Not necessarily distinct sets but sets having distinct indices p.
[Journal London Math. Soo. 35 (1960), 85-90}
460 COMBINATORIAL ANALYSIS

86
P. Erdos and R. Rado
(> a2, l)-system contains a A(> a)-system. In what follows a and 6
denote arbitrary cardinals, and 6+ is the next larger cardinal to 6.
Theorem I.
(i) If a, 6 > 1 then every (> b+bbab+l, ^b)-system contains a A(> a)-
system.
(ii) If a > 2 ; 6 > 1; a-\-b ^N0 then every (> ab, ^b)-system
contains «A(> a)-system.
Theorem II. For every a, b such that a, b > 1 there exists a (ab+l, 6)-
system which does not contain any A(> a)-systern.
Theorem III. If 1 sg a, 6 < N0 and
c = Ma» + »(l-sJ--^-2-...-=^¾ (1)
V 2!a 3!a2 61a''-1/ x '
then every (> c, $; by system contains a A(> a)-system.
Remarks. 1. It follows from II that I(ii) is best possible, in the sense
that, for a > 2; 6 > 1; a-f-6 > N0 not every (ab, $; 6)-system contains some
A(> a)-system.
2. The (ab+l, 6)-system of Theorem II will be constructed explicitly.
3. For a = 2; 6 = 2 the result III is best possible. For we have c = 12,
and the following (12, 2)-system does not contain any A(3)-system.
01, 01, 23, 23, 04, 04, 14, 14, 25, 25, 35, 35, where xy = {x, y).
However, for a = 3 ; 6=2 Theorem III is not best possible. By II we see
that III is best possible except for a factor between 1 and 6!.
It is not improbable that in (1) the factor 6! can be replaced by cxb, for
some absolute positive constant cv Such a sharpened version of III would
have some applications in the theory of numbers, and in fact these
applications originally gave rise to the present investigations.
Before proving Theorem I we establish a simple lemma which is at the
root of a large number of combinatorial arguments.
Ramification Lemma. Let a0 be an ordinal, c0,cv ...,ca be cardinals;
let S be a set and M(s0, slt..., ia) be a subset of S defined for a < a0 and
su,..., sa e S, such that M (sQ, ..., sa)\ ^ca. Let V be a set of "vectors"
(s0, s,, .... 40) such that s0, ..., sa<jsS and
sazM (s0, ..., 4)/or a<a0. (2)
Then |F|<coCl...4o.
Proof. For every subset S' of S choose a representation of the form
8' = Oo> h, ..., <fc}#, where k is the initial ordinal belonging to |$'|.
461 SYSTEMS OF SETS

Intersection theorems for systems of sets.
87
Define <f>(S',t) for teS' by putting <f>(S',tK)=K {x<k). Now let
s= (s0, ..., sao)eV. Then, by (2), we can define an ordinal k(ol, s) by putting
ic(a,«) = ^(jf(«0, ...,4),¾) (a<«0),
and a vector >ft(s) = (k(0, s), /c(1, s), ..., /c(a0, s)).
Then |/c(a, s)| < ca, and s^s' implies >fi(s) =£>fi(s') as oan be seen by
considering the least a with sa ^6 sa'. We conclude that
Proof of Theorem I. We suppose that the system
2:XV (veN)
is a (| N\, $; 6)-system which does not contain any A(> a)-system, and our
aim is to deduce that
1-^1^6 + 6^ + 1. (3)
Throughout the proof the letters a, /3 denote ordinals such that
|aj, | j8[ $; 6, and [x, v denote elements of N. A subset N' of N is called a
A-set with kernel K if the system X„(veN') is a A-system with kernel K.
Put X = S (i; eN)X„ and choose an object 0 such that 00X. Well-order
the sets X and iV as well as the set of all subsets of N.
We define elements fa(v) as follows. Let a0 be fixed, and suppose that
fa(v) has already been defined for a < a0 and for all v, and that
/»sX„+{0} («<«„; „e2V).
Let v0sN. We now proceed to define faa(vQ). Put, for any functions
g0(v), ..., ga (v) defined for v eN and for any x0, ...,xa(j eX-j-{0}
F (g0,..., ga<s\ x0, ., xj = Nil («.< a.){v :g» = ¾}.
Let N'aN; KcX-\-{8}. Put
#(JV') = I,(veN')X,
and define r(iV',Z) to be the first subset N" of N such that (i) N" <=JV',
(ii) 2V" is a A-set with kernel if, (iii) N" is maximal such that (i), (ii) hold.
Then, by hypothesis about S,
|r(iv, /o|<«.
Put ^(a0, ,„) = f(/0, ...,/l0;/„K), .-./..(¾)).
^ (ao> fo) = {/a ("o): a < ao).
N*(k0, Vo) = r(iV(a0, „„), K(a0, „„)) .
Then v0&N(ol0, v0); #*(«,, ^0)C^K "o)! |Ar*(aQ, k0)| <■?■
462 COMBINATORIAL ANALYSIS

88
P. Ebdos and R. Rado
Case I. 0eK(ai>,vo). Then put fXl>(v0) = 9.
Case 2. d$K (a0, v0).
Case 2a. v0eN*(a1), v.). Then put/ao(f0) = 0.
Case 26. v0iN*(ai), v0). Then, by (iii) above, N*(a0, v0)-\-{v0} is not
a A-set with kernel K (a, v0). If we now assume that iV* (a0, v0) = 0 then this
last fact implies that K(a.0, 1¾)^^^ which, however, is false. Hence
iV*(a0, v0) ^0, and there is a first element v1 of N*(a0, v0) such that
X„ XH ^K(a0, v0). Since we are in Case 2, we have i£(a0, 1¾)0^^.
Since vi sN*(ol0, v0) we have K(a.a, v0)<=-XVi. Hence K(ai), njjc^I^,
and we may define fa (v0) to be the first element of the set XHXVi—K(a.Q, v0).
This completes the definition of fa(v). We have, in Case 2b,
fa,(v0)zX^H(N*(*0,Vo))- (4)
Let v sN. If for some a, we have/a(v) ^6 6 then Case 2b applies to a,
and hence also to each /3$; a; the elements fy(v) (/3$; a) are therefore
distinct elements of Xv. Hence, in view of | Xv | ^ b, there is /3,, such that
fa(v) eX„ for a < /J,, and fa(v) = 8. Then Case 2a applies to j8„, and we
have yeiV* (/3,,, v). This shows thatf
N = ~L(v&N)N*(flv, v), \N\^a\{N*(P„ v) : vzN}\.
We now prove that on the right hand side N* may be replaced by N.
Let N(^, p) = #(£„, ,). Then f,sN(^, rf = JV(&, ,) ; /» = /„(„) eX„
for a < jSj, ; /?„ > /?„ and hence, by symmetry, j8„ > j8 . Therefore j8 = /J,,
if (/^, ^) = K(P„ v) ; tf*^, ^) = tf*(j8„, v).
Thus I {N*(P„ v) :v e JV} | < | {iV(j8v, „) : v zN} j.
For any a and any x0, ..., xa sX put
(?(»0, ...,4) = ^(/0, ...,fa ;z0, ...,4),
j/(*„,...,4) = #(r(£(*0, ...,4), {*„,...,4})).
Then 2V(a, ,)=0(/0(,,), ..„/>)).
Let a0 be fixed such that a0 e {j3„ : vzN}. Choose any v with /3,, = 0¾.
Then N(P„v)=G(Mv),...,fat(v)).
Hence N{p„, v) is determined if the vector s— (/0(f), •• -,/^(^) ^8 known.
Denote by S the set of all such vectors s, i .e. the set of all s which correspond
to choices of v such that /3,, = 0¾. Let now s= (s0, ..., sao)e$ ; a<a0.
We proceed to show that sa eM(s0, ..., 4)-
t We remind the reader that \{N*(pv, k) : v e A-'}| denotes the number of distinct sets
N*[e„ v).
463 SYSTEMS OF SETS

Intersection theorems for systems of sets.
89
We can choose vsN such that fiv = a0, and sfi =fp(v) for all /3 < a0. Then
sa =/»e#(jV*(a, „)) = #(r(iV(a, „), £(«, „)))
= #f r(ff(/0M, ...,/»), {/„(.), ...,/1(0})]
= M{80 4).
In addition, we have
\M(S0, ...,4)|=|#(iV*(a, ^))|<6[AT*(a, OK 6a.
Hence, by the ramification lemma, when v ranges through all values for
which jSj, has the fixed value a0, there arise at most (ba) a"1 distinct vectors
(/o(")> -./«») • We deduce that
\N\^a\{N*(Pv,v):veN}\^a\{N(PP,v):vzN}\
= aS(|a0K6)|{JV{jSJ,,i-):veiV;j8v = a0}|
<aS(|a0| <6)(6a)la:oi <^a(ba)bb+,
which proves (3) and so establishes I(i).
Part (ii) of Theorem I follows from (i). For if a > 2; 6 > 1; a+6 > N0;
then b+bbab+x = ab.
Proof of Theorem II. Choose sets A, B such that \A\ = a; \B\=b, and
let J1 be the set of all mappings of B into A. Consider the system
2 :*(*,/)= (*,/(*)) :zs£} (teA;feF).
We consider the members of S as indexed by the pairs (t, /). In fact, they
do not depend on t. Then S is a (a6+1, 6)-system. Let us assume that S
contains a A(> a)-system S' with kernel K, say the system
S':X,=X(fpI/p) (PzB).
Then \R\>a and
(<„>/,) ?*=('.,./<r) for {**}*=*• (5)
Let .¾ f, 5. Then | {/ (a): /> e -B} | =¾ j A \ = a < | B |, and hence there is
{/°.r> CTJ # ^-^ witn //>«(*) =A(*)- Then, for any pzR,
fi>Ax) —fP(x)> so that/p is independent of p. Since|{fp :p ei?}|< I ,41< | R\
there is {Pl, cr ,},,<=£ with tpi = t^. But then (ipi, /pJ = (^, /^) which
contradicts (5). This proves Theorem II.
Proof of Theorem III. Let 1 < ft, 6 < N0. By Theorem I there exists a
least number d, where d < N0, such that every (> d, ^. 6)-system contains
464 COMBINATORIAL ANALYSIS

90
Intersection theorems for systems or sets
a A(> a)-system. Denote this number by f(a, b). We have to show that
/K b) <c,
where c is defined by (1).
There is a least number </>(a, b) such that every (> </>, $; 6)-system
2:X, (^zM)
which satisfies X ^=XV for {//., v} ^ M, contains a A(> a)-system. Clearly,
</> $;/. Also, <j>(a, 1) = a. We first show that
/(a, 6)<a</>(a, 6). (6)
Let S':X„ (vsN)
be a ( > Ckj>(a, b), $; 6) -system which does not contain any A(> a)-system.
We have to deduce a contradiction. Let
v0eN;K(v0)={v:vsN;Xp = XH}.
Then X„ (vsK(v0y\ is a A-system, and therefore \K(v0)\ $;a. Hence, if
{Xv: v s N} = {X, :? s M], X, ^ X„ for {/,, „}„<= Jf,
then [ M | > </> (a, 6), and it follows from the definition of </> that the system
X (jj, e M) contains a A(> a)-system. This is the required contradiction.
There is a (</>(a, b), $; 6) -system
2 :X„ (^siV),
where X =fcXv for /x ^ v, which does not contain any A(> a)-system,
Let iV0 be a maximal subset of N such that XAX„ = 0 for {/x, vJ^ciVo.
Then \N0\ <a, since X„ (vsNQ) is a A-system. Put X* = S(^ e JV0)X„.
Then we can choose elements
x,
sXX* (nzN-N,,)
A P
Let | el*. Then there is v0($) sNQ with $ eX ^). Then the system (of
sets of at most b— 1 elements)
X„o(t)-{^} ; X,-{^} (^ zN-N0; x„ = $)
does not contain any A(> a)-system since any such system S" would yield
a A(> a)-system contained in S if we add to each member of S" the element
£. Hence l+|{/x:/xeiV—iV0; »;A = £}| <</>(a, 6—1),
^(o,6) = |iV| = |^0|+|iV-^0|<o+S(|eX*)|{iLi.>eiV-^0;^=^|
<a+ (</>(a, 6— 1)— l)&a = — a(6 — l)-fa6<£(a, 6—1),
</>(a, 6) _ 6-1 </>(a, 6-1)
6!a* ^ 6!a''-1 + (6-l)!a''-1'
By means of 6—1 successive applications of this inequality we obtain
(f>(a, 6) _ 6-1 _ 6-2 1_ <t>(a, 1)
6!a* ^ 61a*-1 (6—1)! a*-2 '" 2!a+ 11a '
In view of (6) and <j>(a, 1) = a this is the desired result.
The University,
Reading.
465 SYSTEMS OF SETS

INTERSECTION THEOREMS FOR SYSTEMS
OF FINITE SETS
By P. ERDO'S, CHAO KO (Szechuan), and R. RADO (Reading)
[Received 13 August 1961]
1. Introduction
E. Spebneb (1) has proved that every system of subsets av of a set of
(m\
) elements,
where p = [%m]. This note concerns analogues of this result. We shall
impose an upper limitation on the cardinals of the av and a lower
limitation on the cardinals of the intersection of any two sets av, and
we shall deduce upper estimates, in many cases best-possible, for the
number of elements of such a system of sets av.
2. Notation
The letters a, b, c, d, x, y, z denote finite sets of non-negative integers,
all other lower-case letters denote non-negative integers. If k ^1, then
[k, I) denotes the set
{k, k + l,k + 2,...,l— 1} = {t: k < t < I}.
The obliteration operator A serves to remove from any system of elements
the element above which it is placed. Thus [k, I) = {k,k-\-l,...,l}. The
cardinal of a is \a\; inclusion (in the wide sense), union, difference, and
intersection of sets are denoted by a c b, a-\-b, a—b, ab respectively, and
a—b = a—ab for all a, b.
By S(k, I, m) we denote the set of all systems (a0, a1,.„, an) such that
av c[0, m); \av\ < I (v < n),
a^iav ¢^ \a^av\ > k (p < v < n).
3. Results
Theorem 1. If 1 < I < \m; (aQ,...,dn) e S(\,l,m), then n < ).
If, in addition, \av\ < I for some v, then n < I ).
Theobem 2. Let k < I < m, n > 2, (a0,...,dn) e 8(k,l,m). Suppose
that either „, , , . , , 7 , ^ , ,, %
21 ^ k-\-m, \av\ = I (v < n) (1)
or-\ 21 < 1+ra, \av\ ^ I (v < n). (2)
f The condition \av\ ^1 I \s in fact implied by (a0,..., &n) e S(k, I, m).
Quart. J. Math. Oxford (2), 12 (1961), 313-20.
466 COMBINATORIAL ANALYSIS

314 P. ERDOS, CHAO KO, AND R. RADO
Then (a) either (i)
\a0...dn\ > k, n <
or (ii) \a0...dn\ < k < I < m, n ^¾
(6) »/ m > /c+(Z—&)( J , then n <£.
Remark. Obviously, if \av\ = I for v < n, then the upper estimates
for n in Theorem 1 and in Theorem 2 (a) (i) and (6) are best-possible.
For, if k ^ I ^ m and if a0,..., an are the distinct sets a such that
[0,i)cac[0,m), \a\ = I,
then (a0,...,dn) e S(k,l,m), n =
4. The following lemma is due to Sperner (1). We give the proof
since it is extremely short.
Lemma. If
n0 > 1, a„c[0,ra), |a„| = l0 (v < n0),
then there are at least n0(m — ^)(^+1)-1 se-ts b such that, for some v,
v<n0, a„c6c[0,m), \b\ = l0+l. (3)
Proof. Let % be the number of sets b defined above. Then, by
counting in two different ways the number of pairs (v, b) satisfying (3),
we obtain n0(m —10) $^ w1(Z04-l), which proves the lemma.
5. Proof of Theorem 1
Case 1. Let |a„| = I (v < n). We have m > 2. If m = 2, then 1=1;
/m \\
n = 1 ^ I . Now let m ^ 3 and use induction over m. Choose,
for fixed I, m, n, the av in such a way that the hypothesis holds and,
in addition, the number
/K dn) = S0+... + Sn
is minimal, where sv is the sum of the elements of a,,. Put .4 = {a/, v < n}.
If 21 = m, then [0, m) — av$ A and, hence
JSfow let 2l < m.
m — k
Z-fc
m-fc-l\/As.
z—jfc—1 A*/'
m — k
l-k
m — k'
l-k
467 SYSTEMS OF SETS

ON INTERSECTION THEOREMS 315
Case 1 a. Suppose that whenever
ra—leae.4, Ae[0,ra) — a,
then a — {ra— 1} + {A} e A.
We may assume that, for some n0 < n,
m-leo, (v < n0), m—\$av (n0 < v < n).
Put bv = a„ —{ra—1} (v < n0).
Let /j. < y < n0. Then
\afi+av\ < ^1 < m,
and there is A e [0, ra)— a —av.
Then b^ + iX} e A, b^ = (b^ + {X})b, = (^+{A}K ^ 0
and therefore
l-\ > 1, (&„,...,&„,) eS(l,Z-l,ro-l).
Since 2(Z— 1) < ra— 2 < m — 1 we obtain, by the induction hypothesis,
I. Similarly, since
(ano,...,dn) e S(l,I,m—l), 21 ^ ra — 1,
[in — 2\
we have n—nn < . Thus
0 W-i/
> „ /ra — 2\ /ra — 2\ (m — 1
w = nn4-ln — «n) ^ , -I- =
°^v o; ^ \l-2j^\l—l) \l—\
Case 1 6. Suppose that there are a e A, A e [0, ra)—fl such that
ra —lea, a —{ra—l}-f{A} <£ A
Then A < in—1. We may assume that
ra— lea„, A <£ a„, 6„ = a„—{ra—1}+{A} <£ .4 (v < w0),
ra —lea,,, A <£ a„, c„ = a„—{ra—1} + {A} e .4 (ra0 < v < nx),
ra— 1 e a„, A e a^ (w1 ^ v < w2),
ra— 1 ^ a„ (n2 ^C v < n).
Here 1 ^ w0 ^ wi ^ w2 =¾ n- Put ^,, = a^ (no ^ y < w)- We now show
that (b0,...,b„)eS(l,l,m). (4)
Let /jl < v < n. We have to prove that
h^^bv, bilbv^&. (5)
If /x < y < w0 or w0 ^ [l <v, then (5) clearly holds. Now let /j, <n0 ^.v.
Then b^ ¢4, bv = ave A, and hence b^ ^ bv. If n0 ^ y < ^, then
c„e^4, and there is a e a^c^. Then a ^ A, a ^ ra— 1, and aebixbv.
If w1 < y < «2, then A e b^ bv. If, finally, n2 < y < «, then there is
468 COMBINATORIAL ANALYSIS

316 P. ERDOS, CHAO KO, AND R. RADO
p e a av. Then
peav, p < m— 1, peb^b^
This proves (5) and therefore (4). However, we have
f{b0>->bn)-f(a0,...,dn) = 7i0(-[ra-l]+A) < 0,
which contradicts the minimum property of (a0,..., dn). This shows that
Case 1 b cannot occur.
Case 2: min(y < n) \av\ = l0 ^.1.
If l0 = I, then we have Case 1. Now let /0 < I and use induction over
I—10. We may assume that
K\ = lo (v < no), \av\>lo {n0^v<n), (6)
where 1 ^ n0 ^ n. Let b0,..., bni be the distinct sets b such that (3) holds
for some v. Then
(?o+l)-(m-/0) <2(Z-l)+l-m <0, (7)
and hence, by the lemma, n1 ^ n0. Also,
(b0,...,bni,an<j,...,dn) e S(l,l,m).
Hence, from our induction hypothesis,
n < ^ + (71-71,,) < I ?_i
and Theorem 1 follows.
6. Proof of Theorem 2
Case 1: h = 0. Then
2Z < 1+ra, ]a„| < ? (y < w), (a0 dn) e S(Q,l,m).
Now (a) (ii) is impossible, and (a) (i) is identical with (b), so that all we
. Again, we may assume (6), where
^o ^ h 1 ^ no ^ n- If Z0 — Z then
Now let Z0 < I and use induction over I—/0. Let 60,..., fcni be the distinct
sets b such that (3) holds for some v. Again (7) holds and, by the lemma,
n1 ^ n0. We have
(b0,..., bni, a„o dn) e £(0,1, m)
and, by our induction hypothesis,
/ Yfl\
n ^. 7i1 + (ti—n0) ^.
J
This proves the assertion.
469 SYSTEMS OF SETS

ON INTERSECTION THEOREMS 317
Case 2: k > 0. We separate this into two cases.
Case 2a. Suppose that (1) holds. Put \a0...dn\ = r. We now show
that, if r ^ k, then (i) follows. We may assume that
a0...dn = [m — r, m).
Put a„[Q,m—r) = c„ (v < w).
Then (c0)..., c„) e £(0, Z-r, TO-r),
2{l—r) — {m — r) = 2/-r—m < 2l — k — m < 0.
Hence, by Case 1,
lm — r\ lm — r\ lm — k\ (m — k
U ^ (, l-r) = ^ra-Z/ ^ l^m—Z/ ^ [ l-k
so that (i) holds. We now suppose that (i) is false, and we deduce (ii).
We have \a0...dn\ = r <k^ KaJ < |a0| < 1,
and therefore 21 ^ k-\-m < l-\-m, k < I < m.
There is a maximal number p ^ n such that there exist £> — n sets
«„,..., ^ satisfying (a0,...,dp) e 8(k,l,m). (8)
Put A' = {a/, v < p). We assert that
{a0,...,dp)$ S(k+l,l,m). (9)
For otherwise \a av\ > k (p < v < n). Let aei'. Then we can choose
a' c [0, m) such that
|a' | = I, \aa'\ = I— 1.
Then, for every b e A', we have
\a'b\ > \ab\ — 1 Js £
and hence, since p is maximal, a'e.4'. By repeated application of
this result we find that
[0,1), [m—l,m)eA', k < \[0,l)[m-l,m)\ = l—(m—l) < k,
which is the desired contradiction. This proves (9), and hence there
are sets a, b e A' such that \ab\ = k. Since \a0...dp\ ^ \a0...dn\ < k,
there is c e A' such that \abc\ < &. Denote by T the set of all triples
{x,y, z) such that x c a, y c b, z c c, \x\ = \y\ = \z\ =■ k, \x-\-y-\-z\ ^ I.
Put (f>(x,y,z) = {d:x+y+zcde A'}. Then, by (8),
A' = 2((2:,2/,2)6 T) <f>{x,y,z).
If (x,y,z) e T and s = |rr—|—«/-J—z|, then s > k since otherwise we obtain
the contradiction
k > |afcc] > |xz/z| = |x| = k.
470 COMBINATORIAL ANALYSIS

318 P. EKDOS, CHAO KO, AND R. RADO
Hence
which proves (ii).
Case 2b. Suppose that (2) holds. We may assume (6), where l0 < I;
1 =¾ no ^ n- If 'o = h then Case 2 a applies. Now let l0 < I and use
induction over I—10. Let b0,..., bni be the distinct sets b satisfying, for
some v, the relations (3). Then (7) holds and hence, by the lemma,
n1 ^ n0. Also, since l0 < I < in, so that m — l0 ^ 2, we have, by
definition of the b
b0...bni = a0...dm, \b0...f>niaH<t...dn\ = \a0...dn\ < k.
Since (b0,...,6ni,anij,...,dn) e S(k,l,m),
it follows from our induction hypothesis that
lm — k—\\ll^
n < ni + (n-n0) < , l_k_l mj
It remains to prove (b) in Case 2. If k = I, then (b) is trivial. If
k < I and m ^ &-f-(Z—&)( I , then
'm — k\ (in — k—l\m — k/in — k—l\{l\3
l—kj \l—k—ljl—k^\l—k—l)\kj
so that (6) follows from (a). This completes the proof of Theorem 2.
7. Concluding remarks
(i) In Theorem 2 (6) the condition
in > £ + (Z-£)Q3,
though certainly not best-possible, cannot be omitted. It is possible for
(a0,..., dn) e S(k, I, in), k ^.1 ^m
to hold and, at the same time, n > I ). This is shown by the
following example due to S. H. Min and kindly communicated to the
authors. Let a0,..., dn be the distinct sets a such that
a c [0,8), \a\ = 4, |a[0, 4)| = 3.
Then
n = 16, (a0,..., a15) e S(2, 4, 8), (™_ *) = Q = 15 < n.
471 SYSTEMS OF SETS

zc<»<«'>(?)(£»)-*z<»<*>(!
In this case
ON INTERSECTION THEOREMS 319
A more general example is the following. Let r > 0 and denote by
a0,..., dn the distinct sets a such that
ac[0,4r), \a\ = 2r, |a[0, 2r)\ > r.
Then (a0,...,aj e £(2, 2r, 4r),
and we have
2r\l 2r \ , ^ ,, ^ „ > /2r\/ 2r \ l/2r\
A/Ur —Aj
l/4r\ l/2r\2
2\2rJ"2\
'm—k\ _ /4r— 2\
.Z-fc] ~~ Ur-2/'
and, for every large r, possibly for every r > 2, we have I ) < n.
We put forward the conjecture that, for our special values of k, I, m,
this represents a case with maximal n, i.e.
If r>0, (a0,...,dn)eS(2,2r,4r),
then n<i(^)-CT
(ii) If in the definition of S(l,l,in) in § 2, the condition a d av$. a
is replaced by a ^ av and if no restriction is placed upon |a„|, then
the problem of estimating n becomes trivial, and we have the result:
Let in > 0 and av c [0,ra) /or v < n, and a^ ^ a„, a av ^ 0 /or
/j, <C v < n. Then n ^ 2m~1, a»d <Aere are 2m-1 — n subsets an,..., d2m-i
of [0,m) such that a^ ^ a„, afJbai, ^ 0/or /^ < v < 27"-1.
To prove this we note that of two sets which are complementary in
[0,to) at most one occurs among a0,..., dn> and, if n < 2"1-1, then there
is a pair of complementary sets a, 6 neither of which occurs among
a0,..., dn. It follows that at least one of a, 6 intersects every av> so that
this set can be taken as an.
(iii) Let I ^ 3, 21 ^ in, and suppose that
ayc[0,m), \av\ = I for v < n,
and
an 7^ av> anav 7^ 0 f°r /j- < v < n, and a0...dn = 0.
We conjecture that the maximum value of w for which such sets a,,
can be found is n0, where
3?
1—2
,V0
472 COMBINATORIAL ANALYSIS

320 ON INTERSECTION THEOREMS
A system of n0 sets with the required properties is obtained by taking
all sets a such that
ac[0,ra), |a[0,3)| > 2, \a\ = I.
(iv) The following problem may be of interest. Let h =¾ m.
Determine the largest number n such that there is a system of n sets av
satisfying the conditions
a-fj. ^ «„, \a^av\ > k (p < v < n).
If m-\-h is even, then the system consisting of the a such that
ac[0,m), \a\ > %(m-\-k)
has the required properties. We suspect that this system contains the
maximum possible number of sets for fixed m and k such that m-\-k
is even.
(v) If in (ii) the condition a a, ^ 0 (/j, < v < n) is replaced by
a ava ^ 0 (/j. < v < p < n), then the structure of the system av is
largely determined by the result:
Let m ^ 2, a„ c [0,ra) /or v < n, a^^ av for /j, < v < n, and
afxavaP 7= 0 /or /"■ < v < P < n- Then n < 2"1-1, and, if n = 2'"-1,
f/jen a0a1...dn ^ 0, so that the av are all 2m~x sets a c [0,m) which contain
some fixed number t (t < m).
For there is a largest p (1 ^ p ^ n) such that
a0a1...dpe{a0,a1,...,dn}.
If p = n, then a0...dn = av ^ 0 for some v < n. If jj < n, then any
two of the w 4-1 distinct sets
a0a1...ap, a0> alt..., an
have a non-empty intersection and hence, by (ii), »4-1 < 2m-1.
Different proofs of (v) have been found by L. Posa, Gr. Hajos, G. Pollak,
and M. Simonovits.
REFERENCE
1. E. Sperner, Math. Z. 27 (1928) 544-8.
473 SYSTEMS OF SETS

COLLOQUIA MATHEMATICA SOCIETATIS jANOS BOLYAI
4. COMBINATORIAL THEORY AND ITS APPLICATIONS, BALATONFURED (HUNGARY), 1969.
On a lemma of Hajnal-Folkman
by
P. Erdos
Budapest, Hungary
HAJNAL and FOLKMAN [3], [2], independently of each other,
proved the following Lemma: Let | f | = 2n-l, Atc tf, |A-| = n be subsets
of iP so that to every element x of if there is an A^ not containing x. We
define now a graph as follows: xe•i, y e $ are joined if for some A; they
are both contained in A; . The Lemma asserts that this graph contains a
complete graph of n + f vertices. We are going to generalise and extend this
Lemma in various directions and establish some connections with RAMSEY' s
theorem. First we have to introduce some notations.
The basic elements of an r-graph are its vertices and the r-tuples
formed from some of its vertices. K Cn) is the complete r-graphof n vertices
and all its (r) r-tuples.
For r = 2 we obtain the ordinary graphs. Let if be a set. A family
of subsets A -v c $ defines an r-graph Cj.(r)(tf; fi^ ,...) as follows: The vertices
are the elements of f, an r -tuple belongs to our graph if and only if it is a
subset of one of our A's. Such r-graphs were, as far as I know, first studied
in [l] in a context that differs from this. We say that the family can be
- 3U -
474 COMBINATORIAL ANALYSIS

represented by i vertices if there are i. elements x);...,X' of $, so that
all the A' s contain one of the x,-' s, 1 s j s i. . The symbol ( m , n, L , r) —> u
means that if |^| = m£ n and A^cf, |AJ > n is any family of subsets which
cannot be represented by i. vertices, then £^( j1; A.,...) contains a K(r)(u.).
(m,n,t,r) -(-» a means that (m,n,i,r) —» u. does not hold (i.e.
there are sets A-ci, \$\ = m, |A.| 2 n which cannot be represented by i
vertices but G C "j"; A,,-..) does not contain a K (u)). fCl<,t) are the so-
called Ramsey numbers, f (k,£) is the smallest integer, so that every graph of
■fCk.O vertices either contains a k(E) or its complementary graph contains a
K(£) (in the complementary graph, two vertices are joined if and only if they
are not joined in the graph).
Trivially (m,n,i.,r) —► n always holds and the only interesting
cases occur for u > n . Clearly the following monotonicity relations hold:
(1) ( m, n, i., r ) —> u implie s ( m, n, i, r ) —» a if i. > I
(2) (m,n,i,r) —* a implies Cm', n, L, r) —> a if m > m'> u
(3) (m,n,i,r) —>u implies (m,n,i,r') if r,< r
The Lemma of Hajnal and Folkman can be expressed in our notation
as (2n-1, n, 1,2)->■ n + 1. Clearly (2n, n,1,2) 4* n + i (it suffices to take two
disjoint n -tuples in J\ |tfl = 2n ), also for every mi n (m,n,1,2)-)-*n + 2
(take all n-element subsets of ^, I Jl = n+1 ). On the other hand we prove the
following generalisation of the Lemma of HAJNAL and FOLKMAN:
THEOREM. Let i > 1 . Then
(4) (2rn-l-2,n,i,2) —* n+L .
For i = 1 this is the Lemma of Hajnal and Folkman. To prove (4)
for i > "1 , we use induction with respect to L .
- 312 -
475 SYSTEMS OF SETS

Assume that (4) holds for i-i and every n. Let \$\ = 2n + L-2
and let X: be any element of ^. Consider the family of all the A' s contained in
i?— jc.- . They cannot be represented by i. - 1 elements, hence by our induction
hypothesis (2n + i.-3 , n, i-1,2) —>- rn-i-1, thus for every x: C^Z\S- a-, A, ,-■■)
contains a complete graph K^Cn+i-'l) which is contained in C^^Ctf, A,,...) .
Denote the set of vertices of this graph by F:, F: c •S-i- . Clearly the family
of sets Fj cannot be represented by one element, thus by the Lemma of
HAJNAL and FOLKMAN (and (2)) we have (2n+ L - 2 , n + i, - M , 2) —^ n + i,
or (^2)(;?; F,,.-.) contains a K2 ( n+L), but since C^2)(tf; F-,,. ■ ) is clearly
a subgraph of C^CJ1; A^,... ), this completes the proof of our Theorem.
Our Theorem is the best possible. To show this observe that
(5) (2rn-t,-2,n,L,2)4*n+i + 1,
(6) C 2n 4- L- 1 , n, 1,2) 4* n-t-T-tii] ,
(7) C2n + [ -2, n.L-1,2) 4-> n-t-i.
(5) is obvious, it suffices to consider all n-element subsets of
(7) immediately follows from (6). (6) is slightly less obvious.
Assume first that i. = 2J+-1. Let the elements of ;P be x, ,y, , t = 1, ■- , n + 1 .
Let the A:, |A:| = n be all subsets of I? which contain at most one of the
elements x,,y,, t = ■(,..., n+j .
Clearly 0(2)( tf ; A, ,-■■) does not contain a K2(rn-j+0 and the
A' s can not be represented by 2j-H elements; this proves (6) for odd i.
Assume next L = 2j + 2 . We then have to show (2n+ 2j+-*l, n , 2j +. 2, 2.) -(-> n+j + 'l .
Let the elements of tf be the residues mod. (2n + 2j + 0. The sets At, Atc $, lA,] = n
are those n-element subsets of !f which do not contain two consecutive residues.
Clearly Q(i3(;P; A,,...) does not contain a X2( n+j +-1 ) thus, to complete our
proof, we only have to show that the A' s are not represented by 2j + 2 residues.
- 313 -
476 COMBINATORIAL ANALYSIS

Let |U| = 2j + 2 be a set of 2j+2 residues; we show that <3-Vl
must contain an A . Without loss of generality, we can assume that 1 is in U .
Let if, be the set of odd residues excluding 1 and if2 i-s tne set °f even
residues. |ul = 2j + 2 implies that either | ^ n Ul 6 j or | ^ n U I < j .
Assume without loss of generality | &, n U. I 1 j . But then | ^ - ^ n U | i n
or ^-U contains an A , as stated. This completes the proof of (6). It seems
certain that (6) is not the best possible.
Several unsolved problems can be posed. Denote by A(n,i) the
smallest integer for which ( A (n, i) , n , L., 2") 4-» n + 1. (6) and our Theorems
show A(n.O = 2n, A(n,2) = 2n + 1 . I conjectured A(n,3) > 2n + 2 ,
in other words I conjectured
(8) C2n + 2, n,3,2) -^- n + 4 .
For n=2 (8) is Ramsey' s theorem (a graph of 6 vertices either
contains a triangle or a set of three independent vertices). HAJNAL and I
proved (8) for n = 3 and recently SZEMEREDI has proved (8) for all n.
HAJNAL and I proved A(n,3) < 3n i.e. we proved
(9) (3n + 1, n,3,2) -f> n + 1.
To prove (9), let if be the set of residues modOrn-l) and the
A's are all the sets of n consecutive residues. Perhaps (3n,n,3,2) —* n-M
holds.
It is clear that many further problems can be posed.
We just state one more tirival result:
(10) (m, n.i., r) —»• u implies for every t >0 (m+t, n, L-t-t ,r)—>u.
The simple proof of (10) we leave to the reader.
- 314 -
477 SYSTEMS OF SETS

Let us now establish tho connection of our symbol with the RAMSEY
numbers. Let n = 2 denote, say, by g(i) the smallest integer for which
(11) (g(i),2,C,2) -f> 3
holds. (11) means that there is a graph of g(i) vertices which contains no
triangle, and for which the complementary graph contains no K2(g(i) - I) and
g(i) is the smallest integer with this property in other words, g(i) is the
smallest integer for which
(12) g(C) < { (gcO- i,3).
It seems hopeless to determine -f ( k,3), even to get an asymptotic
formula is probably very difficult, thus the determination of g( L) is no doubt
very difficult.
It would be interesting to determine the largest integer m for
which (m,nj,3) —> n + 1 holds.
One final remark. HAJNAL and I proved (11,3,6,2) -+-»£■.
REFERENCES
[1] P. ERDOS, A. HAJNAL and E. MILNER, On the complete subgraphs of
graphs defined by systems of sets, Acta Math. Acad. Sci. Hung.
17 (1966) 159-229.
[2] J. FOLKMAN, An upper bound for the chromatic number of a graph (this
book).
[3] A. HAJNAL, A theorem on k-saturated graphs, Canadian Journal of Math.
17 (1965) 720-724.
- 315 -
478 COMBINATORIAL ANALYSIS

COLLOQUIA MATHEMATICA SOCIETATIS jANOS BOLYAI
4. COMBINATORIAL THEORY AND ITS APPUCATIONS, BALATONFURED (HUNGARY), 1969.
On a problem of Moser
by
P. Erdos and J. Komlos
Budapest, Hungary
Let f(n) be the largest integer with the following property:
Every family Fn of n sets contains a subfamily F^ of f(n) sets
so that the union of two sets of Fn' never equals a third .
Moser asked for the determination or estimation of f (n). A result
of Kleitman [2] shows that f (n) < cn/Vlog n . J. Riddell who communicated
this problem to us pointed out that f (n) > V7T.
The proof is easy. Let A1 ,..., An be any family of n sets, and
order them by inclusion. By the theorem of Dilworth [l] either there is a
chain of length > V~n~ or a family of i \TrT incomparable sets. In any case we
obtain -fCn) 2 V7T , which proves Riddell' s result. Now we prove the following
THEOREM
\ V7T < ffn) < 2 fl^ + 4.
These three sets are assumed to be pairwise different.
- 365 -
479 SYSTEMS OF SETS

The lower bound has just been proved. Thus we only have to prove
the upper bound.
Define the positive integer k 2 3 by the relation
and consider k equidistant points on a circle. By an arc of length i, we shall
mean a set of i neighbouring points. In the case I < k the meaning of the
notion "endpoints of an arc" is clear. It is easy to see that the total number of
arcs of length I for — ■S. i < k is at least [ J, so we can take n arcs, whose
lengths are between -jj- and k . These form our family Fn . Let us choose a
subfamily F^ of S sets from the family Fn , so that the union of two sets of F„
never can be equal to a third one. We show {n< 2k , which, together with (1),
proves the theorem.
We say that an arc (in F^ ) is minimal with respect to one of its
endpoints if it does not contain any other arc (in Fn' ) with the same endpoint.
For every point there are at most two arcs which are minimal with respect to
this point (one "to the right" and one "to the left"), the number of minimal arcs
is thus at most 2 k .
But all arcs in F„ are minimal, since if one of them (say A) was
not minimal with respect to any of its endpoints (say X and Y) then we should
have the relation
(2) A = Ax v Ay
where Ax resp. Ay denote the shortest arcs contained in Fn with endpoints *
resp. y ((2) follous from — s i < k) .
This contradiction proves our statement.
We remark that by the same argument we get that at most 2k arcs
can be chosen from among the arcs of length 1 with 2 - i< 2 (t is
arbitrary), if the requirement is the same as above. Thus, a subfamily F of
- 366 -
480 COMBINATORIAL ANALYSIS

2
the family F of all arcs (F contains k -k + 4 elements), having the property
that the union of two of them is never equal to a third one, can contain at most
2 k Log (k+O elements..(log denotes the logarithm based on 2 .)
The arcs of lengths 2-1, t = 2,... show that a family of k log k
arcs can have this property.
Probably f (n ) = ( & + 0(-1 ID \fn , but we cannot prove this and have
no conjecture about the value of c .
REFERENCES
[1] R. P. DILWORTH, A decomposition theorem for partially ordered sets,
Annals of Math. 51(1950), 161-166.
[2] D. KLEITMAN, On a combinatorial problem of Erd<5s, Proc. Amer. Math.
Soc. 17 (1966), 139-141.
367
481 SYSTEMS OF SETS

146
EXTREMAL PROBLEMS AMONG SUBSETS OF A SET
Paul Erdos
Daniel J. Kleitman
Massachusetts Institute of Technology
The subsets of a (finite) set form a lattice and in fact a
Boolean algebra. The following concepts are natural to them.
A. Intersection
B. Union
C. Disjointness
D. Complement
E. Containment
F. Rank (size)
In this paper we survey the present status of a number of
problems involving maximal or minimal sized families of subsets
subject to restrictions involving these concepts.
Problems of this kind arise in a large number of contexts in
many areas of mathematics. For example, the divisors of a square
free number correspond to the subsets of the prime divisors, so
that certain number theoretic problems involving divisors of
numbers are of this form. Efficient error correcting codes and
block designs can be considered as extremal collections of subsets
satisfying restrictions of thid kind.
Since the concept of set is as basic in mathematics as the
concept of number, one can also investigate the properties
considered here for their ovm sake as one considers similar problems
in number theory. Thus we might ask, "What sort of limitations
482 COMBINATORIAL ANALYSIS

147
are imposed upon families of subjects of a set by simple
restrictions on intersection union, rank, and/or containment
among members of the family?"
Questions of this kind have one additional value. Since
the concepts involved are all easily understood by
non-mathematicians results and elegant proofs in this area have tutorial
value as illustrations of the power of mathematical method that
are accesible to the layman.
To facilitate reference, we divide the problems considered
here into five areas. These are
I. non-intersection
II. size limited intersection
III. intersection and rank limitations
IV. containment limitations
V. union and intersection restrictions
VI. miscellany
Problems and results in these areas are described in the
corresponding section below.
483 SYSTEMS OF SETS

148
I. Non-Disjoint Families
Let S be a finite set having n elements: (!sl = n.).
Among the simplest restrictions that can be placed on families
of subsets of S is that no two are disjoint. Thus if F = {A^}
i = 1, ...,A with A^S we may require that A.nA./^ for all
With this one restriction there are several questions tha+
can be raised. Among these are:
a) How large can F be?
b) If F is "maximal" in that no subset of S can be
added to it without violating the restriction, how
small can F be?
c) How many maximal F's are there of any given size?
d) How many F's are there of any size?
These four kinds of questions can be raised not only about
families subsets restricted as is F above, but also about
families satisfying variants of the restriction.
Among possible variant restrictions of the same general
kind are:
I-, Let F be as defined above, and let G consist of
the minimal members of F that is the members of F not
contained in others.
In Let Got be the union of k families each restricted
as was F above.
Io Let Go^. be a family containing no k members that
are pairwise disjoint.
484 COMBINATORIAL ANALYSIS

149
1^, Let G^, be a family such that the intersection of
every k members is non-empty.
We now describe some results.
No collection of non-disjoint subsets can contain a set and
its complement. Thus our family F can have at most half of the
subsets: |F!<2n~ . A maximal family F contains every set
containing any member. Since every set disjoint from A is
contained in A's complement, if A cannot be added to a
maximal family F, A is already in it. Thus all maximal
families consist of exactly 2n~ subsets, exactly one of A
or A" for each A.
Thus questions a) and b) are easily answered for families
satisfying the non-disjointness restriction satisfied by F above.
The number of maximal families satisfying this restriction on the
other hand has not as yet been determined very well.
There exist several levels of inaccuracy in estimates of
quantities of this kind. Some of these are listed here. One
can have:
1. An exact formula
2. A convergent formula (convergent for large n to the
exact result)
3. An asymptotic formula (ratio to exact result is
convergent)
k. An asymptotic formula 'or the logarithm
In addition, one can obtain bounds upon any of these levels,
one as well as any others.
485 SYSTEMS OF SETS

150
We can easily find a level 4 expression for the total
number of families F; it is n(F) = 22 (i+oC1))-
The argument will be described below.
The analogous result for the number of maximal families is
01
di^
probably 2 but this has not been proven. It is
however a lower bound and an upper bound of 2 is
easily obtained.
To illustrate the kind of reasoning that can be employed to
obtain estimates of this kind we sketch the argument here.
A maximal F can be characterized by its minimal members.
That is, we can define G(F) to be the family consisting of
those members of F which contain no others, and G(F)
determines F. The family G(F) is then what is sometimes called
a "Sperner family" or an "antichain'i no member of G contains
another. (We discuss Sperner families in Section IV).
Some information is available about the number of Sperner
families, from this an upper bound to the number of maximal F's
(I,)'1"'1"
can be obtained, the bound being 2
To obtain a lower bound we divide the n/2 element subsets
of S into those containing a given element a , and the rest
(the rest here are the complements of the members of the former
collection). There are 2 collections Q, made up of
486 COMBINATORIAL ANALYSIS

151
n/2 element subsets containing a Each of these determines
a collection F, with F consisting of all sets with more than
S elements, those G- element sets containing a in Q and
complements of the £ element sets containing a not in Q.
The argument for n odd is similiar.
We expect that the kind of argument used to yield the estimate
(1^)(1+0(1))
2 for the number of Sperner families can be applied
to show that the number Sperner families which contain no disjoint
(l^iuomi
members is 2 . Any maximal family having 2
gn-1
members has 2 subsets. The total member of subsets of
all maximal families hence, the total number of F's is no more
^-^)(1-(1)) 2-l(l+o(D)
than 2 ^ which is of the form 2^ 1-^1-^ J
as stated above.
The other restrictions (1,,...,1^) have not all been
investigated in as much detail. We first present the existent
results on all these problems. Open problems are then listed.
I-, The properties of G(F's) are essentially the properties
n-1
f[n-l])
l to \ y /,
of maximal F's. They range in size from 1 to \27, the
number of them can be estimated as discussed above. They are
all maximal.
Ig The number of members'- ' in the union of k F's has
been shown to be no more than 2n-2n~ . This bound can be
487 SYSTEMS OF SETS

152
acheived by letting the k families be all subsets containing
a. for l<j<k.
J
("2 I
I., Bounds1 J on the size of Go^(n) a family containing
no k disjoint members have been obtained. For n = mk-1
these bounds are realizable, for other values they seem to be
slightly higher then the best possible results. These results
can be obtained by noticing that for any partition of S into
k blocks, at least one block must be outside of any Go^.(n).
This fact for any gives set of block sizes leads to limitation
on the number of members of G^, (n) of these sizes. Manipulation
of the limiting identities yields the results mentioned above.
Snallest size of a maximal G^,, (n) is no more than
2n-2n~ . This might be conjectured to be the exact result.
Ij, Among the maximal P's are families consisting of all
subsets containing some single element. Such families have the
property that all intersections are non-empty. Thus the
restriction (on Gjiv(n)) that every k members have non-vanishing
intersection does not reduce the maximal size of &ny(n) below
2 ~ . There are two natural questions which arise here. What
is the maximal size of Giik(n)'s in which there exist (k+1)
members whose intersection vanishes? Also what is the minimal
size of. a maximal G^, (n)? E.C. Milner ■■ ' 3¾ J nas some results on
the first of these questions. The second is open.
We now list some open problem in this area.
1 . What is the number of maximal families no two members
of which are disjoint?
488 COMBINATORIAL ANALYSIS

153
2 . How small can a family be that is maximal with respect
to the property that is the union of k different maximal families
no tv/o members of which are disjoint? It is asymptotic to 2n
for large nc.
3 . How many such families are there?
4 . Does the smallest maximal G?, have 2 -2 members?
5°. What are the exact upper bounds on Gok(n)?
6°. What Is the minimal size of a maximal Gi,>.(n)?
7 . What are more exact estimates on the number of families
of each type indicated?
489 SYSTEMS OF SETS

154
II. In the problems described so far the basic restriction
was that intersections not vanish. Such restrictions can be
replaced by size limitation on intersections. Thus we could
instead require that no A and A. in F satisfy
lA,f"IA | > k !A UA |-|A,nA | > k
!A,nA.| sk JA.UA.I-IA DA .( *k
IA nA I ^ k
|A,nA.! = k
The entire range of problems considered above can be raised
about families defined by each of these restrictions. The
generalization which most retains the flavor of section I is the
first. A maximal sized family Fv(n) restricted by it, consists
of all subsets having —s—- or more elements, with (-i/?7 v. -i \)
other sets if n+k is odd. 'That this is the largest possible
size for F. (n) was proven by Katona "■ -5'. Few of the other
problems have been examined under this restriction.
The opposite restriction that subsets do not intersect
"too much" is vaguely related to packing and coding problems.
The number of members of size > k of a family restricted so
that no two members satisfy |A.nA.! > 1 is at most (. ) and
J
is achieved by choosing all subsets of size k. If we let f
be the number of members of such a family having q elements.
n
We obtain E f (.) s (£) as a size restriction.
i=k
490 COMBINATORIAL ANALYSIS

155
The coding problem can be described as the study of families
limited by the restriction that the "symmetric difference" between
any two members be no less than k. The symmetric difference
between A. and A. is A.UA. - A.OA.. There are many results
on the maximal size of codes under these restrictions and on
constructions of optimal codes. Many of these are described in
for example, Berlekamp'- *.
Another problem of this general kind is: how large can a
family of subsets of S be if the symmetric difference between
members is always s q < n? For even q, it has been show that
maximal size families consist of any set a and all other whose
'n-l>
odd qi^ij o
symmetric difference with it is s q/2. For odd qV-^j—/ of
the subsets differing from a by -^£— may also be included *-' J,
491 SYSTEMS OF SETS

156
III. Another important class of problems involve families
of subsets of a given size subject to intersection restrictions
of the kinds already discussed.
P. Erdos, C. Ko, and R. Rado*- * shov/ed that the maximal size
of a family of subsets of S satisfying
1° all subsets are of size sksn/2 (with Is! = n).
2° no two are disjoint, no one contains another,
is (k~-i),the optimum being achieved by choosing all k elements
sets which contain a given element.
m-1
([n/2j.
» ' f) of sue
If 2k = n there are a large number (2 v- ' /) of such
families. If 2k<n, however, the maximal sized family is
unique up to permutation of the elements.
The minimal size of maximal family here may or may not be
2k-l
Ik )•
Among the questions that" have been raised in this area are:
1 What is the largest family if one excludes families all
of whose members contain some element?
2 Given two families such that the members of one all
intersect the members of the other, and subject to the member
size limitation described above what can be said about their sizes?
The following somewhat more general result has been obtained
in this direction'-"':
Let F and G be two families of subsets of S, with
the members of F having k elements and the members of G q
492 COMBINATORIAL ANALYSIS

157
elements. Let k+q be no bigger than n; if k is no more
n+1
than —p— then F can have k or fev;er members as long as
no member contains another; the same possibility for G is
allowed. Then, either
|F| * (¾) or !G! < (£j)
Milner'- ' -* has certain results on the first problem above.
For sufficiently large n and given k the family
consisting of all k element subsets including one particular
k
element is far larger (of the order of c ^-,- as opposed to
k-1
c' n « i t.) than any other. Under these circumstances it is
easy to answer many of the related questions that arise here.
Thus, for sufficiently large n for fixed k and q we
can show that
A. The number of members in the union of q sets of
k-element non-disjoint subsets of S with |S] = n, is no
greater than
(k-l) + (k-2) ••• + (n-q)
B. The number of members of a set of k element subsets
of S under the restriction that no (q+1) are pairwise disjoint
is bounded in the same way'- .
493 SYSTEMS OF SETS

158
C. The number of members of a set of k element subsets
of S under the restriction that the intersection of each pair
has at least q elements in it Is at most (v-Iq) •
One might conjecture that similar results hold so long as
2ksn-q+l for A and B, and that the best result for C is
the maximum over m of £ (2m~q) (P4"q"2m, ). Results of this
n_0 nH-p' vk-m-p-l''
kind have not yet been obtained.
A related problem, also as yet unsolved, is due to Kneser'- ',
How many families of k-element subsets of S, each consisting
of subsets which are not disjoint from one another, are
necessary to cover all k-element subsets? The answer appears
to be n-2k+l (if this number is at least one).
Restrictions of the following kind
subset size = r
size of intersection s q
represent packing problems, or coding problems involving words
of "fixed weight". Problems of the form
subset size = k
intersection size = q
describe such structures as projective planes (q=2) Steiner
494 COMBINATORIAL ANALYSIS

159
systems and designs. There exist a vast literature on such
questions. Neither class of problem v;ill be considered here.
Erdos, Ko, and Rado'- J conjectured that if |S] = 4k and
F consists of subsets of size 2k of S which overlap by at
least two, then max |F| = ((^)-(^)2)/2-
495 SYSTEMS OF SETS

160
IV Containment Restriction
In this section we consider families of subsets that
are subject to containment restrictions. The prototype of such
restriction is that satisfied by a "Sperner family" or antichain,
T12 1
that no member contain another. Sperner1- 'J in 1927 showed that
such a family could have at most ( ^ ) members. Lubell
[13]
£]'
in 1959 and independently Meshalkin ■■ * in I963 obtained a
somewhat stronger restriction. If f. is the number of
k-element members of a Sperner family of subsets of S, with
|S] = n then the inequality
k=0 K K
holds. Equality can only occur if f. = 1 for some value of
k. Sperner's result is a corollary of this inequality, since
it is trivial that
fc=0 k [£] k=0 k k
Lubell's argument is so simple that we repeat it here. A
maximal chain is a set of n+1 subsets of S totally ordered
by inclusion. Each k element subset occurs in the same
proportion (1/(1,)) of maximal chains. Since no chain can contain
more than one member of a Sperner family, the sum of the
proportion of maximal chains containing each member cannot exceed
one, which is the Lubell-Meshalkin inequality.
496 COMBINATORIAL ANALYSIS

161
The same argument implies that the maximal number of members
in a family which has at most q member in common with any chain
is the sum of the largest q binomial coefficients. This result
follows from the inequality
which must be satisfied by such a family. Lubell's argument
can be applied in many other contexts. Thus, by its use,
along, with certain additional arguments the following general-
risi
ization1- Ji has been obtained. Let f be any function defined
in the members of any partial order and let F be a family which
has at most k members in common with any chain in the partial
order. Let G be a permutation group defined on the partial
order which preserves f (for g in G, f(gA)=f(A)) and is a
symmetry of the partial order (AsB if and only if g Asg B
for every g in G). Then the maximum value of the sum of f ovei
the members of F is acheived for some F which is the union of
orbits under G. That is, there is an F such that
E f(A) s r_f(A)
A6F A6F
with F the union of complete orbits under G. Lubell'- ' has
obtained still further generalizations of his result.
The following questions have also been raised about Sperner
families. Let F be a Sperner family, let G be the family
connecting of all subsets which contain at least one member of F,
497 SYSTEMS OF SETS

162
and let GT be the family of all subsets ordered by inclusion
with respect to at least one of F.
How large can |F! be given ]G|? Given |G !? If
|F|>( ), how many pairs A, B with A=>B must there be in F?
The following results along these lines have been obtained
1. If |Fl>(£) for k< n/2, then | G, I > T. (")[17j
K g=0 g
2. !F!/|Gll<([n72])/2:
n[l8]
3. The number of "containment pairs" is minimized if F
consists of all subset having [£], [£] + 1, [§]- 1, [£] + 2, ...
elements and of the remaining members of F all have a number
f 191
of elements given by the next entry on this listL y*.
The minimal number of "containment triples" has not been
found as yet, although one could guess the same conclusion.
Spemer's conclusion can be obtained, when the restriction
of non-containment is relaxed considerably. Suppose, for example'- J»P^J
that S is the union of two disjoint sets T-, and T?
S = T-,UT2 T-^Tg = )6
and suppose that F is restricted such that if A=>B for
A, B6F then A-B^ and A-B^Tg. Then |P|s( £ ), that is
Spemer's bound still applies with these weakened requirements
on F. An interesting unsolved problem is the analogue of this
for S --- T-,UTpUT^, all t's disjoint; under these circumstances
the analogous restriction on F is not sufficient to get the
498 COMBINATORIAL ANALYSIS

163
same bound on |Fl. One can ask, what is the best bound? Also,
what are the weakest additional restrictions necessary to impose
upon F to get back to the Sperner bound in this case? One can
also ask, what analogue of Lubell's inequality can be obtained
for the S = T-,UTg problem?
Katona^32J, SchonheiirJ21 \ and Erdosf22J have obtained
further generalizations of Sperner's theorem.
The number of Sperner families of subsets of S has been
investigated by many authors beginning with Dedekind. The best
recent resultL J is that this number is greater than
([n])(l+c n l°g n)
Katona'- ■" and Kruskal'- -5^ have considered a related question.
Given an f member family F of k-element subsets of S. Let
G consist of the (k+1) element subsets which contain one or
more members of F. How small can |G| be given f? His result
is an exact one: f can be uniquely expressed as
<2i)<?2m£3).-.<£„)
r~ r-3 r
with ri>r2>...rm. Then |g|>( k2)+(k!2)+ • • -+{£m) •
ri4i
MeshalkinL J has obtained a result on families of partitions
of an n element sets into k labelled blocks restricted so
that no blocks properly contains a block with the same label.
The result, the largest k-nomial coefficient, is really a
corollary of the Lubell-Meshalkin identity.
499 SYSTEMS OF SETS

164
V. There are a number of problems that have been studied
which involve intersection restriction involving three or more
subsets. The following set of limitations have been considered
a) F-, is limited in that no three members A..B..C satisfy
AUb = C (AOB = C would be equivalent)
b) Fp obeys the restriction that no four members A, B, C, D
satisfy AUB = C, Af"lB = D
c) No three members of A, B, C of F_ satisfy AUB = C
or Af"IB = C
d) No three members of Fh satisfy AUB^C (equivalently
AOBCC)
e) No three members of F,- satisfy AUB<=c
f) No 2 members of Fgk form a Boolean algebra under
union and intersection
g) Given any k members A-,...A, of F7^ the intersection
A^ApHA^,. . . nA, is nonempty and the .same restriction holds of any
or all A .' s are replaced by their ::omplements .
J
h) Given two disjoint members of Fo their union
is a nonmember: AUB = C, Af~lB = j5 i.i excluded.
Results on these areas have beei as follows:
a) The restriction AUb £ C l.ould seem to limit F-,
[26]
to ( n Ul+cn ) members. The best limitation1 J obtained has
been) ( nV1+c/n )
500 COMBINATORIAL ANALYSIS

165
b) Under the restriction AUB / C or ADb / D, F2
Is of
[27]
can have c 2 n ' members. Upper and lower bounds of this
form have been obtained; they may or may not be equal'
c) The restriction stated above probably requires that
F can have at most ( _ \+l members for n even. Clements <- ■"
has found examples having this many members.
d) The number of members of Fu is exponentially small
compared to 2 . Little is know about this limitation.
e) Under AUBj^C the size of F,- cannot exceed
Yl + —) which bound can be achieved.
f) Little is known beyond the case (b) above for this
restriction.
g) This problem has been considered by Joel Spencer.
[10]
For k = 2 it is resolved: the bound is ( ~, \ . For k = 3
upper and lower bounds of the form Cr with l«cs2 have
been obtained. They are not close to one another. This
restriction includes that of (d) namely A-,riAp^A^ for k>3.
h) Roughly speaking, under these restriction the family
G can contain all sets having ^- to -^- elements. Best
results have been obtained for n = 3k+l. For n = 3k, 3k +2
there is a slight gap between the best bound and best existing
results .
Another set of related problems are due to P. Erdo's and
f2Ql
L. MoserL -7J. Rewards for their solution are available from the
former author.
501 SYSTEMS OF SETS

166
'find bounds for f(n) = the least number of subsets of a
set A of n elements such that every subset of A is the
union of two of the f(n) subsets. It Is easy to prove that
/2 2n < f(2n) £ 2 2n.
We offer $25. deciding (with proof) whether f(2n) is > or
< (1.75)2" for sufficiently large n."
"Find bounds on f(n) = the largest number of subsets
A-,,A„, . . ,A„/ >, of a set of n elements such that the ( ^ ) sets
A.UA.j l<isf(n)j are distinct. We can prove that for large n
(1 + €,)n < f(n) < (1 + € )n,
where 0<6-,<Pg<lj and offer $25. for finding fp C2 wi_fcn
<VV 1.01."
502 COMBINATORIAL ANALYSIS

167
VT. Another kind of problem involves families of sets of a
specified size out of a not necessarily specified set.
Two problems of this kind are:
1 Suppose that no three subsets have pairwise the same
intersection, and they are of size k. How many can there be?
2 Suppose that any subset which interests all members
of the family contains at least one member. How few members can
the family have?
The property mentioned in 2 called "Property B", has been
extensively studied. For n = 3 one can find a 7 member family
with this property. For n = 4 the smallest family size is
unknown but probably around 20. Erdos ^ ■"» •■ ■" has an upper
bound of c n22n and Schmidt^31-' a lower bound of 2n(l + 4/n)-1.
These results have recently been improved slightly by Herzog
and Schonheim'- .
The best bound for problem 1 here is probably of the form
c . The best result obtained so far for an upper bound has been
of the form k! c1^3^'37-'.
503 SYSTEMS OF SETS

168
REFERENCES
D. Kleltman, "Families of Non-Disjoint Subsets", journal
of Combinatorial Theory, Vol. 1, 153-155 (I966).
D. Kleitman, "Maximal Number of Subsets of a Finite Set No.
k of Which are Pairwise Disjoint", Journal of Combinatorial
Theory, Vol. 5, p. 152 (1968).
E.C. Milner, "A Combinatorial Theorem on Systems of Sets",
J. London Math. Soc. 43(1968),204-206.
P. Erdos, "On a Combinatorial Problem III", Canad. Math.
Bull. 12(1969), 413-416.
G. Katona, "intersection Theorems for Systems of Finite Sets",
Acta Math., Acad. Sci. Hung. 15(1964), 329-337.
E. Berlekamp, "Algebraic Coding Theory", McGraw-Hill Book
Company, N.Y. (1968).
D. Kleitman, "On a Combinatorial Conjecture of Erdos", J.
of Combinatorial Theory, Vol. 1, 209-214 (I966).
P. Erdos, Chao Ko, R. Rado, "Intersection Theorems for
Systems of Finite Sets", J. Math. Oxford, Sec 12(48)
(1961).
D. Kleitman, "On a Conjecture of Milner on k-Graphs with
Non-Disjoint Edges", Journal of Combinatorial Theory,
(1968).
Private communication.
R. Kneser, see Aufgabe 360 Jahresbericht d. Deutschen
Math. Vereinigung 58(2) (1955).
E. Sperner, "Ein Satz uber Untermengen einer endlichen
Menge", Math. Z. 27(1928), 544-548.
D. Lubell, "A Short Proof of Sperner's Lemmas", Journal of
Combinatorial Theory, 1 (I966), 299.
L.D. Meshalkin, "Generalization of Sperner's Theorem on the
Number of Subsets of a Finite Set", Theory of Probablility
and its Applications, 8(1963), 203-204. (English translation
D. Kleitman, M. Edelberg, D. Lubell, (to appear).
D. Lubell, (to appear).
504 COMBINATORIAL ANALYSIS

169
D. Kleitman, "On Subsets Contained in a Family of Non-
commensurable Subsets of a Finite Set", Journal of
Combinatorial Theory, 7, 131-183 (I969).
D. Kleitman, "On a Lemma of Littlewood and Offord on the
Distribution of Certain Sums", Math. Zeitschr. 90, 251-259
(I965).
D. Kleitman, "A Conjecture of Erdbs-Katona on Commensurable
Pairs Among Subsets of an n-Set", in Theory of Graphs,
Proc. Colloq. held at Tihany, Hungary, I966, Akademiai
Kiado, 1963, 187-207.
G. Katona, "On A Conjecture of Erdo's and a Stronger Form of
Sperner's Theorem", Studia Sci. Math. Hungary 1 (1966)5
59-63.
J. Schonheim, "A Generalization of Results of P. Erdos and
G. Katona", Journal of Combinatorial Theory, (1970) (to
appear).
P. Erdos, J. Schonheim, to appear in Proc. of Balutonfured
Conference, 1969.
D. Kleitman, "On Dedekind's Problem: The Number of Monotone
Boolean Functions", Proc. of the Am. Math. Soc, Vol. 21,
No. 3, PP. 77-682 (I969).
G. Katona, "A Theorem on Finite Sets", in Theory of Graphs,
Proc. Colloq. held at Tihany, Hungary, 1966, Akademiai
Kiado, 1968, 187-207.
F.B. Kruskal, "The Number of Simplices in a Complex",
Mathematical Optimization Techniques, Edited by R.
Bellman, 25I-278 (1963).
D. Kleitman, Proc. AMS (to appear).
P. Erdos, D. Kleitman, Proc. AMS (to appear).
D. Kleitman, "On Families of Subsets of a Finite Set Containing
No Two Disjoint Sets and Their Union", Journal of
Combinatorial Theory, Vol. 5, No. 3 (1968).
P. Erdos, L. Moser, Proc. Calgary Conference (I969).
P. Erdos, "On a Combinatorial Problem II", Acta. Math.
Acad. Sci. Hung. 15 (1964), 445-449-
W.M. Schmidt, "Ein Kombinatorisches Problem von P. Erdos
and H. Hajnal", Acta. Math. Acad. Sci. Hung. 15(1964),
373-374.
505 SYSTEMS OF SETS

170
G. Katona, "Spemer Type Theorems", Dept. Statistics UNC,
(196).
G. Katona, "A Generalization of Some Generalizations of
Sperner's Theorem", Dept. Statistics UNC, (I969).
P. Erdos and R. Rado, "Intersection Theorems for Systems 0
Sets," J. Lond. Math. Soc, 35 (i960), pp. 85-90.
H.L. Abbott, "Some Remarks on a Combinatorial Theorem of
Erdos and Rado", Can. Math. Bull., Vol. 9, No. 2, 1966,
pp. 155-160.
H.L. Abbott and B. Gardner, "On a Combinatorial Theorem of
Erdos and Rado", (to appear).
H.L. Abbott, D. Hanson, N. Sauer, "Intersection Theorems
for Systems of Sets", (to appear).
E.C. Milner, "intersection Theorem for Systems of Sets",
Quart. J. Math. Oxford (2), 18 (1967), 369-84.
506 COMBINATORIAL ANALYSIS

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 28, No. 1, April 1971
ON COLLECTIONS OF SUBSETS CONTAINING
NO 4-MEMBER BOOLEAN ALGEBRA
PAUL ERDOS AND DANIEL KLEITMAN
Abstract. In this paper, upper and lower bounds each of the
form c2n/n1'4 are obtained for the maximum possible size of a
collection Q of subsets of an n element set satisfying the restriction
that no four distinct members A, B, C, D of Q satisfy A \JB = C and
ACiB = D.
The lower bound is obtained by a construction while the upper
bound is obtained by applying a somewhat weaker condition on Q
which leads easily to a bound. Probably there is an absolute
constant c so that
max | Q | = c2"/n"4 + o(2"/«"4)
but we cannot prove this and have no guess at what the value of c is.
1. Introduction. A collection of square free natural numbers which
contains no four distinct integers a, b, c, d satisfying l.c.m (a, b)
= c, g.c.d (a, b)=d naturally corresponds to a collection of sets of
prime factors such that no four A, B, C, D satisfy A^JB = C, and
AC\B = D'. Bounds on the maximal size of the latter kind of collection
thus lead to bounds on the former and hence (see Erdos, Sdrkozi and
Szemeredi [l ]) to bounds on sums taken over such collections.
In this paper we derive such bounds, which are of the form c2n/»"4.
Analogous results for the corresponding problem when the integers
are not required to be square free are indicated. This case corresponds
to a collection C of sequences of integers of length n (Si, 52,---, 5„)
satisfying 5,-C(5 max),-, such that C contains no four distinct
sequences {SI}, {5?}, {Sf}, {#} with Max (5(\S?)=S?, Min (5}. 5?)
= 5j for alH.
2. Upper bound. Let Q be a collection of subsets of an n element
set 5 which satisfies the restriction that no four members A, B, C, D
of Q satisfy A KJB = C and AC\B = D.
Then, if T is any subset of 5 and W, X, Y, Z are distinct and
satisfy WCXCT, YCZCS-T\t is not possible that WKJY, XKJY,
WKJZ and XKJZ are all in Q as
(X \J Y) r\ (W VJ Z) = W VJ Y, {X \J Y) \J {W \J Z) = X \J Z.
Received by the editors November IS, 1969.
A MS 1970 subject classifications. Primary 0SA99-
Key words and phrases. Bounds on collection size, sizes of subset families.
Copyright © 1971. American Mathematical Society
87
507 SYSTEMS OF SETS

88
PAUL ERDOS AND DANIEL KLEITMAN
[April
In consequence of this fact, if X\, Xi, • • ■ , Xk and Y\, F2, ■ • ■ , Ym
are all distinct and the X's and Y's are both totally ordered by
inclusion with XiCXkCT and YidYmdS—T, there cannot be two X's
and two Y's whose four unions (XJU Y,) are all in Q. Thus, if we
define the zero-one matrix Af.-y such that Af,-/=1 when XiUYj^Q
and Mn = 0 otherwise, Mi,- cannot contain a 2X2 submatrix all of
whose entries are 1. The total number of entries of M<,- which are 1
is the number of members of Q of the form Xi^J Yj.
The general question of the maximal number Mt,{k, m), of +1
entries in a kXm matrix containing no tXs submatrix all of whose
entries are 1 is a well-known problem of Zarankiewicz [2] and [6];
in the case arising here £=s = 2, a good bound is a known result due
to (Reiman [3]); namely
M22(k, m) £h[k+ (-¾2 + 4m.(m - 1)£)"2J,
M2,(k, m) ^i[m+ (m2 + 4k(k - l)™)1'2].
We may apply this result to the problem at hand by partitioning
the subsets of T and of S—T into blocks each totally ordered by
inclusion. For each pair of blocks F, G of subsets of T and S—T
respectively, we may apply the result above to deduce that no more
than Af22(/, g) members of Q can consist of the union of a member of
F with one of G (where/ and g are the number of members of F and
G respectively).
The maximal size of Q is therefore no more than
(2) £ pT(f)pa-r(g)Mn(f, g)
l.Q
where pr(f) and ps-r(g) represent the number of blocks in the
partitions of the subsets of T and of 5— T having respectively/ and g
members.
It may be noticed that this bound makes use of a condition
somewhat weaker than the original condition on Q. We only here exclude
one of four subsets A, B, C, D from Q when /lWB = Cand AC\B = D
if C—AQT and C — BdS—T or vice versa, for some fixed subset
T of 5. Moreover the exclusion is only effective if both CC\T and
DC\T be in the same block in the partition of the subsets of T into
blocks each totally ordered by inclusion, and similar remarks hold
for Cf\S—T and DC\S—T. In general the restriction will differ for
differing choices of T. Families which are invariant under changes of
T which maintain its size, are those which contain all members of
each of several sizes. For such families the restriction obtained for
each T of fixed size are all the same.
508 COMBINATORIAL ANALYSIS

197'] SUBSETS CONTAINING NO 4-MEMRER BOOLEAN ALGEBRA 89
We here choose | l'\ = \njl\. Below we take n divisible by 4 for
notational convenience.
It is well known that the subsets of an n/2 element set can be
partitioned into Q/l) blocks each block totally ordered by inclusion.
Moreover the number of such blocks of size 2g + l can be made equal
+ _ ( n/2 \ _ / n/2 ^
L0 \n/i + q) \n/i+q+U-
We may therefore set
/ n/2 \ / n/2 \
(3) M2?+1)=A«(2?+D= /1,)-(/,. , J
\n/4 + q/ \;j/4 -f <? + 1/
in expression (2) above. It is not possible to partition the subsets
into as many blocks that are less regular in size. It can be seen from
the expression (1) for M^(j, g) that the size restriction (2) is
maximally restrictive when block sizes are maximally unequal, so that
this partition will be the most useful for our purposes.
If, instead of subsets of an n element set we were concerned with
collections of sequences (Si ■ ■ ■ Sn) of integers satisfying SiQS,
max for each i, we could proceed in the fame manner. If the indices
/i • • • n were divided into two blocks, T and T', sequences resinned
to blocks T and T' play the role played above by subsets of T and of
5— T. If such sequences are divided into totally ordered blocks under
the natural orderings ({s}±s{<} if Si^t,- for all i) the results are
identical to those given above, namely the size of Q can be no greater
than
T,pT(f)pT'(g)M22(f,g)
/.a
where now pr(f) (and similarly pr'(g)) represents the number of
totally ordered blocks of sequences restricted to T having/ members
in a partition of all such sequences into such blocks. This limitation
can be estimated by the same means used for the subset case.
Straightforward manipulation of (2) and (3) yields that an upper
bound on the size of our family is
,_o W-l + q/ V m=o V«/4 + m)I
which quantity behaves as ca2nn~U4.
3. Lower bound. We can construct collections Q satisfying the
constraint under consideration here as follows. If A, B, C, D are
distinct and satisfy A^JB = C, AC\B = D, they must also satisfy the
conditions:
509 SYSTEMS OF SETS

90
PAUL ERDOS AND DANIEL KLEITMAN
\ A\ + \B\ =~ \C\ + \ D\, \C\ > \ A\, | 51 > | Z?|.
Thus if we construct a collection of subsets, each of which contains
mi elements for some i, the collection will satisfy our constraint if the
mi's satisfy mi+mj^mk+mi for m^m*, m^mi.
It is known [5] that there is a sequence of integers l^Ui< ■ ■ ■
<uk<nli2, £ = (l+o(l))»1/4 satisfying Ui+Uj5*u, + u,. Set mt
= [»/2]+w;. Clearly mi+mj^mr + m,. Let 5 satisfy \S\ =», and let
Q be the collection of all subsets of 5 having mi elements for all
/=1, 2, • • ■ , k. Clearly no four elements of Q satisfy AKJB = C,
AC\B = D and further
£iW/ W/ \[n/2] + [n"»]/
^ (1 +o(l))n>'«(. " r J
\[n/2] + k'2]/
> c2n/w"4
which establishes our lower bound.
Analogous results may be obtained by use of the same arguments
in the case of sequences of integers mentioned earlier in the paper.
References
1. P. Erdos, A. S£rkozy and E. Szemerddi, On the solvability of the equations,
[o,-, aj\ = ar and (o), a'-)=a'r in sequences of positive density, J. Math. Anal. Appl.
15 (1966),60-64. MR33 #4035.
2. K. Zarankiewicz, Problem P 101, Colloq. Math. 2 (1951), 301. See also: R. K.
Guy, A problem of Zarankiewicz, Proc. Colloq. Theory of Graphs (Tihany, 1966),
Akad. Kiad6, Budapest, 1968, pp. 119-150.
3. I. Reiman, Vber ein Problem von K. Zarankiewicz, Acta Math. Acad. Sci.
Hungar.9 (1958), 269-273. MR 21 #63.
4. D.J. Kleitman, On a lemma of Liltlewood and Offord on the distribution of certain
sums, Math. Z. 90 (1965), 251-259. MR 32 #2336.
5. P. Erdos and P. Tur£n, On a problem of Sidon in additive number theory, and on
some related problems, J. London Math. Soc. 16 (1941), 212-216. See also: P. Erdos,
J. London Math. Soc. 19 (1944), 208. MR 3, 270; MR 7, 242.
6. R. K. Guy and S. Zn£m, A problem of Zarankiewicz, Recent Progress in
Combinatorics, Academic Press, New York, 1969, pp. 237-243.
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
510 COMBINATORIAL ANALYSIS

Chapter 11
Block Designs
For a given n > k > I and (n, k, /) tactical configuration is a family of
^-element subsets of an n-set such that every /-set is contained in exactly one k-set.
An important combinatorial problem is to determine for which (n, k, /) such
tactical configurations exist. For example, if k = 3 and /=2 the (n, k, I)
designs are called Steiner Triple Systems. An (n2 + n + l,n + l,l) design
is equivalent to a projective plane of order n.
It is easy to show that if such a tactical configuration exists it will contain
exactly (,")/(*) /t-sets. In these papers the question of how close one may come
to a tactical configuration is looked at. In particular, set M{n, k, /) equal the
minimum number of A>sets to cover every /-set and m(n, k, I) the maximum
number of A>sets to cover no /-set twice. If a tactical configuration exists then
M{n,k,l) = m{n,k,l) = (?)/(?).
It is conjectured in [412] that for k, I fixed
lim M(n, k, /)(0/(0 = Km m(n, k, /)(0/(0 = 1-
This is shown for 1 = 2, all k and for /= 3, k = p -\- 1 (p a prime power).
The general conjecture is still open.
Further results on [412] have been obtained by Richard Wilson (Ohio
State University, unpublished).
Papers in Chapter 11
[243] (with A. Renyi) On some combinatorial problems
[412] (with H. Hanani) On a limit theorem in combinatorial analysis
511 BLOCK DESIGNS

On some combinatorical problems.
In memoriam Tibor Szele.
By P. ERDOS and A. RENYI in Budapest.
Introduction.
Let Ck(n) denote the least number of such combinations of order k of
n different elements, that any two elements are contained in at least one
combination (£,/7=2,3,...). Such a system of combinations will be called
a (k, /?)-system. Clearly we have „ G (/7) g _ , as there are pairs-
in any combination of order k and each of the „ possible pairs must be
contained in one of the G(/7) combinations. Thus we have
If for some values of k and n there is equality in (1), we say that an
optimal (k, /7)-system exists. It is well-known, that if k = P-\-\ and n = Pr +
+ P*"1-\-■ ■ ■-\-P-\-1 where P is a power of a prime and r^l an arbitrary
integer, there exists an optimal (k, /7)-system. This has been proved —
according to our knowledge — first by Th. Skolem (see [1]). There exist also optimal
(P, Pr) systems, if P is a power of a prime and r ^ 1. These facts are nowadays
utilized in constructing balanced incomplete block designs (see [2]). An
optimal (k, /7)-system is clearly a balanced incomplete block design of n varieties
into . * blocks of k plots each, such that every variety occurs with
every other variety exactly once in the same block. It seems that up to now
interest was focused on optimal (k, «)-systems and the asymptotic behaviour
of G;(/?) for n~* °o has not been investigated. In § 1. of the present paper
we prove that* if k = P is fixed, where P is a power of a prime, we have
/on .• c*(") 1
(2) hm —
/7(/7-1) k(k—\)
513 BLOCK DESIGNS

P. Erdos and A. Renyi: On some combinatorical problems. 399
i. e. there exists a sequence of asymptotically optimal (k, /?)-systems for each
fixed k=^P where P is a prime power. (2) is valid also for k = P-\-\ where
P is a prime power. The proof is analogous to that given in the present
paper for k = P, only Lemma 1 is used instead of Lemma 3. It can be proved
by the same method that the limit lim —r^iT==^,t exists for any k^\,
n->• at Tlyl 1 )
but we do not know the value of yk for other values of k than mentioned
above. However it can be proved that
<3) Hm k(k-\) lim -~^%-1
i.e. that lim k(k—\)yk=i. These results together with a simple but ingen-
ious method of proof, which has been formulated and used by T. Szele
in his thesis [3], are applied in § 2. to prove a conjecture which has
been recently proposed by the second named author [4]. Let Dk(n) denote
the length of the shortest sequence formed from the digits 1,2,..., n in
which any two digits i and j (1 ^i<J^n) are at least once to be found
in such position, that they are separated by at most k numbers. It has been
proved, in [4], that ■
1 ^ .. Dk(n) ^ — Dk(n) 1
-^ lim V^- § lim ^- -^ —
and it has been conjectured, that
(4) lim \^- = uk
exists for k = 2, 3,.. .; however the existence of (4) is proved only for k = 2
and k = 3, the proof for k=3 being due to N. G. de Bruijn; in these two
cases the limit is -yr—^-. We prove that the limit (4) exists for all k ^ 2;
however our method does not lead to the determination of the value of ft(:.
§ 1. The asymptotic behaviour of Ck(n).
Let us put
(5) ck(n)- *<*-'>C*(/0
We shall prove
Theorem 1.
n(n-l)
lim 0,(/7)=1 for k = P
where P is a power of a prime.
514 COMBINATORIAL ANALYSIS

400
P. Erdos and A. Renyi
The proof requires a number of lemmas, some of which are well-known:
and are stated only for convenience.
Lemma 1. (SkOlem): If k = P-{- 1, where P is a power of a prime-
and n = Pr + Pr~l-{ \-P-\-\, where r is arbitrary, we have ck(n)=l,
i. e. there exists an optimal (k, n)-system; this system can be chosen in such
a way that it contains a subsystem which is an optimal k, ——- ysystem.
For the proof see [1] or [2] p. 109—111.
Lemma 2.
ck(n) ^ ck(q)ctl(n) (k<q< n).
Proof of Lemma 2. Let us form from the numbers 1,2,..., n a (q, n)-
system consisting of Cq(n) combinations of order q. From each such
combination let us form a (k, q) system consisting of Ci-(q) combinations of
order k. Thus a (k, n) system is obtained. Thus we have Ck(n) ^ Ck(q)Cq(n).
Multiplying this inequality by —~ ^r we obtain the assertion of Lemma 2.
Lemma 3. If P is a power of a prime, and r 2= 1, we have cP(Pr) = 1,.
i. e. there exists an optimal (P, Pr)-system.
Lemma 3 can be deduced from Lemma 1 (see [2] p. 112). Let us
consider an optimal (k, n)-system, for k = P+ 1, n = Pr + P,'~1 -\ \-P+l,
which exists according to Lemma 1, and which contains as a subsystem an
optimal \k,-T—r- -system. Let us omit from the given (k, rc)-system the
mentioned \k,j_—- -system ; it is easy to see that any one of the remaining
combinations contains exactly one element of the omitted subsystem. Omitting
the mentioned element from each of these combinations, we obtain an optimal
-\,n—-r—- -system, i.e. an optimal (P, Pr)- system, as /i:—1=/3 and
n—\ pr
k—\
Optimal (P, .Pr)-systems can also be constructed directly, without using
optimal (P+l,Pr + Pr~l -\ \-P+ l)-systems. We give here only the
construction of optimal (p, />2)-systems if p is prime. Let us represent the p-
elements by all pairs (;,/) of residue classes mod/? (;',/ = 0, 1,...,p—1).
Let us consider to any two residues h and k mod/? the combinations
G* :(0,h),(\,h + k),...,(r,h+ rk), ..., (p-I, h + (p—l)k).
Thus we obtain p2 combinations; let us consider besides these for any residue
h mod/? the combination
Ch:(h,0),(h,\),...,(h,p-\).
515 BLOCK DESIGNS

On some combinatorical problems.
401
Thus we obtain altogether p2+p combinations of order/*, which together form
an optimal (/?, /?2)-system. As a matter of fact if (x, y) and (u,v) are two
pairs of residue classes mod/? then if x = u these two pairs of residue classes
are both contained in Cx", if x=\=u the two pairs of residue classes are both
contained in the combinations C,,k, where h and k are determined by the
congruences h-\-xk~y (mod/?), h + uk=r, (mod/?), which have a solution
owing to x ^ u (mod p) namely k = — (mod /?) and h = —' (mod p).
In what follows P shall denote a fixed number which is the power of
a prime; a1(fl2)... will denote positive constants, depending eventually on
P or s but not on n.
Lemma 4. If 0 < t■ < -j- we have
ck(n)^ck(N)(l+St)
if
N(\—t)^n^N (yVg2).
Proof. We have G(n)^G(A/) for n s N and thus
and if 0 < « < -j- we have -r, ^-vr = 1+8? which proves Lemma 4.
4 (1—2s)
Lemma 5. (Ingham) : If pn denotes the sequence of primes, (n = 1,2,...),
we have /?„_i — /?„ < p" 16 /or p,, > a1; (See [5]).
Lemma 6. // /?;; denotes the k-th prime number and A2 > A s a,
we have
Max ^-^1 + !
A^-^A, Pk ' A,""
Lemma 6. follows easily from Lemma 5.
Lemma 7. There exists to any t with 0 < t■ < -j- arbitrary large
numbers B for which
cP(n) ^ 1 + 32* for B^n^ 2B1.
Proof. Let 0<£<-- and an integer a3 be given. Let us choose an
integer r such that A=Pr ^a3. By Lemma 3 cP(A)= 1 and thus by Lemma 4
cP(n) ^ 1 4-8« for A(l— t) ^ n ^ A. If a3 is sufficiently large, there is at
least one prime in the interval (A(l—e),A). Let <?i < q2 < • •• < qs denote the
primes in the interval (A(\—t),A). It follows from Lemmas 2 and 3 that
if m is any fixed integer, m^2, we have Cr(q?)^l+8s as cP(^,-)=1 +8«
516 COMBINATORIAL ANALYSIS

402
P. Erdos and A. R6nyi
(/=1,2, ...,5); thus by Lemma 4
cP(n) =S(l+8«)2s 1+32^ for q]n(\—e) ^ n <q?.
Now, if we choose m in such a way that the intervals (q7(\—s),Q?) are not
disjoint, i.e. g"+i(l—s)<qT, it follows, that cP(n) =£ 1 +32^ for g'i"(l—*) ^
1
n ^ g.r. Now ¢£1(1 —e) < tf" (/=1,..., 5) is satisfied if -^- < f l
q> 11—* J
(1^/^5). As by Lemma 6. -^^-<l+~r\n \TTnr '* fl3 's sufficiently
Qi \Ayi *))''
1 I 1 v'"
large, this is true if 1 -\--rj-r, tt^t?< \-\ and thus if a3 is sufficiently
(.4(1—s)yib \l—vl J
large, this is true if m<A14. Thus if a3 = fl3(f) is sufficiently large, we have
ck(n) s 1 +32« for 04(1-^) + .41116)'"s n s (4—41' 1B)"'. Now the intervals
[(A(\ — €) + Any, (A—An "T] are not disjoint, if m > (log A)-, if «3 is
sufficiently large. Thus it follows that
Cp(n) ^\+32e for (A(\—e) + Aniliy]0zAr-^n^(A—Ani,i)Aii
and thus a fortiori for e-<l0^>3s n g eA+ As the interval (e2a°g^, ^'')
contains an interval (B, 2Br), provided that a,, is sufficiently large, Lemma 7 is
proved.
Lemma 8. If cP(n) ^ a for B ^ /ig 2B- we /wve cP(rc) g « 1 + -^-r
/0/- B-^n<2BA if B^at.
Proof. Let :^, 71», ..., :Tt denote the primes in the interval (B, 2B1);
then by Lemma 2 and 3 cP(rrf) ^ Cp(rr,) s «. By Lemma 6 we have
and thus we have
4^(1+^)^1+^ iffl^fll
-<+] 11 — ^^ I < ^
24
It follows by Lemma 4. that cr(n) gj «1 1 + „. ,„ I for
, IB I
As
^(i-7^)-(^+^ifiKi-^)<^
and
.-¾ ^ (2B-—2Biyy ^2 B*
if a4 is sufficiently large, Lemma 8 is proved.
517 BLOCK DESIGNS

On some combinatorical problems.
403
Now we are in position to prove Theorem 1. It follows from Lemma 7
that for any s 0<js-- we can find a number B, which can be chosen
greater than an arbitrary given number, such that cP(n) ^ I-\-32s for
Bmn^2B2. It follows by Lemma 8 that
Cp(/i)^ (1+32*)/7(1 +^ for any /7 i, £.
As
3[,+&)s,+-&
it follows that
(6) UHT cp(n) ^(1+32*-) f 1+-^
As «>0 can be chosen arbitrarily small and B arbitrarily large, (1) and (6)
implies
lim cP(n) = 1.
Thus Theorem 1 is proved.
Now let k denote an arbitrary number and P the greatest prime power
^ k. Clearly we have Ck(n) s Cp(n) as any (P, n)-system can be transformed
into a (k, n) system, by adding arbitrary k—P elements to each combination
of the given (P, rc)-system. Thus it follows from Lemma 3 that if k is arbitrary
^ , , k(k—X)
where P is the greatest prime power si, As P>k—kll-'1& for k^a^ it
follows that
\im ck(n)^ 1+-^¾
and thus
(7) Iim(Iimc*(/i))=l.
fc-> CO 71-> CO
It is not difficult to prove by the same method as applied in proving
Theorem 1 that lim ck(n) exists for every k, and ihus lim can be replaced
H-> CO
by lim in (7).
§ 2. Application of a lemma of T. Szele.
In his paper [3] Szele has used the following simple but often very useful
Lemma 9. If a„ is a sequence of real numbers, which is „almost mono-
tonically decreasing", i. e. if a„ ^ am(\ + «) for any e>0 and any m & m0(s),
if n ^ n0(i, m), further a„ is bounded from below, then lim an = a exist.
518 COMBINATORIAL ANALYSIS

404
P. Erdos and A. Renyi
Lemma 9 may be proved as follows. It follows from our supposition
that for any s >0 and m ^ ma(s) we have
(8) lim an s a„,(\ +e)
»->- co
and thus
(9) lim an ^ (1 +e) lim am ;
"-»■<» ,„ -,. co
as ^ > 0 is arbitrary, (9) implies that lim a„ exists.
Ti ->- CO
Now let Dk{n) denote the length of the shortest sequence, con sisting
of the digits 1,2, ..., n, which has the property that any two digits i and j
(1 ^i<j^n) occur somewhere in the sequence in such a position that they
are separated by not more than k elements of the sequence. We may restate
the definition of Dk(n) in the language of the theory of graphs. Dk(n) is the
length of the shortest directed path in the complete graph of n points, which
has the property that from any point of the graph we may reach any other
point in not more than k-\-\ steps, by going along the path always
according to the given direction or always in the opposite direction. Then we have
clearly
Dk(n) ^ C«(/I)A(m) (k<m< n)
(10)
and thus
(11)
Thus
(12)
and n ^ n
o(s,
D
m),
i. e.
m- m — 1
Dk(m) m [. . 2aa , ., __
ml m —11 m'
(13) ^#g(l+,)A(m)
m-
if m ^ m0{s) and n i= na(e, m). Applying the Lemma of Szele this implies
that lim —'-—^- exists.
n ->- co ft
It should be mentioned that the authors of the present paper have
applied the lemma of Szele with success to other combinatorical and number-
theoretical questions too.
519 BLOCK DESIGNS

On some combinatorical problems.
405
Bibliography.
11] E. Netto, Lehrbuch der Combinatorik, 2. Aufl. Noten von Th. Skolem p. 328—329.
(2] H. B. Mann, Analysis and design of experiments, New-York, Dover, 1949.
[3] T. Szele, Kombinatorikai vizsgalatok az iranyitott teljes graffal kapcsolatban, Math.
Phys. Lapok 50 (1943), 223—256.
[4] A. Renyi, Neh^ny kombinatorikai problemarol, meiyek a lucernanemesft£ssel kapcsolato-
sak, Mat. Lapok, 6 (1955), 151—163.
[5] A. Inqham, On the difference between consecutive primes, Quart. J. Math. Oxford 8
(1937), 255—266.
(Received October 20, 1955.)
520 COMBINATORIAL ANALYSIS

On a limit theorem in combinatorical analysis
By P. ERDOS (Budapest) and H. HANANI (Haifa)
Given a set E of n elements and given positive integers k, 1,(1'■%k^n), we
understand by M(k, I, n) a minimal system of £>tuples (subsets of E having k elements
each) such that every /-tuple is contained in at least one fc-tuple of the system.
Similarly we denote by m(k, /, n) a maximal system of fc-tuples such that every
/-tuple is contained in at most one set of the system. The number of Ar-tuples in these
systems will be denoted by M(k, /, ri) and m(k, /, n) respectively.
Further we denote
rk\ (k>
fi(k, /, n) = M(k, /, n) ■ -)-( , v(k, /, n) =m(k, /, n)
,// \l
Trivially
(1) ' v(k,l,n)?k\?ku(k,l,n)
holds. It can also be easily verified that the equalities in (1) can hold only if
n — h
l—h
(2) -) (- = integer, (h =0, 1, ..., I- 1),
'k — h\
J-hJ
(see e. g. [4]). So far it has been proved that under condition (2) the equalities in
(1) hold for / = 2, k = 3, 4, 5 (see [5]) and for / = 3, k = 4 (see [4]). R. C. Bose suggested
that perhaps the equalities in (1) hold for / = 2 and every k if n satisfying (2) is
sufficiently large.
On the other hand it has been already conjectured by Euler [2] and proved
by Tarry [9] that for / = 2, k = 6 and 11 — 36 the equalities in (1) do not hold though
the condition (2) is satisfied.
For general n the problem has been solved completely by Fort and Hfdixnd
[3] for the case / = 2, Ar = 3.
Erdos and Rf.nyi [1] proved that for every k
(3) limn(k,2,n) = yk
exists with
(4) lim yk --= 1
521 BLOCK DESIGNS

P. Erdos and H. Hanani: On a limit theorem in combinatorical analysis 11
and moreover that for k=p and k = p + 1 (where p is a power of a prime)
(5) yP=yPHi = i.
It can be easily seen that the two statements
(6) limfi(i,/,n) = l, lim v(k, I, n) — 1
are equivalent and it may be conjectured that (6) holds for every k and /.
We shall prove that (6) holds for / = 2 and every k and also for / = 3 and fc = p + 1.
Theorem 1. For every integer k (k^2):
(7) l\mn(k, 2, n) = lim v(k, 2, n) = \.
Proof. By (6) it suffices to prove
(8) limv(fc, 2, n) = l.
We fix the integer k and assume that
(9) limv(it,2,»)=l-£.
We show that for every positive integer d
(10) lim v(k, 2, dn) = lim v(k, 2, n) = 1 - e.
Trivially
(11) lim v(k, 2, dn) Slim v(fc, 2, n).
Further let ? = dn -+ r, (r < rf) then
m(k, 2, t) ^m(k, 2, dn)
iind therefore
Consequently
v(k, 2, ,)-v(*. 2,dn).^^ .
lim v(k, 2, 0 Slim v(k, 2, dn)
and from (11), (10) follows.
Suppose that n = kg where g is a multiple of (A:!)2. Divide the set E having n
elements into k sets E; (i= 1, 2, .... k) of g elements each. It is well known [8, 5]
that there exist g2 ^-tuples such that each of them has exactly one element in each
Ei and any two of them have at most one element in common.
We form the system m(k, 2, n) by taking the mentioned g2 fc-tuples and further
by taking all the /r-tuples of the systems m(k, 2, g) constructed on each of the sets
Et {1=-1.2,..., k).
522 COMBINATORIAL ANALYSIS

12
P. Erdos and H. Hanani
If g is sufficiently large we have by (9), v(k, 2, g) > 1 — \z and thus
fc(fc-l)V 2
v(k,2, n)~
^+*f£4 M-
1 2k
n(n— 1)
which contradicts (10).
Theorem 2. If p is a power of a prime then
(12) limn(pJr\, 3, n) = \\mv(p + 1,3, n) = \.
Proof. We shall use the notion of a finite Mobius geometry introduced by
Hanani [6]. If p is a power of a prime then a Mobius geometry MG(p, r) is a set
of />r + 1 elements forming a Galois field in which circles are defined as bilinear
transformations of any line of the corresponding finite Euclidean geometry EG(p, r\
to which the additional element °° has been adjoined. It is proved that any triple
of elements in MG(p, r) is included in exactly one circle and that every circle has
p + 1 elements. Using this construction our proof will be basically on the same lines
as the proof of the theorem for / = 2 given by Erd6s and Renyi [1] except for a
simplification.
By (6) it suffices to prove
(13) limv(/?+-l,3, n) = 1.
For ii = pr + \, MG(p,r) exists and therefore
(14) v(/> + l,3,/>r + n = i.
By a simple computation it can be verified that to every £>0 there exists an 7
depending on e only such that
(15) v(p + l,3,n)>l-c, (// + 1^-=//(1+-7)).
Take all the prime-powers q{
(16) pr = q0<qx <q2<... <?, S//(l+7).
By the theorem of Hocheisfx and Ingham [7] we have for // sufficiently large
07) ?i+i-?i<tf'8.
For every /, (/ = 0, 1, ..., t) form the Mobius geometrices MG(qt,s) where 5 runs
through all the integers between (log q0)2 and q\,Ar. We have
v(?,+1,3,^+1) = 1, (/ = 0, 1, ...,0
and by (15) and (16)
(18) v(/7+l,3,<?f + l)gv(4ri+l,3,?f+l)-v(/? + l,3,4ri+l)>l-£,
(/ = 0,1,...,?; (logq0)2Ss^q'<<).
From (17) it follows that for .s-'^V4
(19) W+n)>qi+1
523 BLOCK DESIGNS

On a limit theorem in combinatorial! an.il>sis
13
■and therefore for n satisfying q]- //:.-.-.7--. ,, (/ = 0. 1 /) it follows from (15) and
(IS)
(20) v(p-r\.3,.n.)--\-2i:
and consequently (20) holds for e\ery 11 satisfying qf-,- ny^q].
Considering q.;q0 - 1 + ',/? and .s - (log <70): it follows (q,!qoy -q„ and therefore
4cV'^C {Oogq0V---sr-q'0^_
Consequently (20) holds for every 11 satisfying
(21) gru»E</oi-i+' . „ q\>,\JJ.
Denote b\ 1,. the inletsal defined in (21). It remains to be pro\ed that for sufficiently
large r the intervals 1,. overlap. This means that
(fji-^ IjKlogp'- H:j-H ._:q[<l, '1
which is evident.
Bibliography
[1] P. Erdos and A. Renyi. On some combinatorial problems. Puhl. Mulh. Debrecen 4 (1956).
398-405.
[2] L. Euler. Recherches sur une nouvelle espece de quarres magiques. L. Euleri Opera Omnia.
Teubncr, ser. 1. vol. 7 (1923). 291—392.
[3] M. K. Fort. Jr. and G. A. Hedlund, Minimal coverings of pairs b> triples. Pacific ./. Math.
8 (1958), 709-719.
[4] H. Hanani. On quadruple systems, Cauad. J. Math. 12 (1960). 145-157.
[5] H. Hanani, The existence and construction of balanced incomplete block designs. Ann. Math.
Statist. 32 (1961). 361-386.
[6] H. Hanani, On some tactical configurations, Cunuil. J. Math. 15 (I963J. 702-722.
[7] A. Ingham, On the difference between consecutive primes. Quart. J. Math. Oxford Ser. 8 (1937).
255-266.
[8] H. F. MacNmsh. Eulcr squares. Aim. of Math. 23 (1922). 221 227.
[9] G. Tarry. Lc problcme des 36 officiers, C. R Assoc. Franc. Ay. Sci. 1 (1900). 122- 123 and 2
(1901). 170-203.
(Received January 25, 1962)
524 COMBINATORIAL ANALYSIS

Chapter 12
Tournaments
In these papers, tournament refers to round robin tournaments where every
player plays every other player and there are no draws—in mathematical
terms, complete directed graphs. Topics on Tournaments by J. W. Moon
includes the results of these papers and many other similar results. In addition,
ES covers these papers.
Papers [399] and [424] involve the existence of tournaments in which
every k players are beaten by some player. Paper [450] concerns the optimal
ranking of players in a tournament. The results of [450] were improved by
Joel Spencer, "Optimal ranking of tournaments," Networks 7, 135-138.
Papers in Chapter 12
[399] On a problem in graph theory
[424] (with L. Moser) A problem on tournaments
[450] (with J. W. Moon) On sets of consistent arcs in a tournament
525 TOURNAMENTS

ON A PROBLEM IN GRAPH THEORY
By P. Eedos
1. Suppose there are n towns every pair of which are connected
by a single one-way road (roads meet only at towns). Is it possible
to choose the direction of the traffic on all the roads so that if any
two towns are named there is always a third from which the two
named can be reached directly by road? It will follow from the
proof of (1) that no such choice is possible if n < 6. With seven
towns the choice can be made. Let T0, Tlt ..., T6 denote the seven
towns; take as outgoing roads from Ta the roads leading to T^.-^,
Ta+2 and Ta+i, with the convention that Ta+h denotes Ta+h_7 if
a -(- b > 7. Because the differences between the numbers 1, 2, 4 are
all the numbers ±1, ±2, ±3, it follows that ifO<?;<?7-(-A<6
then h or h — 7 is among these differences and consequently that
527 TOURNAMENTS

ON A PROBLEM IN GRAPH THEORY
221
both Tv and Tv+h are included among Ta+1, Ta+2, Ta+i for some a:
if h = 1 or 3 take a = v -\- 6, if h = 2 or 6 take a = w -f- 5, if
A = 4 or 5 take a = v -\- 3.
The problem can be generalised by requiring that every k towns
(instead of every two) can be reached directly from a suitable
(k -\- l)th. Assuming that the problem is soluble for every k, we
then have the problem of finding the least possible value of n. If we
denote this least value by f(k), it is trivial that/(1) = 3; we have
indicated why/(2) = 7. The formula/(¾) = 2^+1 — 1 fits all these
cases and it may well be correct for all k. In this note we shall prove,
by simple counting arguments, that
f{k) > 2*+* - 1 for £=1,2,..., (1)
and
lim sup f{k)2~kk-* < log 2, (2)
*
the meaning of (2) being that if e is any positive number then Kc
exists such that
f(k) < 2¾2 log (2 + e) whenever k > Kc. (2.1)
The problem was recently put to me by Professor Schiitte in its
graph-theoretic form: If 3?(Ti) is a complete directed graph, with n
vertices, which has the property that for every k vertices of 2?(re>
there is at least one vertex from which edges go out to each of the
k, we shall say that rS(ri) has the property Sk. Schiitte's problem is
to show that for every k there is a 2?("> with the property Sk and to
find the least possible n for a given k.
It will be convenient to use the following terms. If E is a set of
vertices in a complete directed graph, and x is a vertex not in E,
then x will be called deficient for E if at least one edge starts in E and
ends at x; x will be called efficient for E if every edge joining x to a
vertex in E starts at x and ends in E. To say that ^(re) has the
property Sk means that for every E in 2?(n) with k elements there
is at least one x which is efficient for E.
2. Proof of (1). The proof is by induction. We know (1) holds
when k = 1. The existence of/(¾) for all k will follow from the
proof of (2) (see § 3). Suppose now that (1) holds for k = in — 1
where m is some integer exceeding 1; we have to prove that (1)
holds when k = m. Suppose it does not and let ^("> be chosen with
the property Sm and n < 211^1 — 2; we show that this leads to a
contradiction. For each vertex x of eS(rC) let ^("'(x) denote the set of
starting points of all edges of eS(n) which end at x. Since £/(re) has
\n{n — 1) edges, at least one S?(re)(x) has \{n — 1) or fewer elements.
Let | be a vertex for which ^(n)(|) has N elements with N <
[\(n — 1)]. Since n < 2m+i — 2, we have N < 2'" — 2. There are
now two possibilities, (i) N > m — 1, and (ii) N < m — 1.
528 COMBINATORIAL ANALYSIS

222
THE MATHEMATICAL GAZETTE
Suppose (i). We show that ^(n)(|) has then the property Sm__1,
which implies that N > 2m — 1 and so contradicts N < 2m — 2.
Let E be any set of m — 1 elements of ^(n>(f); since ^(n) has the
property Sm it includes a vertex r\ which is efficient for the set
E U (|); but this means that r\ is the start of an edge ending at f
and consequently tj e ^(n)(l); hence ^(n'(|) has the property Sm_v
Next suppose that (ii) holds. Add to ^(n'(|) any m — 1 — N
vertices of ^(n) other than f to obtain a subgraph S?(m-1> which has
the property Sm_1 since the m — 1 vertices of g^"1-1) together with
f form a set of m vertices in ^(n> for which, by hypothesis, there
exists an efficient vertex tj; as before, r\ e g^"1-1). It follows from
the induction hypothesis that m — 1 > 2m — 1, which is impossible.
In either case we reach a contradiction. When we have proved
(see § 3) that f(k) exists for all k, this will complete the proof
of (1).
3. To prove (2) it is convenient to use the language of probability.
Suppose k is given and V is any chosen set of n vertices. The
\n(n — 1) joins of the vertices in V can be directed in 2"(re_1"2 ways
to give a complete graph ^(Ti>. We have to show that if n is large
enough at least one ^(n) has the property Sk. Suppose that for a
given value of n none of the graphs ^(n) has the property Sk. Let E
be any one subset of V with k elements. The probability that any
one chosen x in V — E is deficient for E is (1 — 2~k) since there are
2k ways of directing the edges joining x to elements of E and only
one of these ways makes x efficient for E. Hence the probability
that all n — k vertices in V — E will be deficient for E is (1 — 2~k)"~k.
ln\
Now E itself can be chosen in I I ways and the probability that
one or more of these E has no efficient vertex is at most pn =
0
(1 — 2~k)n~k. Since, by hypothesis, none of the graphs has the
property Sk, pn > 1. Now
hence
i.e.
/n\ nk
and 1 — 2~k < exp ( — 2~k
n
k
1 < Pn < 7T exP ((^ ~ n)2~k) < nk exp ( — n'2~k),
l
1 log n
(3) «» > exp (2-¾). or —k <
Now (log n)jn decreases as n increases if n > 3, and so if e is any
positive number and
n > 2¾2 log (2 + e),
529 TOURNAMENTS

ON A PROBLEM IN GRAPH THEORY 223
(3) implies
W*<^Y with 3 = 2,:fc21og(2 + e)>
i.e.
fc log (2 + e) < log {2¾2 log (2 + e)}
or
log (2 + C) < log 2 + -log (fc2(2 + e)).
This is however impossible if fc is large enough, and this contradiction
establishes (2.1); the existence of/(fc) itself is a consequence of the
contradiction implied by (3) for all sufficiently large n.
We have used the language of probability in this proof because
it seems to make it more intuitive; it wovild not be difficult to
reformulate it in purely combinatorial terms.
University College, London. P. Erdos
530 COMBINATORIAL ANALYSIS

A PROBLEM ON TOURNAMENTS
P. Erdos and L,. Moser
(received November 14, 1963)
By a tournament we mean the outcome of a round-robin
tournament in which there are no draws. Such a tournament
may be represented by a graph in which the n players are
represented by vertices labelled 1,2,. . . , n, and the outcomes
of the games are represented by directed edges so that every
pair of vertices is joined by one directed edge. We call such
a graph a complete directed graph. One can also represent
such a tournament by an nXn matrix T = (t ) in which t
is 1 if i beats j, and 0 otherwise, so that T is a (0,1)
matrix with t + t = 1 for i 4 j and (by definition) t = 0.
ij J1 ii
In the summer of 1962 K. Schutte asked P. Erdos
the following question: Does there exist for every k, a
complete directed graph such that for every k vertices
x , x , ..., x there is one vertex y such that the edges
(x.y), i = 1, 2, . . . , k, are all directed away from y ? Erdos [1
1 2 k
proved that, provided n > (log 2 + € ) k 2 (e a positive
constant which can be taken arbitrarily close to 0 if k is
large enough), there do exist complete directed graphs with
this property. He also proved that such graphs do not exist
k+1
with n < 2 - 1. It is not obvious, and as far as we know
it has never been proved, that if such graphs exist for a given
n then they must also exist for every m > n.
At the seminar of the Canadian Mathematical Congress
in Saskatoon in August, 1963, H. Ryser asked the following:
Canad. Math. Bull. vol. 7, no. 3, July 1964.
351
531 TOURNAMENTS

Is it true that in every tournament matrix, there is a set of 4
or fewer columns, such that every row has at least one 1 in
at least one of these columns. L,. Moser showed that the
answer is no and in fact showed that for every large n, there
are tournament matrices in which for every set of
[log n - 21og log n] columns there is some row which has no
1 in any of these columns. He also showed that there does
exist, in every nXn tournament matrix, [log (n+1)] columns
such that every row has a 1 in at least one of these columns.
He further observed that for n > n (k, £ ) there are nXn
o
tournament matrices in which for every k columns there are
£ rows such that the kXi submatrix determined by these
columns and rows consists entirely of zeros. It is easy to see
that our results, which were obtained independently, are closely
related. By our methods we can obtain, almost without any
essentially new ideas, somewhat stronger results.
Consider a tournament on n players 1, 2, . . . , n. Pick
k of them, say x , x , ..., x . Clearly one of the other
k
players, y, can obtain 2 different sets of results with the
players x , x x . Now we prove
12 k
2 k
THEOREM 1. Let n > (log 2 + € ) k 2 . Then there
exists a positive a = a(e ) so that for each £ < k and every
choice of I players x , x . . . . , x , each of the 2 classes
12 £
in which the remaining n-£ players are divided (two players
are in the same class if they perform in an identical way against
£
the players x , x , ..., x ) contains more than an/2 players,
for all but o(2 ) of the tournaments.
By a slightly more complicated calculation we can prove
THEOREM 2. For every r| > 0 there is a c = c (r|)
2 k * l
such that for n > c k 2 and any £ < k players x , x , . . . , x ,
£ £ * 2 l
each of the 2 classes contains (1 + 6)n/2 players, where
|6| < r\, for all but o(2 ) tournaments.
352
532 COMBINATORIAL ANALYSIS

Theorem 2 can also be stated as follows. For every
r] > 0 there is a c = c (r\) such that in almost all tournaments
on n players, for every set of 4 players x , x , ..., x ,
each of the 2 classes will contain (l+6)n/2* players,
|&| < r|, provided t < log n - 21og (log n) - c .
Proof of theorem 1. The total number of tournaments
of n players is 2 Thus it will suffice to show that
the number of tournaments which do not satisfy the conditions
of theorem 1 is o(2 ). Further, a simple argument
shows that it will suffice to prove the theorem for I = k.
The k players x , x , . . . , x can be chosen in ways
and, as already stated, there are 2 classes into which the
remaining n-k players are decomposed. Let us fix our
attention on a particular set of k players x , x , . . . , x
12 k
and a particular class (i. e. , y is a member of the class if
he wins against a fixed subset of the x' s and loses against
the complementary subset). Let us determine an upper bound
for the number R(t) of tournaments in which our class contains
exactly t players.
First of all, only the games between x , x , ..., x
and the remaining n-k players are restricted by our conditions
so we have I l-k(n-k) unrestricted games and these yield for
12 l|n|_k#n_k>
R(t) a factor 2 . Next, the t players may be chosen
from the n-k players in J ways, and for the games
between the t players and x , x , . . . , x the outcomes are
determined. Finally, the games between x , x , ..., x
k
and any one of the remaining n-k-t players can go in 2 - 1
ways, since the only excluded case is if such a player is in the
given class with respect to x , x , ..., x . Hence
12 k
^)-k(n-k)|n-k| n.k.t
(1) R(t) < 2 I t J (2 - 1)
353
533 TOURNAMENTS

Since we are assuming t < [an/2 ] = L, , and since the k players
can be chosen in I ways and there are 2 classes, the total
number of tournaments S which do not satisfy the conditions of
theorem 1 fulfills the inequality
L
(2) S< [1) 2 2 R(t) .
_lk/ t = 0
To obtain an upper bound for S we note first that for k large,
Inl k k
12 < n and that in the range 0 < t < L,, R(t) is increasing
with t. Hence using (1) and (2) we obtain
and
(4) s<nkH2(2)|nj..(n.L,/2k,-kL
e "2
Our theorem will be established if we can show that
S=o(2n(n-1)/2) or
l*\ k+1 I*) ?-kL ^k ' #41
(5) n It2 e = o(l) .
Now, note that I 12 < (ne/L2 ) < (e/a) so we must still
prove only
/n-L
k+lfe\L \2k .
(6) n (-) e =o(l)
From
n > (log 2 + € )k2 2k
354
534 COMBINATORIAL ANALYSIS

we find
> (1 + € )k 2k
log n 1
and
(7) (k+ 1) logn <i- (1 - e2)
where € and e are positive numbers depending on € .
Taking logarithm of the left hand side of (6) and using (7) it is
seen that it only remains to prove that
(8) ^ - L(l - log a) - (1 - € ) 2-
2k L 2k
Since L, =na/2 and a(i - log a) — 0 as a — 0 the required
result follows.
We suppress the proof of theorem 2 since it is similar
to that of theorem 1.
By the method used in the proof of theorem 1 we can also
prove
THEOREM 3. Let e < - , n > n (e , k). Consider all
k o
r 2-e
incomplete tournaments on n players who play [n ] games.
The number of tournaments is | |. Almost all of
these tournaments contain, for each k players, at least one
player in each of the 2 classes.
Theorem 3 is not very far from being best possible since
., , . 2-1/k
if the number of games is en then we can show that for
almost all tournaments there are k players for which there is
no player who plays with all of them.
We conclude with two problems:
355
535 TOURNAMENTS

Problem 1. What is the minimum number of edges in a
graph of n vertices so that it can be directed in such a way
that to any k vertices x , x , ..., x there is a vertex y
12 k
such that all edges (x.,y), i = l,2,...,k are directed from
x. to y ? Of course we must assume here that n is large
enough that some complete directed graph has the required
property.
Problem 2. Let n > k. What is the smallest number
E(n;k) for which there is an ordinary graph of n vertices and
E edges in which for every set of k vertices, there is some
vertex, joined to each of these k.
We have solved this problem and hope to return to it.
REFERENCE
1. P. Erdos, Mathematical Gazette, 47 (1963) pp. 220-223.
University of Alberta
356
536 COMBINATORIAL ANALYSIS

ON SETS OF CONSISTENT ARCS IN A TOURNAMENT
P. Erdos and J. W. Moon
(received August 30, 1964)
1. Introduction. A (round-robin) tournament T consists
of n nodes u , u , . . . , u such that each pair of distinct nodes
12 n
u and u is joined by one of the (oriented) arcs u u or u u .
i J i J J i
The arcs in some set S are said to be consistent if it is possible
to relabel the nodes of the tournament in such a way that if the
arc u u is in S then i> j. (This is easily seen to be equiva-
i J
lent to requiring that the tournament contains no oriented cycles
composed entirely of arcs of S. ) Sets of consistent arcs are
of interest, for example, when the tournament represents the
outcome of a paired-comparison experirnent [l]. The object in
this note is to obtain bounds for f(n), the greatest integer k
such that every tournament T contains a set of k consisted
n
arcs.
2. A lower bound for f(n). In this section we show ■
for all positive integers n,
(1) f(n)>[|]. [^] ,
where, as usual, [x] denotes the largest integer not exceeding
This is trivially true when n = 1; suppose it has been
established for all n such that 1 <n <m - 1, and consider any
1
tournament T . Since such a tournament has a total of —m(m-l)
m 2
arcs, there must exist some node, say u , from which at least
m
Canad. Math. Bull. vol. 8, no. 3, April 1965
269
537 TOURNAMENTS

[—m] arcs issue. By definition, the tournament defined by the
remaining m-1 vertices contains a set S of at least f(m-l)
consistent arcs. It is clear that the arcs issuing from u and
B m
the arcs in S are consistent; therefore, appealing to the
induction hypothesis, it follows that T contains a set of at least
m
rm. rm- 1 , ,m rm , rm+l
consistent arcs. This suffices to complete the proof of (1) by
induction.
3. An upper bound for f(n). In this section we show
that for any fixed positive £ and all sufficiently large values
of n,
(2) f(n) < ^~ Cz) ■
Let e > 0 be chosen. In a tournament T there are n.'
n
ways of relabelling the nodes and N =(-.) pairs of distinct nodes.
N
Hence, there are at most n! ( ) such tournaments whose largest
set of consistent arcs contains k arcs. So, an upper bound for
the number of tournaments T which contain a set of more than
(1 +£ )N/2 consistent arcs is given by
<n! N2N([(l+£)N/2])([N^]y
(3)
, N (N-[N/2])(N-[N/2]-l) . . . (N -[(l+£)N/2] + 1)
" n- ([N/2] + 1) ([N/2] + 2) . . . [(l+e)N/2]
2
N -e N/4
< n! N2 e
270
538 COMBINATORIAL ANALYSIS

The last inequality of (3) follows from a simple computation
_ }£
using the fact that 1 - x < e for 0 < x < 1. But for all
sufficiently large n the last quantity in (3) is easily seen to
N
be less than 2 , the total number of tournaments with n nodes.
Hence, there must be at least one tournament T which does
n
not contain any set of more than (1 + e)N/2 consistent arcs.
This proves (2), by definition. With a more careful analysis
of inequality (3) this argument actually implies that
(4) f(u) < 1/2 (£) + (1/2 + o(l)) (n3log n)1/2
It would be desirable to obtain a better estimate for f(n).
The argument employed in the preceding paragraph
illustrates the usefulness of probabilistic methods in extremal
problems in graph theory, for while we can easily infer the
existence of a tournament with a certain required property we
are unable to give an explicit construction actually exhibiting
such a tournament in general.
4. A more general problem. Let G(n,m) denote an
incomplete tournament, or oriented graph, with n nodes and
m arcs. Let f(n,m) denote the greatest integer k such that
every incomplete tournament G(n,m) contains a set of at least
k consistent arcs. If it is assumed that n log n/m-»0 as n and
m tend to infinity then it can be shown, by arguments similar
to those used above, that
(5) lim f(n,m)/m = 1/2 .
n-«-oo
REFERENCE
1. M. G. Kendall and B. Babington Smith, On the method of
paired comparisons, Biometrika, 31 (1939) 324-345.
University College London
271
539 TOURNAMENTS

Chapter 13
Information Theory
Paper [405] considers the following problem: given n coins of two different
known weights how many weighings are required to determine the weight of
each coin? Further work on the problem was done by D. G. Cantor and W. H.
Mills, "Determination of a subset from certain combinatorial properties,"
Canad. J. Math., 75(1966)42-48, and by B. Lindstrom, "On a combinatorial
problem in number theory," Canad. Math. Bull, 5(1965)477-490. The
problem also appears in ES. While [405] involves probabilistic techniques, the
results of Cantor, Mills, and Lindstrom are constructive.
Paper in Chapter 13
[405] (with A. Renyi) On two problems of information theory
541 INFORMATION THEORY

ON TWO PROBLEMS OF INFORMATION THEORY
by
Paul ERDOS and Alfred RENYI
§ 1. Introduction
The first prob lem1 which will be discussed in this paper can be formulated
as follows: Suppose we are given n coins, which look quite alike, but of which
some are false. (For instance suppose that the right coins consist of gold, while
the false coins consist mainly of silver and are covered only by a thin layer of
gold.) The false coins have a smaller weight than the right coins; the weights
a and b < a of both the right and false coins are known. A scale is given by
means of which any number 5i n of coins can be weighed together. Thus if we
select an arbitrary subset of the coins and put them together on the scale, then
the scale shows us the total weight of these coins, from which it is easy to
compute the number of false coins among those weighed. The question is what
is the minimal number A(n) of weighings by means of which the right and false
coins can be separated? It can be seen by an elementary
information-theoretical argument that (denoting by log2x the logarithm with base 2 of x)
(1-1) A(n)^ r—rr-
log2(7l+ 1)
As a matter of fact, the amount of information needed is log2 2" = n bits,
because the subset of the coins consisting of the false coins may be any of the
2" subsets of the set of all n coins; on the other hand if we put k ^ n coins on
the balance, the number of false coins among them may have the values
0, 1, . ■., h and thus the amount of information given by each weighing can
not exceed log2(& -j- 1) 5a l°g2(w + 1)- Thus s weighings can give us at most
s log2(w + 1) bits, and thus to get the necessary amount of information (that
is n bits) it is necessary that s log2(w + 1) should be not less than n; thus we
obtain (1.1). On the other hand, a trivial upper estimate is
(1.2) ' A(n) ^ n
because if we put the coins one by one after another on the scale then
clearly these n weighings are sufficient. The inequality (1-2) is best possible for
n = 1, 2 and 3, but already for n = 4 we have A(4) = 3. As a matter of fact
1 This problem was proposed for n = 5 by H. S. Shapiro [1] and for arbitrary
n by N.J. Fine [2].
229
543 INFORMATION THEORY

230
ERDOS—RfiNYI
if we label the 4 coins by the numbers 1, 2, 3, 4 then the following 3 weighings
are always sufficient: we put first the coins 1, 2, 3 on the scale, then the
coins 1, 3, 4 and finally the coins 1, 2 and 4. Let the number of false coins
among the coins 1, 2, 3 be/1; that among 1, 3, 4 be/2 and that among 1, 2, 4 be
/3. The following table gives us the false coins:
/l
0
1
1
1
0
2
2
1
2
1
1
3
2
2
2
3
h
0
1
0
1
1
1
2
2
1
1
2
2
2
3
2
3
/a
0
1
1
0
1
2
1
2
1
2
1
2
3
2
2
3
1
—
+
—
—
—
+
+
+
—
—
—
+
+
+
—
+
2
—
—
+
—
—
+
—
—
+
+
—
+
+
—
+
+
3
—
—
—
+
—
—
+
—
+
—
+
+
—
+
+
+
4
—
—
—
—
+
—
—
+
—
+
+
—
+
+
+
+
Note that among the possible 64 triples /1; /2, /3 (0^/,-^13, 7=1,2,3)
only 16 are possible and each corresponds to a different distribution of
the false coins.
It is easy to see that in general one has
(1.3) A(nm) ^ A(n)-m
(because if we have nm coins we may determine by A(n) weighings from each
group of n coins the false ones). Thus from the above example one gets
A(4n) ^ 3ra
and as A(n) is evidently monotonic, we obtain
(1.4) 4(n)^|-^.J + 2
where {z} stands for the least integer ^ x
544 COMBINATORIAL ANALYSIS

ON TWO PROBLEMS OF INFORMATION THEORY
231
It may be guessed2 from this that one has
Aln)
(1.5) hm -^ = 0.
This is in fact true; moreover we shall prove in § 1 that the lower estimate (1.1)
gives the correct order of magnitude of A(n). We shall prove namely in § 2
(Theorem 1) that for any b > 0 we have for n ^. n0(b)
(1.6) A^^il+d)7^!
\og2n
It remains an open question whether the limit
A(n)\oe.,n
(1.7) hm —y——?i—= a
„^+» n
exists, and if it exists, what is its value? We shall prove in § 4 (Theorem 3)
that
(1.8) liminf^re)1°g^>2.
n-> +'
It follows from (1.8) that if the limit a in (1-7) exists one has 2 ^ a 5i
^ log2 9 ^ 3,17. We shall prove in § 5 that if the problem is modified so that we
are contented with finding a method of weighing which leads to the separation
of the false coins with a prescribed probability p < 1 which may be arbitrarily
near to 1, (supposing that all the 2" possibilities have same probability) then
2n
(1 4- e) weighings are sufficient for anv e > 0 if n is large (Theorem 4).
log2 n
Let us return now to the original problem of determining A(n). This
problem may be formulated as follows: We have to guess an unknown sequence
of n digits, each digit being equal to 0 or 1. We have the right to select arbitrary
,,testing" sequences of zeros and ones of length n and with respect of each
such sequence we arc told what is the number of places in which a 1 stands both
in the sequence to be guessed and in our testing sequence. The minimal number
of testing sequences by means of which the unknown sequence can be uniquely
determined whatever it may be, is equal to A(ri).
This reformulation of our first problem shows its connection with the
second problem which will be discussed in this paper and which is as follows:
Suppose we want to guess an unknown sequence of n digits, each digit being
either 0 or 1. Information concerning the unknown sequence may be obtained
in the following way: We have the right to select arbitrarily ,,testing"
sequences of digits consisting of zeros and ones and we are told the number of
places in which the two sequences coincide. Let B(ri) denote the minimal
number of sequences by means of which we can determine the unknown sequence,
whatever it may be. The problem is to determine the asymptotic behaviour of
B(n). Clearly we have
(1.9) B(n) ^
log2(/2 + 1)
This conjecture was stated in [2].
545 INFORMATION THEORY

232
EB.D(3S— RENYI
The inequality (1.9) is obtained by a similar information-theoretical
argument as that which leads us to (1.1). We have here the same trivial upper
estimate
(1.10)
B(n) ^ n .
As a matter of fact, if we select the n testing sequences 11 . . .1, Oil . . . 1,
1011 ...1,..., 111... 101 then if k is the. number of places in which the
sequence 11...1 and the unknown sequence coincide, then the number of
coincidences between the sequence Oil . . . 1 with the unknown sequence is
either k — 1 or h + 1 according to whether the first digit of the unknown
sequence is 1 or 0. Thus by the 2nd, 3rd, ..., n—th testing sequences we can
determine the first n — 1 digits of the unknown sequence; the last one can b&
determined because the total number k of ones in the unknown sequence is
known from the first comparison. We have clearly B(n) = n for n = 1, 2, 3, 4
and -8(5) = 4 as can be seen from the following example: using the testing
sequences
11111
11100
01010
01101
we can guess any sequence of 5 zero-or-one digits. As a matter of fact we get
for the number of coincidences the following values for the 16 sequences
consisting of not more than two ones:
sequence
00000
00001
00010
00100
01000
10000
00011
00101
01001
10001
00110
01010
10010
01100
10100
11000
11111
0
2
2
2
2
2
2
2
2
2
2
coincidence
11100
2
1
1
3
3
3
0
2
2
2
2
2
2
4
4
4
with
01010
3
2
4
2
4
2
3
1
3
1
3
5
3
3
1
3
01101
2
3
1
3
3
1
2
4
4
2
2
2
0
4
2
2
It is unnecessary to try the other 16 sequences with 3 or more onesr
because these are obtained by replacing 1 by 0 and 0 by 1 in the above 16
sequences, and this changes the number of coincidences from x to 5 — x.
546 COMBINATORIAL ANALYSIS

ON TWO PROBLEMS OF INFORMATION" THEORY
23a
We shall prove for B(n) the same inequality as obtained for A(n); viz. we
obtain in § 3 (Theorem 2) for any d > 0 that for n >. n^d)
(1.11) B(n) ^ (1 + d)7^9-.
\og2n
Here again the question remains open whether the limit
(1.12) hm -1^—— =/g
exists, and if so what its value is. We shall show in § 4 (Theorem 5) by the same
method which we have used to prove Theorem 3 that
,. . ,.i?(w) log, n
(1.13) liminf—y—±—e^->2.
Thus if the limit /3 in (1.12) exists, then certainly 2 ^, @ < log2 9 ^ 3,17. W&
shall prove also in § 5 that if we modify our second problem so that we want
to determine a fixed but unknown sequence of n zero-or-one digits by th&
number of coincidences with certain testing sequences and we are contented
with finding it with probability p < 1 which may be arbitrarily near to 1 then
2n
(1 + e) testing sequences are sufficient, (Theorem 6), for any e > 0 if re is
log2w
large enough.
Finally let us mention the following geometric interpretation of both
problems. To any sequence of zeros and ones there corresponds a vertex
of the unit cube Cn of w-dimensional space. The function B(n) can be
interpreted as follows: B(ri) denotes the least number such that by selecting B(n)
suitable chosen vertices of Cn each vertex of Cn is uniquely determined by its
distances from the chosen B(n) vertices.
Now let us interpret any sequence of n zeros and ones as a vector of the
re-dimensional space leading from the origin to one of the vertices of Cn. With
this interpretation A(n) denotes the least number such that by selecting A(n)
vectors vv v2, ■ ■ ■ , »A(n) leading to suitably chosen vertices of Cn each vector
v leading to a vertex of Cn is uniquely determined by its projection on the
A(n) chosen vectors, i. e. by the A(n) numbers (v, %), ■ • ■, (v, vA^n)) where
(v, w) denotes the inner product of the vectors v and w.
We prove our Theorems 1 and 2 by the same method, consisting in a
random selection of the testing sequences.
§ 2. An upper estimate for A(ri)
Our first problem can be formulated as follows: What is the least value
A(ri) of s such that there exists a matrix M having srows and n columns and
consisting of zeros and ones, such that if we select an arbitrary subset e of the
set i? of the columns of M, and form the row-sums of the submatrix M(e)
consisting of the selected colums of M, and denote by ve the column-vector
consisting of these row-sums, then the vectors ve and ve- are different if e and
e' are different subsets of E. We shall call such a matrix an 4-matrix.
547 INFORMATION THEORY

234
EKDOS—RfiNYI
Thus A(n) is the least value of s such that there exists an 4-matrix
with n columns and s rows. Clearly the matrix corresponding to the example
given in the introduction for n = 4 is the 4-matrix
1110
(2.1) 1011
1101
The 2i = 16 possible column-vectors ve are in this case
0111022121122233
0101112211223223
0110121212122323
and all are different, thus the matrix (2.1) is in fact an ^4-matrix.
In order to estimate A(n) we shall prove that if we choose M at random
so that the sn elements of M are independent random variables, each taking
on the values 0 and 1 with probability — , then the random matrix M will have
the required property with positive probability (in fact with probability
71 lot?. 9
tending to 1 for n —»- + °°) provided that s > (1+(5) —-----
log2w
This can be proved as follows: Let Psn(A) denote the probability that a
random matrix M of order sXn is an ^4-matrix, and put Qsn (4) = 1— Psn(A).
Let A(ev e2) (where % and e2 are different subsets of the set E of columns of M)
denote the event that the row-sum vectors vei and vei are identical. Evidently
if vei = vei and the sets e1 and e2 are not disjoint, then putting e[ = e± — e1 e.2
and e2 = e2 — eie2> we nave v^ = ve^. (Here and in what follows the product of
sets denotes their intersection and the difference e—/ of two sets e and /
denotes the set of elements of e which do not belong to/). It follows that is M
is not an ^1-matrix, then there exist disjoint subsets % and e2 of the set of its
columns such that vei = ve . Thus we obtain that
(2.2)
Q.,„(^U 2 P(A(ev^))
where the summation has to be extended over every pair of disjoint subsets
e1 and e2 of the set E of the columns of M and 0 denotes the empty set.
It follows that
(2.3)
Q.,n(^)
By the well known identity
■I
we obtain
( = 0
QsAA)
<
2
. m\n(k1,k.l)
v
k1 -|~ k2
„*1!**
h
Ok,
548 COMBINATORIAL ANALYSIS

ON TWO PROBLEMS OF INFORMATION THEORY
235
It follows
(2.4)
As
(2.5)
further
we obtain
(2.6)
s+l
r=l V I k=Q
<
{k = 0,1, . ..,r)
<
fr~+ 1
- for r = 3, 4, . . . (this follows easily by induction)
r=3 K' ' {r + l)2
Kow we choose s ~ ■ As we have
log2w
2
nloglogn
2r = 0(2 |0S2" )
rS
log? n
where c > 0 is a constant, it follows that
(2,7)
*<'>* 2 ;
9r
0(1)
logSn
(r + 1)*
Taking into account that > 2r = 3", we obtain
r-0 V r I
(2.8) QIiB(4) ^2-("*3-T) + -<">
provided that s = s(n)
(2.9)
a n
where a > 2 log., 3 = log2 9. It follows
log2w
lim Qs(n),n(4) = 0
Now clearly if Qs(n)i„ (A) < 1 then PS(n)A (A) > 0; as P, „(4) is the
probability that the random matrix M is an 4-matrix, it follows that if 6 > 0 and
s >
°2 then for n^nQ(b) there exists an 4-matrix of order sxn
\og2n
which implies A(n) ^ s. Thus we proved the following
Theorem 1. For any 8 > 0 we have for n 2i n0 (8)
(2.10)
A(n) ^ (1 + <5)
n log2 9
log2w
549 INFORMATION THEORY

236
ERDOS—UfiNYI
§ 3. An upper estimate for B(n)
Let u = (e-^ e2, . . ., sn) and u' = (s[, s'2, . . ., e'n) be row-vectors
consisting of n components, each of which is either 0 or 1. Let us put c(u, u') =
n
= n—^ (e, — e'i)2 Thus c(u, u') is the number of ,,coincidences" of the
vectors u and u', i.e. the number of columns of the 2x« matrix
e, ... s„
which consist of equal elements. By the usual notation || u ||2 = J£ ef we
i = i
have
(3.1) c(u, u') = n—|| u— -w' ||2.
Let now M be a matrix having s rows and n columns, and consisting of elements
0 and 1. Let uv . .., us be the rows of M interpreted as vectors. Let u be an
arbitrary row-vector consisting of n components which are either 0 or 1. Let U
denote the set consisting of all 2" such vectors u. To any u there corresponds a.
vector Wu consisting of the numbers c(u, m1), . . ., c(u, us). The matrix M will be
called a 5-matrix if the 2" vectors Wu corresponding to different elements u
of U are all different from each other. Thus if M is a 5-matrix, then each
vector u 6 U is uniquely determined by the s numbers c(u, %), . . ., c(u, us)
(and thus also by the distances \\u— uj ||, 7 = 1, 2, . . ., s). Let Psn(B)
denote the probability that the random matrix M of order sXn (whose
elements are independent random variables each taking on the values 0 and 1 with.
probability — should be a 5-matrix, and put Qsn(B) = 1 — Psn(B).
Let u = (e1( . . ., en) and u = (e[, . . ., e'n) be two arbitrary different
row-vectors consisting of n components each of which is either 0 or 1. Let
H and H denote the set of those indices k (1 ;£_& gL n) for which ek = 1 and
ek = 0, respectively and similarly let H' and H' denote the set of those
indices k (1 5i k gL n) for which £^=1 and £^ = 0 respectively. Let kuk2,Jc3 and ki
denote the number of elements of the sets HH', HH', HH' and H • H'
respectively. Let Uj = (#^, ..., #;n) be the 7-th row of the random matrix M and let
ljlt lj2, lj3_a.tia_lja denote^the number of those indices k which belong to the sets
HH', HH', HH' and HH' respectively and for which ftjk = 1. Clearly we have
c(u, Uj) = c(u', Uj) if and only if
f/i 4~ '/2 + ^3 73 4-^4 74 = f/i 4- ijz 4- ^½ f/2 4- kt ljt
that is if
(3.2) 2(lj2-lJ3) = h — i3-
It follows that a necessary condition for c(u, Uj) = c(u', Uj) is that k2—k3
should be even, and further that
k,+ k.
(3.3) Q,JB)< V 1 -
V I s,n\ I — ^ lc\lr\(n h k \\ \ 9k, + k,
l^k, + k,Sn
550 COMBINATORIAL ANALYSIS

ON TWO PROBLEMS OF INFORMATION THEORY
237
Now
k3
"•2 "-3
and thus
[t], ,
(3.4) Qiin(5)sV"2^
■*-V 2n
r=l
It follows similarly as in § 2 that if s = s(n) ^- a —-— where a>log29 then
log2w
(3.5) lim Qmn(S)^0.
Clearly if Qs^:n(B) < 1 then Ps^n)n(S) > 0 and thus there exists a 5-matrix
of order s(n) x'n and therefore B{n) < s(n). Thus we have proved the following
Theorem 2. For any d > 0 and n S: n^d) one has
<3.6) S(n) ^ (1 + d)7^2- .
log2w
§ 4. Lower bounds for A(n) and -B(w)
In this § we prove the following results
Theorem 3. One has
,,,-, ■ r A(ri) log, n
{4.1) lim inf ~U—^-^2.
n-> + ~ n
Theorem 4. One has
{4.2) lim inf *(")»°g."^2.
Proof of Theorem 3. Let M be now an arbitrary 4-matrix of order sXn.
Let us divide the row-vectors of M in two classes. A row-vector u of M belongs
to the first class if it contains less than h elements equal to 1, where h =
= ]pnXogn; otherwise it belongs to the second class. We shall give first an upper
estimate for the number of different column-vectors ve consisting of the row-
sums of the submatrix M(e) of M consisting of the columns of M belonging to
the set e\ here e is an arbitrary subset of the set E of columns of M. Clearly
any component o£ve corresponding to a row belonging to the first class may
take on at most h different values. On the other hand if a row Uj of M belongs to
the second class, and contains m ones (m ^. h) then the number of possible
choices of the subset e of the columns for which the sum of the elements of the
row Uj standing in the selected columns does not lie between the bounds
— £ J Ym log m (where the positive constant X will be chosen later) is equal to
(4.3) 2"~m ^
> X ym log m
551 INFORMATION THEORY

238
EKDOS—HENYI
According to the Moivre—Laplace theorem we have
' 2m
(4.4)
2
> a I'm log m
0
rrv-
Let us call a subset e of the set E of columns a ,,bad" subset, if there exists a
row Uj belonging to the second class and containing m ones, for which the sum
of the elements of w; standing in the columns belonging to e lies outside the
bounds — i h][m\ogm. Otherwise we call e a ,,good" subset. Clearly by
(4.4) if N denotes the number of bad subsets we have
»2"
h2i'
O
(4.5) N = Ol
Thus if X2 >. 1 we have
(4.6) N = 0\
2"
nW->) (log ny
2"
log «. J
On the other hand, denoting by V the number of different values of the vector
ve if e runs over the good subsets, we have
(4.7) V ^ hx[2^Ynk>gn]s-x^ (2lpi log n)
As M is by supposition an 4-matrix, the inequality
(4.8) V^2" — N
has to be valid, which implies by (4.6) and (4.7)
1
(4.9)
(2ZYn\ogn)s ^ 2" 1 — 0
(log n
Thus we obtain that the inequality
(4.10) s^_
2n
\og2n+0(loglogn)
holds, from which Theorem 3 immediately follows.
Proof of Theorem 4. Theorem 4 can be proved in a similar way as we
proved Theorem 3. The only difference is that the distinction between rows of
the first and second class is now unnecessary. Let M be a 5-matrix of order
sXn. Let U denote again the set of all possible rows of n elements each of
which is equal either to 0 or to 1. Let denote the rows of the
matrix M. An element uoiU will be called ,,bad" if there is a row Uj of M such
that the number of coincidences c(u, uj) of u and w;, does not lie in the interval
— ± XYn log n\ otherwise u will be called ,,good". If JVdenotes the number of
,,bad" elements u of U we have by Chebyshev's inequality
2"
(4.11)
N=0
log n
552 COMBINATORIAL ANALYSIS

ON TWO PROBLEMS OF INFORMATION THEORY
239
On the other hand if W denotes the number of possible values of the
vector wu = (c(u, wt), . . ., c(u, us)) where u runs over the „good" elements of
U, we have
(4.12) W ^ (2 fn log r,
As M is a 5-matrix, we have
(4.13) W^2" — N.
Thus it follows from (4.11) and (4.12) that
, ,> 2n
(4.14) s>
log2 n + O(loglog n)
which proves Theorem 4.
§ 5. Discussion of a modified form of both problems
Let U denote again the set of all possible sequences of length n consisting
of zeros and ones. Let M denote again a matrix of order sxn consisting of
zeros and ones; let uv . . ., us denote the rows of M. Let (w, u') where u 6 U,
u' € U denote the inner product of the vectors u and u', i.e. the number of
places in which 1 stands both in u and u''. Let c(u, u') = n —||w — w'||2 denote
the number of coincidences of the vectors u and u', i.e. the number of places in
which the same number stands both in u and u'. A matrix M will be called
a p-A matrix (resp. a p-B matrix) where 0 < p < 1 if by choosing at
random an element u of U (so that each of the 2" elements of U has the same
probability to be chosen) the probability that u is uniquely determined by the
sequence of numbers (u, w-J. (u,u2),.. ., (u,us) (resp.. by the sequence c^.Wj),
c(u, u2), ..., c(u, us)) exceeds p. Let sA(n, p) and sB(n, p) respectively denote
the minimal value of s for which a p-A matrix resp. a p-B matrix M of
order sxn exists. Then the following results hold:
Theorem 5. For any fixed p with 0 < p < 1 one has
(5.1)
Iim
SAin,P)
2n
Uog2 nj
(5.2;
Theorem 6. For any fixed p with 0 < p < 1 one has
sB(n,p)
Iim
2n
log2 nj
1 .
Proof of Theorems 5 and 6. Let M denote the set of all sxn matrices the
elements of which are zeros and ones. Let PA(M) and PB(M) resp. denote the
probability that by choosing at random an element u of U this element should
be uniquely determined by the sequence (w, w^, (u, u2), ■ ■ ■, (u, us) resp. by the
sequence c(u, u-^), c(u., u2), ..., c(u, us) where %, . . , us denote the rows of the
553 INFORMATION THEORY

240
EKDOS—B.15NY1
matrix M. Clearly the assertions that sA(n, p) iL s and sB(n, p) ±L s resp. are
equivalent to the assertions that
(5.3)
and
(5.4)
max PA(M) ^ p
max PB(M) ^ p .
MitA
Evidently if PA(M) and PB(M) denote the mean value of PA(M) and
PB(M) when 31 is chosen at random so that 31 may be equal to any element of
M with the same probability , then
(5.5)
and
(5.6)
max PA(3I) ^ PA(M)
AfgM
max PB(3I) ^ PB(31) .
AfgM
Thus if we prove that for a certain value of s we have
PjM)^P
and
PB(M) ^ P -
it follows that the inequalities sA(n, p) ^ s and sB(n, p) ^ s hold.
Let A(u, M) denote the event that the row vector u € U is uniquely
determined by the sequence (w,Wj), . . ., (u, us) and B(u, M) the event that the
row vector u (. U is uniquely determined by the sequence c(u, wa), . . ., c(u, us)
where uv . . ., us denote the rows of M. Then evidently
(5.7)
and
(5.8)
Pa(3I) = P(A(k,31))
PB(M) = P{B(u,M))
where on the right hand side of (5.7) and (5.8) u is a randomly chosen element of
U and M a randomly chosen element of M. Let us put
{5.9) l — P(A(u,M)) = QA{s,n)
and
(5.10) l — P(S(u,M)) = QB(s,n).
We obtain by a similar argument as that used in § 2 and § 3 resp.
ln — k\
(5.11)
QA(s,n)
2
n\ 1 -^-, lie
«Z\i
fc=0 ' * ' 2"/t7^ol *
1
I
2'+i
554 COMBINATORIAL ANALYSIS

ON TWO PROBLEMS OF INFORMATION THEORY
241
and
(5.12)
and therefore
" / \ "I 7 \
Q^n^2 ,h 2
. . .it 2'' . -«—< , i i
I 7'
f* +7\is
* + 7
91+/
" /„ i '
•*<•■->* 2 :?2G 2
2'
and
— (K^ft
[2?
I
22'
Using again the inequalities ^
n , / ? \ < 2<_
)/r+i
it follows that
(5.13)
and
(5.14)
H (^ + i)s'2-:
[t1
n
lB(s,n)^y til
<H (-21+ 1
f-i (2«+ l)s/2
Thus if s
an
lo
we, obtain
g2<
(5.15)
and similarly
nil- —\ o/"loglogn\
QA(s, re) ^2^ 2' " '"S" ^
„(t. °),o("t°8'°8")
< 2 V 2 / V logn /
(5.16) QB(s, w)^2"
Thus if a > 2 we have
(5.17) lim QA(s,n)= lim QB(s, n) = 0 .
By (5.9) and (5.10) this implies
(5.18)
lim P(A(v,M))= lim P(B(u,M)) = l
n-)-+oo n —+o=
J(j A Matematikai Kntatd Intezet KozlemiSnyei VIII. A/l-l!
555 INFORMATION THEORY

242
EB.DOS—RENY1
and thus by (5.5)—(5.8) if follows
(5.19) lim max PA(M) = lim max PB(M) = 1 .
As mentioned above this proves Theorems 5 and 6.
(Received July 28, 1963.)
REFERENCES
[1] Shapiro, H. S.: ,,Problem E 1399." American Mathematical Monthly 67 (1960) p. 82.
[2] Fine, N. I.: ,,Solution of problem E 1399." American Mathematical Monthly 67
(1960) p. 697.
Remark, added on September 24, 1963
Since the present paper was given to print, we have been informed that
quite a number of mathematicians worked on the first problem of the present
paper, and obtained results which are closely related to our results. None of
these results are published but some of them are in print. As W. Moseb informed
us, D. G. Cantob has proved in a paper in print in the Canadian Journal of
Mathematics the relation (1.5); in fact he obtained the estimate A{n) =
= 0\ —— . L. Moser (University of Alberta, Edmonton) informed us
that, lie obtained together with Abbot, that
A(n) log, n
(*) limsup - ->-- ;—e2__ ^ iog2 27 .
71^-1 ~ n
While this upper bound is greater by the factor 3/2 than our bound log29, the,
method of proof applied by Abbott and L. Moseb has the advantage that it is
n log 27
constructive; they exhibit effectively A- matrices of size sxn where s^ " .
log2 n
The same result has been obtained by H. S. Shapiro and S. Soderbebg.
Their paper is in print in the American Mathematical Monthly. E. R. Berlekamp
(Bell Telephone Laboratories) has obtained by a method, essentially the same
as our method, that
... n log, 9
A(n) g. 5i_ .
log2«
This result is slightly better than our result (1.6) (by the factor 1 + b). To get
rid of the unnecessary factor (1 + &) one has to use instead of the rough
estimate (2.5) a sharper estimate following from Stirling's formula.
Berlekamp conjectured also that (1.8) holds, and gave a heuristic
argument for his conjecture.
Other proofs of (1.8) have been given by B. Gordon (University of
California, Los Angeles) and L. Moser. E. Mills has also proved that
A(n) = 0\ . Quite recently B. Lindstrom (University of Stockholm)
[log n)
has proved the conjecture (1.7) with a. = 2. His paper will be printed in the
next issue of this journal.
556 COMBINATORIAL ANALYSIS

Part IV
Miscellany

Chapter 14
Random Objects
Paper [324] is the central paper of this section. The properties of a random
graph with n vertices and e edges are considered. The approach used is to add
random edges to an initially empty graph and ask at what point various events
occur. These events include the appearance of subgraphs of certain types, the
increase of chromatic number to certain levels, connectedness, nonplanarity,
and many others. The authors make strong use of probabilistic arguments,
especially the second moment method.
A review of the results of [324] is given, without proof, in [v]. Some of the
results of this section are included in ES. The other papers in this section are
either predecessors to or extensions of the results of [324].
This paper has received a good deal of attention in the social and
biological sciences. For example, the interrelationships among a group (of people or
primates) is sometimes represented as a random graph.
Most properties considered in these papers exhibit a strong threshold
property. That is, fixing n large, for most e the random graph will either
"almost certainly" or "almost certainly not" have the property. There is only
a small band of values for e in which the probability that the random graph
has the property lies between (say) .01 and .99. This "metaproperty" has been
very useful in applications.
Further results will appear in two forthcoming papers by Brown, Erdos,
and V. Sos Turan in the Proceedings of the Michigan Mathematics
Conference (Oct. 1971) and Periodica Mathematica.
Papers in Chapter 14
[309] (with A. Renyi) On random graphs. I
[v] (with A. Renyi) On the evolution of random graphs
[324] (with A. Renyi) On the evolution of random graphs
[337] (with A. Renyi) On the strength of connectedness of a random graph
[414] (with A. Renyi) On random matrices. I
[517] (with A. Renyi) On random matrices. II
559 RANDOM OBJECTS

On random graphs I.
Dedicated to O. Varga, ar rhe occasion of his 50* birt-hday.
By P. ERDOS and A. RENY1 (Budapest).
Let us consider a "random graph" f»,.v having n possible (labelled)
vertices and N edges; in other words, let us choose at random (with equal
probabilities) one of the | ^ > | possible graphs which can be formed from
the n (labelled) vertices Plt P.,,. .., Pn by selecting hedges from the 2
possible edges PiPj (1 ^ '</ = n). Thus the effective number of vertices of
r„, * may be less than n, as some points Pi may be not connected in r,h x
with any other point P7; we shall call such points PL isolated points. We
consider the isolated points also as belonging to r„,x. -T»,.v is called
completely connected if it effectively contains all points Pu P.,, ..., Pn (i, e. if it
has no isolated points) and is connected in the ordinary sense. In the present
paper we consider asymptotic statistical properties of random graphs for
/j_,.-j_oo. We shall deal with the following questions:
1. What is the probability of rn,x being completely connected?
2. What is the probability that the greatest connected component
(subgraph) of rn,x should have effectively n—k points? (/: = 0,1,...).
3. What is the probability that r„,x should consist of exactly k+\
connected components? (/: = 0, 1,...).
4. If the edges of a graph with n vertices are chosen successively so
that after each step every edge which has not yet been chosen has the same
probability to be chosen as the next, and if we continue this process until
the graph becomes completely connected, what is the probability that the
number of necessary steps v will be equal to a given number /?
As (partial) answers to the above questions we prove the following
four theorems. In Theorems 1, 2, and 3 we use the notation
y n log n + c n
(1) K
where c is an arbitrary fixed real number ([x] denotes the integer part of x).
561 RANDOM OBJECTS

P. ErdSs and A. Renyi: On random graphs I.
291
Theorem 1. Let P()(n, N,) denote the probability of F„. .Vc being
completely connected. Then we have
(2)
lim P0(n, jV,) =
tl->- fOD
Theorem 2. Let P,,(n, Nt) (k = 0, 1,. ..) denote the probability of the
greatest connected component of F„.\ consisting effectively of n — k points.
(Clearly Pu(n, K) lias the same meaning as before.) Then we have
(3)
lim Pk(n,Nc)-.
k\
i. e. the number of points outside the greatest connected component of F„. *
is distributed in the limit according to Poisson's law with mean value e -''.
Theorem 3. Let JT,,(n, AQ denote the probability of F„.X(. consisting
exactly of k + \ disjoint connected components. (Clearly lJ0(n, N,_) == Pn(n, N,).)
Then we have
(4)
lim nb(n,Nc)-
k\
i. e. the number of connected components of F,hy diminished by one is in the
limit distributed according to Poisson's law with mean value e-'.
Theorem 4. Let the edges of a random graph with possible vertices
P,, P.,,..., Pn be chosen successively among the possible edges PiPj in such
a manner, that at each stage every edge which has not yet been chosen has
the same probability to be chosen as the next, and let us continue this process
until the graph becomes (completely) connected. Let r„ denote the number of
edges of the resulting connected random graph F. Then we have
(5)
for !/,
(6)
P '
0(n) and
1
n log n
' n
lim P
»'»-
1
n log n
<x
= e
As regards the first question, previously P. ErdOs and H. Whitney have
obtained some less complete results. They proved that if AT> —+ « nlogn
where «>0 then the probability of F„,.v being connected tends to 1 if
562 MISCELLANY

292
P. Erdos and A. Renvi
n—» + oc, but if N< - — ,<•\nlogn with *■ > 0 then the probability of /"„ _v
being connected, tends to 0 if n —► + <». They did not publish their result.
This result is contained in Theorem 1.
Let G,: y denote a graph having n vertices and N edges. Let C(n,N)
denote the number of (completely) connected graphs G„..y. Of course the
solution of the above problems would be easy if a simple formula were
known for C(n, N). As a matter of fact, e. g. the probability Pu(n, N) of l„. .v
being completely connected is given simply by
(7) />.(„,*)-«"•">
Unfortunately such a simple formula is not known. The special case N== n — 1
has been considered already by A. Cayley [1] who proved that C(n, n—-1) =
==- n"'2 (for other proofs see O. Dziobek [2], A. Prufer [3], G. Poly a [4]).
For the general case, it has been shown by R. I. Riddell and G. E. Uhlen-
beck [5] (see also E. N. Gilbert [6]), that
C(«jv^= f^^o+^l
(8) 2 2-^^= log - ,^ „
Relation (8) is a consequence of the following recursive formula for C(n,N):
-m)
(9) I1 2 >\=2&y2C(m + l,M)"
Unfortunately neither (8) nor (9) helps much to deduce the asymptotic
properties of C(n, N). In the present paper we follow a more direct approach.
All four theorems are based on the rather surprising Lemma given below.
Let us call the graph r„, .y, with the n possible vertices P, ,P.,,...,P„
and N„ edges, of type A, if it consists of a connected graph having n — k
effective vertices and of k isolated points (£ = 0,1,...). Any graph -T„,-v,
which is not of type A shall be called to be of type A. Then the following
lemma holds.
Lemma. Let P(A, n, Nt) denote the probability of r„,xc being of type A.
Then we have
(10) hm P(A, n,Nt) = 0.
Thus for a large n almost all graphs /",._V(. are of type A.
563 RANDOM OBJECTS

On random graphs I.
293
Proof of the Lemma. Let M be a large positive number, which will
be chosen later. All graphs G„.yc having the possible vertices P, P.,,..., Pn
and Nc edges, where Nc is defined by (1) shall be divided into two classes:
Those in which the greatest connected component consists of not less than
n — M points shall constitute the class EM, all others the class Eu- Let
9t(£.v, n, A/c) denote the number of graphs G„,xc belonging to the class Em,
i. e. the number of graphs in which the greatest connected component
consists of less then n — M points.
If the graph consists of r connected components having /; points
(/ = 1,2, ...,r) then
2 A = n and £ (¾ §= A/,
and therefore if L =
= max /; we have
L— 1 .
2 =
2A7
and thus L> -; this implies that
n
2N,.
(11) Mr&n-
Therefore we have
(12) lUEx,n,Nf)^ Z 2X
because if the n—5 points belonging to the greatest connected component
of the graph are fixed, the s(n—s) edges connecting one of the points of
this component with a point outside this component can not belong to the
graph. Thus, denoting by P(Eir,n,Nc) the probability of r,hN<, belonging to
the class EM, we have
(13) PiE„,n,K)=*&£jy-m Z,
D 20
564 MISCELLANY

294
P. Erdos and A. Renyi
We obtain for n ^ n0, by using some elementary estimations,
-s(n—s) |
(14)
and
(15)
Nc
e(:s ins
n
2
K
~s(n — s)
si
for 5
n
~2
n
2
\Nj
We obtain from (13), (14) and (15)
(16) P(Ev,n,Nc)^ X
e(3-2<0(»-s)
(ns)\
for 5:
Thus we have
(17)
5! +/^/ s\
lim P(E\ogiogn, n, Nc) ---=0.
for n s: n0.
Thus to prove our Lemma it is sufficient to show that denoting by. ^fiogiog..
the class of those graphs which belong to both A and £iogi0gH and by
P(AE\0g\ogll) the probability that r»,yc belongs to the class AE\og\ogu we have
(18) lim P{AEXogXogn, n, Ne) = 0.
n ->- + CO
Now we have
(19)
log log 1!
P(AEiOgl0gn,n,Nt)^ £
Ml
n — 5
2
Mr'
n
2
K
because if the n—5 points forming the greatest connected component of a
graph belonging to the class AE\og\ogu are fixed, then if r is the number of
edges connecting some of the 5 points outside this connected component we
must have rsl, and the r edges in question can be chosen in v ' ways,
\ r
565 RANDOM OBJECTS

On random graphs I.
295
and the remaining Nc—r edges which connect points of the connected com-
n — 5
ponent having n—5 points can be chosen in less than I l ^ | ways. Thus
\Nc — r,
we obtain
(20) P(AbiogiogH,n,Ne)^-%— Z r,— =
2c —(log log")2
n 7^2 s\ n
which proves (18). Thus our lemma is proved.
Now we turn to the proof of our theorems.
Proof of Theorem 1. Let cK0(n, Nc) denote the number of completely
connected graphs G,,,xc. Then 9L0(n, Ne) is equal to the number of graphs
G„, xc of type A having no isolated points. Now let Tt'0{n, N,) denote the
number of all graphs Gn,s0 (including graphs of type A) having no isolated
point. Then clearly
(21) rC0(n,Nt)^±(-\f[nk)\y 2
As the left hand side of (21) is contained between any two consecutive
partial sums of the sum on the right of (21), and for any fixed value of k we
have
::)(^) ,
(22) lim
- +C0
i)\ k] '
Thus we obtain
N,
TL'0(n, N.) ^ (-1)^-^ =e_
(23) ^^77^==^ *,
But clearly
(24) 0 g ^(N. Nf)-lljn, Nr) ^ p{1> ^ ^
2j
\ Nr
Thus, applying our Lemma, Theorem 1 follows.
566 MISCELLANY

296
P. Erdos and A. Renyi
Proof of Theorem 2. According to our Lemma we have to consider
only graphs of type A. Now the number of graphs Gn,x„ of the type A
having a connected component consisting of n—k points is clearly equal to
L. multiplied by the number of connected graphs Gu.ki lV,- Thus it follows that
(25)
P„(n,Ne)
Pv(n-k,N,).
Taking into account that for a fixed value of k
Nr — ±-(n — k)log(n—k)
lim
with respect to (22) and (2) we obtain the assertion of Theorem 2.')
Proof of Theorem 3. As we may restrict ourselves to graphs of type
A, Theorem 3 follows from Theorem 2 immediately.
1
n log n
+ l=--N+l then be-
Proof of Theorem 4. Clearly if vH =
fore choosing the last edge, we had a disconnected graph G„, v which can
be made completely connected by adding one edge. With respect to our
Lemma we may suppose that G„, iV consists of a connected graph having
n — 1 vertices and an isolated point. As the last edge can be chosen in n — 1
ways among the remaining I ^
1
■N edges it follows from Theorem 2
(26)
provided that
P\r„
/I
n
y n log n
is bounded.
+ 1
-P(n,N)<
') Compare Theorem 2 with the following known result (see [7], Exercise No. 7 of
Chapter IV, p. 134): If N balls are distributed at random in n boxes and Rk{n, N) denotes
the probability that exactly k boxes reniain empty, then we have for any fixed real x and
for £ = 0, 1, ...
li m /?, («, n log n + x n) -= -—'— .
567 RANDOM OBJECTS

On random graphs I.
297
Thus (5) is proved. To prove (6) we have only to sum the
probabilities (5) for l<nx and obtain
'dt.
Theorem 4 is proved.
The following more general questions can be asked: Consider the
random graph /\ .v(,o with n possible vertices and N(ri) edges. What is the
distribution of the number of vertices of the greatest connected component
of r„. .v(„i and the distribution of the number of its components? What is the
typical structure of l\ y(n) (in the sense in which, according to our Lemma,
the typical structure of /\.vc is that it belongs to type ,4)? We have solved
these problems in the present paper only in the case AT(/z) =--- — « log n +en.
We schall return to the general case in an other paper [8].
2
Bibliography.
[1] A. Cayley, A theorem on trees, Quart. J. Pure Appl. Math. 23 (1889), 376—378.
(See also The Collected Papers of A. Cayley, Cambridge 1897, Vol. 13, pp.
26-28.)
[2] O. Dziobek, Eine Formel der Substitionstheorie, S.—B. Berlin. Math. Ges. 17 (1917)
64-67.
[3] A. Proper, Neuer Beweis eines Satzes iiber Perrautationen, Arch. Math. Phys. 27 (1918)
142—144.
[4] G. Polya, Konibinatorische Anzahlbestimmungen fiir Gruppen, Graphen und chemische
Verbindungen, Acta Math. 68 (1937), 145—255.
[5] R. I. Riddell, Jr. and G. E. Uhlenbeck, On the theory of the virial development of the
equation of state of monoatomic gases. J. Chem. Phys. 21 (1953), 2056—2064.
[6] E.N. Gilbert, Enumeration of labelled graphs, Canad. J. Math. 8 (1956), 405—411.
[7] A. Renyi, Valoszinusegszami'tas, (Textbook of probability theory) Budapest, 1954
(Hungarian).
[8] P. Erdos and A. Renyi, On the evolution of random graphs, Publications of the
Mathematical Institute of the Hungarian Academy of Sciences 5 (1960) (in print).
(Received November 19, 1958.)
568 MISCELLANY

ON THE EVOLUTION OF RANDOM GRAPHS
by
P. ErdOs and A. RENYI
Institute of Mathematics
Hungarian Academy of Sciences, Hungary
1. Definition of a random graph
Let En, n denote the set of all graphs having n given labelled vertices Vi,V2,---,
Vn and A' edges. The graphs considered are supposed to be not oriented, without
parallel edges and without slings (such graphs are sometimes called linear graphs).
Thus a graph belonging to the set E„, N is obtained by choosing A' out of the
possible (2) edges between the points Vi, Vi, •••, V„, and therefore the number of
elements of En, N is equal to ( 12 J). A random graph rn,N can be defined as an
element of En, n chosen at random, so that each of the elements of En, ,\ have the
same probability to be chosen, namely 11((2)). There is however an other slightly
different point of view, which has some advantages. We may consider the
formation of a random graph as a stochastic process defined as follows: At time £ = 1
we choose one out of the (j) possible edges connecting the points V'i, l^,"", Vn,
each of these edges having the same probability to be chosen ; let this edge be denoted
by ei. At time t = 2 we choose One of the possible ('2')—1 edges, different from <?i,
all these being equiprobable. Continuing this process at time t = k + l we choose
one of the (2) —k possible edges different from the edges ei, ei, •••, ek already
chosen, each of the remaining edges being equiprobable, i.e. having the probability
1/((2)-^}. We denote by rn, n the graph consisting of the vertices V\, V2, ■■-,
Vn and the edges ey, e2, ■■■, eN.
1) Other not equivalent but closely connected notions of random graphs are as follows:
1) We may define a random graph /'*, .v by dropping the restriction that there should
be no parallel edges; thus we may suppose that ek + l may be equal with probability
1 /(") with each of the (^) edges, independently of whether they are contained in the
sequence of edges e\, e«, -■ ■, ex or not. These random graphs are considered in the paper
[3]. 2) We may decide with respect to each of the f") edges, whether they should form
part of the random graph considered or not, the probability of including a given edge
being p~ A^/fgjfor each edge and the decisions concerning different edges being
independent. We denote the random graph thus obtained by /'** v. These random graphs
have been considered in the paper [4].
569 RANDOM OBJECTS

344
Theorie de l'information
The two definitions are clearly equivalent". According to the second definition
the number of edges of a random graph is interpreted as time, and according to
this interpretation we may investigate the evolution of a random graph, i. e. the
step-by-step unravelling of the structure of rn, y when AT increases.
The evolution of random graphs may be considered as a (rather simplified)
model of the evolution of certain real communication-nets, e. g. the railway-,
road- or electric network system of a country or some other unit, or of the growth
of structures of anorganic or organic matter, or even of the development of social
relations. Of course, if one aims at describing such a real situation, our model of
a random graph should be replaced by a more complicated but more realistic model.
The following possible lines of generalization of the considered stochastic process
of the formation of a random graph should be mentioned here :
a) One may distinguish different sorts of vertices, and / or edges—by a usual
terminology one may consider coloured vertices resp. edges.
b) One may attribute different probabilities to the different edges ; this can be
done e. g. by attributing a weight, We ^ 0 to each of the (!j) possible edges e so
that ) We = l and to suppose that ei is equal to the edge e with probability We and
that after <?i, ei, ■■■, ek have been chosen, ek+i is equal to any edge e not occurring
among the edges eu e2, ■■■, ek with probability ~^r where Sk= J We . An
'7=', (>=1,2, -.,*)
other alternative is to admit that the probability of choosing an edge e — (Vit Vj)
after k other edges have already been chosen, should depend on the number of
edges starting from the points Vi resp. Vj which have already been chosen.
In what follows we consider only the simple random graph-formation process,
described above, i.e. we consider only the random graphs Fn, jv.
Our main aim is to show through this special case that the evolution of a
random graph shows very clear-cut features. The theorems we have proved belong
to two classes. The theorems of the first class deal with the appearance of certain
subgraphs (e. g. trees, cycles of a given order etc.) or components, or other local
structural properties, and show that for many types of local structural properties A
N(n)
a definite "threshold" A(n) can be given, so that if . —> 0 for n—> + oo then
/\{n)
the probability that the random graph /'„, A-(n) has the structural property A tends
to 0 for n—-> + oo while for „ , . —> + 00 for n—> + oo the probability that A,, «(„>
A(n)
has the structural property A tends to 1 for n—> + oo. In many cases still more
can be said : there exists a " threshold function " for the property A, i. e. a proba-
N(ri)
bility distribution function FA (x) so that if lim ——-— — x the probability that ln,Nui
„->+«> A(n)
has the property A tends to Fa (x) for n—> + oo.
The theorems of the second class are of similar type, only the properties A
considered are not of a local character, but global properties of the graph /'„, n
(e. g. connectivity, total number of components, etc.).
In the next S we briefly describe the process of evolution of the random graph
rn, a-. The proofs, which are completely elementary, and are based on the asymptotic
evaluation of combinatorial formulae and on some well-known general methods of
probability theory, are published in the papers [1] and [2].
570 MISCELLANY

P. Erdos and A. Renyi
345
2. The evolution of Pn, n
If n is a fixed large positive integer and A' is increasing from 1 to('2'), the
evolution of Pn, n passes through five clearly distinguishable phases. These phases
correspond to ranges of growth of the number N of edges, these ranges being
defined in terms of the number n of vertices.
Phase 1. corresponds to the range A'(n) — o(n). For this phase it is
characteristic that Pn, A(n) consists almost surely (i. e. with probability tending to 1 for
n —>+oo) exclusively of components which are trees. Trees of order k appear only
when N(n) reaches the order of magnitude n*-i (k = 3, 4,---). If A'' (n) ~ pn * -'
with [>>0, then the probability distribution of the number of components of Pn, a-(»>
which are trees of order k tends for n—* + oo to the Poisson distribution with mean
(2 /if'1 k k~2 N(n)
value -? = — ~r . If l~^T —> + oo then the distribution of the number of
« ■ n*-i
components which are trees of order k Is approximately normal with mean
kk~2 f 2N{n)
Mn = n—rr~[ ) e " and with variance also equal to M„. This result
holds also in the next two ranges, in fact it holds under the single condition that
M„—> + oo for n—> + oo.
Phase 2. corresponds to the range N{n)~cn with 0<c<l/2.
In this case /'„, a(„) already contains cycles of any fixed order with probability
tending to a positive limit: the distribution of the number of cycles of order k in
(2c)k
Pn, k<7<) is approximately a Poisson-distribution with mean value ~k~f- In this
range almost surely all components of Pn, /v(n) are either trees or components
consisting of an equal number of edges and vertices, i. e. components containing
exactly one cycle. The distribution of the number of components consisting of k
vertices and k edges tends for n—>-)-oo to the Poisson distribution with mean value
(2 ce'2cik / k2 kk~3 \
r~. (1 + &+ ;~-\ hTT—^TTJ- In th's phase though not all, but still almost
all (i.e. n — o(n)) vertices belong to components which are trees. The mean number
of components is n — N(n) + 0 (1), i.e. in this range by adding a new edge the
number of components decreases by 1, except for a finite number of steps.
Phase 3. corresponds to the range N(ri)~c n with c ^ 1/2. When N (n) passes
the threshold n/2, the structure of Pn, x(n> changes abruptly. As a matter of fact
this sudden change of the structure of Pn, N(n> is the most surprising fact discovered
by the investigation of the evolution of random graphs. While for N(n)~cn with
f<l,'2 the greatest component of /'„, a>(,o is a tree and has (with probability
tending Lo 1 for n—>-foo) approximately (log n— log log n ) vertices, where
a \ 2 J
a = 2 c — 1— log 2c, for N(n) ~ n/2 the greatest component has (with probability
tending Lo 1 for n—> + oo) approximately n2/3 vertices and has a rather complex
structure. Moreover for N(n) ~ en with c > 1/2 the greatest component of Pn, ,v(n>
has (with probability tending to 1 for n—> + oo) approximately G(c)n vertices, where
571 RANDOM OBJECTS

316
THEOKIE DE L'l.M-'ORMATION
4-00
i c kk~l
(clearly GU2)=0 and Urn G(c)=l).
Except this "giant" component, the other components are all relatively small,
most of them being trees, the total number of vertices belonging to components,
which are trees being almost surely n (1 — G (c)) + o (n) for c ^J 1/2.
As regards the mean number of components2', this is for A'(n) ~ en with
n / ,,. X2(c)\
t'> 1/2 asymptotically equal to —— I X (c) —I, where
(2) AT(c)= ^-^-^2 cC-2«)*=2c(l-C\c))
The evolution of /'„, a(„> in Phase 3. may be characterized by that the small
components (most of which are trees) melt, each after another, into the giant
component, the smaller components having the larger chance of "survival"; the
survival time of a tree of order k which is present in /n, woo with \'{n)~cn,
t">l,2 is approximately exponentially distributed with mean value nj2k.
Phase 4. corresponds to the range N(n)~c n log n with c ^ 1/2. In this phase
the graph almost surely becomes connected. If
n k — 1
(3) lW(-n)=^k log n+ ~1zk~n log log n+yn+°w
then there are with probability tending to 1 for n—> + oo only trees of order t^k
outside the giant component, the distribution of the number of trees of order k
e-2*i/
having in the limit again a Poisson distribution with mean value ———.
Thus for k = l, i. e. for
n
(4) N(n) = — log n+yn + o(n)
/'...Mm consists, with probability tending to 1 for n-> + °°, only of a connected
component containing n —O(l) points and a few isolated points, the distribution of
the number of these being approximately a Poisson distribution with mean value
c~2j. Thus in case (4) the probability that the whole graph /\,, .v(„> is connected
tends to e~c~-'J for n—>-poo and thus this probability approaches 1 as J increases.
This last result has been obtained by us already in 1958 (see [2]). The
probability of /'** being connected has been investigated by E. N. Gilbert (see [4]).
It should be mentioned that the investigation of f **■ can be reduced to that of
rn,x as follows3': /'** can be obtained by first choosing the value k of a random
variable •£> having the binomial distribution P(jQ — k) = \^'iJ\p''{l—pfi'K where
2) The mean number of components of /'*, .v has been investigated in [3]. Our results fur
f',i, v are however more far reaching.
3) This idea has been used by J. Hajek [5] in the theory of sampling from a finite population
who has shown in this way that the Lindeberg-type conditions given by us [6] for the
valMity of the central limit theorem for samples from a finite population are not only
sufficient but also necessary.
572 MISCELLANY

1'. EktWb AM) A. KENY1
3-17
/>=.Y,'G') and then choosing I'n,k. In this way one can -,how that the threshold and
the threshold function for connectivity of /',?* are the same as that of /'n,,v. It
should be mentioned that this does not follow from the inequalities given by
Gilbert [4].
Phase 5. consists of the range .Y(nj~(tt log »0 zv(n) where Zi-(n)—< + °°. In
this range the whole graph is not only almost surely connected, but the orders of
all points are almost surely asymptotically equal. Thus the graph becomes in this
phase "asymptotically regular".
References
Ul P. Erdos-A.Rcnyi : On the evolution of random graphs, Publications of the Mallienialical
Institute of the Hungarian Academy of Sciences 5 (I960) (in print).
\2\ P. ErdOs-A.Kcjnyi : On random graphs, Publications Mathematicae, (> (1959; 290-297.
[31 T. L. Austin, K. E. pagen, W. F. Penney and J. Riortlan : The numher of components in
random linear graphs. Annals of Mathematical Statistics, 30 (1959) 747-754.
[-1J E.N. Gilbert: Random graphs, Annals of Mathematical Statistics, 30 (1959'., 1141-11(4.
L5J J. Fhijek : Limiting distributions in simple random sampling from a finite population,
Publications of the Mathematical Institute of the Hungarian Acadcm\ of Sciences 5 (19()0)
(m print).
[(JJ P. Erdos and A. Rcnyi : On the central limit theorem for samples from a finite population,
Publications oj the .Mathematical Institute of the Hungarian Academy oi Sciences,
4 (1959) 49-61.
RESUME
Soil K„ ,\ l'ensemble de tovis les graphes possedants it sommets donnes et ayain
A arcs. Nous considerons seulenieiit des graphes non-orientes et sans boucles. Un
graph aleatoir /\,,\ est defini comme un element de l'ensemble E„,,v ehoisi au hasard
tel que tons les elements de E„\ ont la meme probability d'etre choisis.
Les auteurs considerent les proprietes probables de /\,,.v quand n et X tends
vers l'indni d'un tel fagon que X—X(n) est une fonction doiinee de n.
573 RANDOM OBJECTS

ON THE EVOLUTION OF RANDOM GRAPHS
by
P. ERDOS and A. RfiNYI
Dedicated to Professor P. Turdn at
his 50th birthday.
Introduction
Our aim is to study the probable structure of a random graph rn N
which has n given labelled vertices Plt P2 Pn and N edges; we suppose
i Tb
that these N edges are chosen at random among the
12
possible edges,
so that all I I = GnN possible choices are supposed to be equiprobable. Thus
if Gn jy denotes any one of the Gn N graphs formed from n given labelled points
and having N edges, the probability that the random graph rn N is identical
with Gn jy is . If A is a property which a graph may or may not possess,
we denote by P„ N (A) the probability that the random graph rn N possesses
the property A, i. e. we put PnN (A) =—2^ where AnN denotes the
number of those Gn N which have the property A.
An other equivalent formulation is the following: Let us suppose that
■n labelled vertices Pv P2, . . ., Pn are given. Let us choose at random an edge
71 \
possible edges, so that all these edges are equiprobable. After
among the
12/
n \
this let us choose an other edge among the remaining — 1 edges, and
continue this process so that if already k edges are fixed, any of the remaining
n\
— k edges have equal probabilities to be chosen as the next one. We shall
study the "evolution" of such a random graph if N is increased. In this
investigation we endeavour to find what is the "typical" structure at a given stage
of evolution (i. e. if N is equal, or asymptotically equal, to a given function
N(n) of n). By a "typical" structure we mean such a structure the probability
of which tends to 1 if n —>- 4- oo when N =N(n). If A is such a property
that lim P„iN(„) (A) —1, we shall say that ,,almost all" graphs Gn^n)
possess this property.
17
574 MISCELLANY

18
EHDOS—B&5YI
The study of the evolution of graphs leads to rather surprising results.
For a number of fundamental structural properties A there exists a function
A(n) tending monotonically to 4- °° for n ->- + °° such that
lim P„,SM(A)
li
Njn)
A(n)
lim —-—
+ <
If such a function J.(«) exists \vc shall call it a "threshold function" of the
property 4.
In many cases besides (1) it is also true that there exists a probability
distribution function F(x) so that if 0 < x < -f °° and x is a point of
continuity of F(x) then
(2)
"m Pn,m„>(A)= F(x) if
JV(n)
lim
«-+» Atn)
If (2) holds we shall say that A(n) is a ,,regular threshold function" for the
property 4 and call the function F(x) the threshold distribution function of the
property A.
For certain properties 4 there exist two functions Ax(n) and A2(n)
A (n)
both tending monotonically to 4- °° f°r n—>- -j- °°, and satisfying lim —^L_i = o,.
such that
(3)
lim P
0 if
n,,V(n)l
if
lim
lirn
JVfn)
^i(n)
42(n)
Nln)
A(n)
Az[v)
(4)
Clearly (3) implies that
lim Pn,NM(A) z
if
if
lim sup -- < 1
iV(7l)
lim inf
«- + » ^x(n)
> 1
If (3) holds we call the pair(41(w), A2(n)\& pair of "sharp threshold"-functions
of the property A. It follows from (4) that if (41(w), 42(w)) is a pair of sharp
threshold functions for the property A then Ax(n) is an (ordinary) threshold
function for the property A and the threshold distribution function figuring
in (2) is the degenerated distribution function
jy*)
for
for
x < 1
X > 1
575 RANDOM OBJECTS

ON TUB EVOLUTION Of B.VNDOM GRAPHS
19
and convergence in (2) takes place for every x =/=1. In some cases besides
(3) it is also true that there exists a probability distribution function G(y)
defined for -°° < y < -+- °° such that if y is a point of continuity of G(y) then
(5) Mm PrhNiJA) = G(y) if lim--U ll ' = y.
n- ■ =0 n-- - A2(n)
Jf (5) holds we shall say that we have a regular sharp threshold and shall call
G(y) the sharp-threshold distribution function of the property A.
One of our chief aims will be to determine the threshold respectively
sharp threshold functions, and the corresponding distribution functions for
the most obvious structural properties, e. g. the presence in rn lV of subgraphs
of a given type (trees, cycles of given order, complete subgraphs etc.) further
for certain global properties of the graph (connectedness, total number of
connected components, etc.).
In a previous paper [7] we have considered a special problem of this
type; we have shown that denoting by C the property that the graph is
connected, the pair GJn) =- w log ??, GJn) =« is a pair of strong threshold
2
functions for the property C, and the corresponding sharp-threshold
distribution function is e-e^2"; thus we have proved1 that putting
-V(n) = — n log n -+- y w-j- o(n) we have
(6) lim Pn,Mn)(C) = e-^iv (- oo < y < + oo).
In the present paper we consider the evolution of a random graph in a
more systematic manner and try to describe the gradual development and
step-by-step unravelling of the complex structure of the graph Pn N when
^V increases while re is a given large number.
We succeeded in revealing the, emergence of certain structural properties
of rn N. However a great deal remains to be done in this field. We shall call in
§ 10. the attention of the reader to certain unsolved problems. It seems to us
further that it would be worth while to consider besides graphs also more
complex structures from the same point of view, i. e. to investigate the laws
governing their evolution in a similar spirit. This may be interesting not only
from a purely mathematical point of view. In fact, the evolution of graphs
may be considered as a rather simplified model of the evolution of certain
communication nets (railway, road or electric network systems, etc.) of a country
or some other unit. (Of course, if one aims at describing such a real situation,
one should replace the hypothesis of equiprobability of all connections by
some more realistie hypothesis.) It seems plausible that by considering the
random growth of more complicated structures (e. g. structures consisting
of different sorts of "points" and connections of different types) one could
obtain fairly reasonable models of more complex real growth processes (e. g.
1 Partial result on this problem has been obtained already in 1939 by P. Eedos
and H. Whitney but their results have not been published.
576 MISCELLANY

20 ERDOr— REXYI
the growth of a complex communication net consisting of different types of
connections, and even of organic structures of living matter, etc.).
§§ 1—3. contain the discussion of the presence of certain components
in a random graph, while §§ 4—9. investigate certain global properties of a
random graph. Most of our investigations deal with the case when N(n) ~ en
with c > 0. In fact our results give a clear picture of the evolution of rn v(r))
when c = —— (which plays in a certain sense the role of time) increases.
In § 10. we make some further remarks and mention some unsolved problems.
Our investigation belongs to the combinatorical theory of graphs,
which has a fairly large literature. The first who enumerated the number
of possible graphs with a given structure was A. Cayley [1]. Next the
important paper [2] of G. Poly a has to be mentioned, the starting point of which
were some chemical problems. Among more recent results we mention the
papers of G. E. Uhlenbeck and G. VV. Ford [5J and E. X. Gilbert [6].
A fairly complete bibliography will be given in a paper of F. Harary [8].
In these papers the probabilistic point of view was not explicitly emphasized.
This has been done in the paper [9] of one of the authors, but the aim of the
probabilistic treatment was there different: the existence of certain types
of graphs has been shown by proving that their probability is positive. Random
trees have been considered in [14].
In a recent paper [10] T. L. Austin, R. E. Fagen, \V. F. Penney and
J. Riordan deal with random graphs from a point of view similar to ours.
The difference between the definition of a random graph in [10] and in the
present paper consists in that in [10] it is admitted that two points should
be connected by more than one edge ("parallel" edges). Thus in [10] it is
supposed that after a certain number of edges have already been selected.
the next edge to be. selected may be any of the possible
\ ~
the n given points (including the edges already selected). Let us denote such
a random graph by f*N . The difference between the probable properties
of rn N resp. T* N are in most (but not in all) eases negligible. The
corresponding probabilities are in general (if the number N of edges is not too large)
asymptotically equal. There is a third possible point of view which is in most
cases almost equivalent with these two; we may suppose that for each pair
of 11 given points it is determined by a chance process whether the edge
connecting the two points should be selected or not, the probability for
selecting any given edge being equal to the same number p > 0, and the decisions
concerning the different edges being completely independent. In this case of
course the number of edges is a random variable, having the expectation
71 \
p; thus if we want to obtain by this method a random graph having in
M J
X
the mean N edges we have to choose the value of p equal to . We shall
2
denote such a random graph by r**N. In many (though not all) of the problems
treated in the present paper it does not cause any essential difference if we
consider instead of rn N the random graph r**N.
edges between
577 RANDOM OBJECTS

ON THU EVOLUTION OF RAXDOM flRAPHrf 21
Comparing the method of the present paper with that of [10] it should
be pointed out that our aim is to obtain threshold functions resp. distributions,
and thus wc are interested in asymptotic formulae for the probabilities
considered. Exact formulae are of interest to us only so far as they help in
determining the asymptotic behaviour of the probabilities considered (which is
rarely the case in this field, as the exact formulae are in most cases too
complicated). On the other hand in [10] the emphasis is on exact formulae resp.
on generating functions. The only exception is the average number of connected
components, for the asymptotic evaluation of which a way is indicated in
§ 5. of [10]; this question is however more fully discussed in the present paper
and our results go beyond that of [10], Moreover, we consider not only the
number but also the character of the components. Thus for instance we
71
point out the remarkable change occuring at N <~— . If N ~nc with c < 1/2
then with probability tending to 1 for n —>■ -\- °° all points except a bounded
number of points of Pn N belong to components which are trees, while for
N ^rtc with c > - this is no longer the case. Further for a fixed value of
n the average number of components of P„iN decreases asymptotically in a
71 71
linear manner with N, when N rg —, while for N > — the formula giving
2 2
the average number of components is not linear in N.
In what follows we shall make use of the sysmbols O and o. As usually
a(n) = o (b(n)) (where b(n) > 0 for n = 1, 2, . . .) means that lim ! = 0,
n^-r°= b(n)
while a(n) = O (b(n)\ means that ----^- is bounded. The parameters on
b(n)
which the bound of •'——I may depend will be indicated if it is necessary;
b(n)
sometimes we will indicate it by an index. Thus a(n) = Oe(b(nj) means that
a n)
b(n)
K(e) where K(e) is a positive constant depending on e. We write
a(n) --^ b(n) to denote that lim -^-^- = 1.
n-^» b(n)
We shall use the following definitions from the theory of graphs. (For
the general theory see [3] and [4].)
A finite non-empty set V of labelled points Plt P2, . ■ ., Pn and a set
E of different unordered pairs (Pit Pj) with P,- £ V, Pj £V, i =f= j is called
a graph; we denote it sometimes by G={V,E}; the number re is called
the order (or size) of the graph; the points P1: P2, . . ., Pn are called the vertices
and the pairs (P,, P;) the edges of the graph. Thus we consider non-oriented
finite graphs without parallel edges and without slings. The set E may be empty,
thus a collection of points (especially a single point) is also a graph.
A graph G2 = {V2, E2} is called a subgraph of a graph G± = {Vx, 2¾
if the set of vertices F2 of G2 is a subset of the set of vertices F1 of G1 and the
set E2 of edges of G2 is a subset of the set E1 of edges of Gv
578 MISCELLANY

22 FiBDOS—BfiXTI
A sequence of k edges of a graph such that every two consecutive edges
and only these have a vertex in common is called a path of order k.
A cyclic sequence of k edges of a graph such that every two
consecutive edges and only these have a common vertex is called a cycle of
order k.
A graph G is called connected if any two of its points belong to a path
which is a subgraph of G.
A graph is called a tree of order (or size) k if it has k vertices is connected
and if none of its subgraphs is a cycle. A tree of order k has evidently k — 1
edges.
A graph is called a complete graph of order if it has k vertices and
edges. Thus in a complete graph of order k any two points are connected
by an edge.
A subgraph G' of a graph G will be called an isolated subgraph if all
edges of G one or both endpoints of which belong to G', belong to G'. A
connected isolated subgraph G' of a graph G is called a component of G. The
number of points belonging to a component G' of a graph G will be called the
size of G'.
Two graphs shall be called isomorphic, if there exists a one-to-one
mapping of the vertices carrying over these graphs into another.
The graph G shall be called complementary graph of G if G consists
of the same vertices Pv P2, . . ., Pn as G and of those and only those edges
(Pt, Pj) which do not occur in G.
The number of edges starting from the point P of a graph G will be called
the degree of P in G-
A graph G is called a saturated even graph of type (a, b) if it consists of
a -\- b points and its points can be split in two subsets T'x and \\ consisting
of a resp. b points, such that G contains any edge (P, Q) with P £ \\ and
Q £ F2 and no other edge.
A graph is called planar, if it can be drawn on the plane so that no two
of its edges intersect.
We introduce further the following definitions: If a graph G has n
2 A7
vertices and N edges, we call the number -■-- the "degree" of the graph.
??
2AT
(As a matter of fact is the average degree of the vertices of G-) If a graph
n
G has the property that G has no subgraph having a larger degree than G
itself, we call G a balanced graph.
We denote by P (. . .) the probability of the event in the brackets, by
M(|) resp. D2(|) the mean value resp. variance of the random variable |.
In cases when it is not clear from the context in which probability space the
probabilities or respectively the mean values and variances are to be
understood, this will be explicitly indicated. Especially M,liN resp. D%N will denote
the mean value resp. variance calculated with respect to the probabilities
■n.N-
579 RANDOM OBJECTS

(7)
OX THE EVOLUTION" OJ? RANDOM GBATHS 23
We shall often use the following elementary asymptotic formula:
k* k'
n
k
Our thanks are due to T. Gallai for his valuable remarks.
valid for I- = otn'!')
k I
§ 1. Thresholds for subgraphs of given type
If X is very small compared with n, namely if N —o (fn) then it is
very probable that rn.N is a collection of isolated points and isolated edges,
i. c. that no two edges of -TniN have a point in common. As a matter of fact
the probability that at least two edges of .TniN shall have a point in common
is by (7) clearly
If however N ^c \n where c > 0 is a constant not depending on n, then the
appearance of trees of order 3 will have a probability which tends to a
positive limit for n —*■ + °°> but the appearance of a connected component
consisting of more than 3 points will be still very improbable. If X is increased while n
is fixed, the situation will change only if K reaches the order of magnitude
of n2-3. Then trees of order 4 (but not of higher order) will appear with a
probability not tending to 0. In general, the threshold function for the presence
fc-2
of trees of order k is nk-[ (k = 3, 4, . . .). This result is contained in the
following
- 1 jg / iS be positive integers. Let
(2)1
,&ki, denote an arbitrary not empty class of connected balanced graphs consisting
of k points and I edges. The threshold function for the property that the random
graph considered should contain at least one subgraph isomorphic with some ele-
Theorem 1. Let k > 2 and I
2- k-
l
ment of &ktl is n
The following special cases are worth mentioning
Corollary 1. The threshold function for the property that the random, graph
k-2
contains a subgraph which is a tree of order k is nk{ (k = 3, 4, . . .) .
Corollary 2. The threshold function for the property that a graph contains
a connected subgraph consisting of k 3r 3 points and k edges (i. e. containing
exactly one cycle) is n, for each value of k.
Corollary 3. The threshold function for the property that a graph contains
a cycle of order k is n, for each value of k >. 3.
580 MISCELLANY

24
EBDOS—BfiNYI
Corollary 4. The threshold junction for the property that a graph contains
a complete subgraph of order k 2: 3 is n \ k~1' .
Corollary 5. The threshold function for the property that a graph contains
a saturated even subgraph of type (a, b) (i. e. a subgraph consisting of a -\- b
2-—-
points Px Pa, Qv . , . Qb and of the ab edges (Pit Qj) is n ab .
To deduce these Corollaries one has only to verify that all 5 types of
graphs figuring in Corollaries 1—5. are balanced, which is easily seen.
Proof of Theorem 1. Let Bk , > 1 denote the number of graphs
belonging to the class ,.<j@kj which can be formed from k given labelled points. Clearly
if Pn,N (Mk.i) denotes the probability that the random graph .TnN contains
at least one subgraph isomorphic with some element of the class .55^,, then
(1.1)
Pn,N(^6V()^
As a matter of fact if we select k points (which can be done in different
\2j
ways) and form from them a graph isomorphic with some element of the class
«Sgu (which can be done in Bkd different ways) then the number of graphs
G„iN which contain the selected graph as a subgraph is equal to the number
71 \
of ways the remaining N — I edges can be selected from the — I other
z
possible edges. (Of course those graphs, which contain more subgraphs
isomorphic with some element of £Skl are counted more than once.)
2-- '
Now clearly if iV = o(n ' \ then by
°n,N(A,l)
■0(1)
whieh proves the first part of the assertion of Theorem 1. To prove the second
part of the theorem let ^") denote the set of all subgraphs of the complete
graph consisting of n points, isomorphic with some element of 3&kl. To any
S^Mkn} let us associate a random variable s(S) such that e(8) = 1 or e(S) = 0
according to whether 8 is a subgraph of rnN or not. Then clearly (we write
in what follows for the sake of brevity M instead of M„n)
(1.2) m [2 ^)) = 2 M(£(N))
*<>
se,®
k.l
581 RANDOM OBJECTS

ON THE EVOLUTION OP RANDOM GRAPHS
25
On the other hand if 8X and 82 are two elements of ,¾¾¾ and if St and
82 do not contain a common edge then
M(6(£f1) a(Sa))
If 81 and iS2 contain exactly seommon points and r common edges (l<Lr^Ll—1)
we have
M(s(81)e(S2)) =
O
N21-r
nU-2r
On the other hand the intersection of jS^ and 82 being a subgraph of jS^ (and S2)
by our supposition that each 8 is balanced, we obtain — <l — i. e. s > —
s h I
and thus the number of such pairs of subgraphs 8X and 82 does not exceed
rk,
Bh 2
m r4
Thus we obtain
(1.3)
M
In — k
\k - j
= 0
2k-
((2^))1 =
«<}
= 2 M(e^)+
Now clearly
n\ Bh
k\2(n- 2fc)!
+ 01
N< \
,2(-/(
2 (
X1
N
582 MISCELLANY

26
ERDOS—BEXYI
If we suppose that
X
co —>- 4- °°
it follows that we have
(1.4)
1(2 M (*(£())*\
S£&
.(")
W J
It follows by the inequality of Chebysheff tbat
n,N
and thus
\2 e{8)- 2 M(e(8))i > > M(e[8))
**$
se<>
- se^g1
(n)
0
(1.5)
n,N
«<}
s6<>
= 0
As clearly by (1.2) if w —>- -)- °° then J^1 M(e(i8))—> -b °° it follows not onlv
«<}
that the probability tbat Pn N contains at least one subgraph isomorphic
with an element of &ki tends to 1, but also that with probability tending
to 1 the number of subgraphs of r N isomorphic to some element of c£k i
will tend to -f °° with the same order of magnitude as co1.
Thus Theorem 1 is proved.
It is interesting to compare the thresholds for the appearance of a
subgraph of a certain type in the above sense with probability near to 1, with
the number of edges which is needed in order that the graph should have
necessarily a subgraph of the given type. Such "compulsory" thresholds
have been considered by P. Turan [11] (see also [12]) and later by P. Erdos
and A. H. Stone [17]). For instance for a tree of order k clearly the compulsory
'fi(]c 2)1
threshold is + IJ f°r the presence of at least one cycle the
compulsory threshold is n while according to a theorem of P. Turan [11] for
(k — 2)
complete subgraphs of order k the compulsory threshold is (n2 — r2) -)-
— \ii J.)
+
where r
(k
1)
k
In the paper [13] of T. Kovabi,
V. T. Sos and P. Turan it has been shown that the compulsory threshold
■for the presence of a saturated even subgraph of type (a, a) is of order of magni-
2- '■
tude not greater than n " . In all cases the "compulsory" thresholds in
Turan's sense are of greater order of magnitude as our "probable" thresholds.
583 RANDOM OBJECTS

0_\' THE EVOLUTION OK EAVDOM GRAPHS 2 I
§ 2. Trees
Now let us turn to the determination of threshold distribution functions
for trees of a given order. We shall prove somewhat more, namely that if
fc-2
N. -~ g nk~} where g > (), then the number of trees of order k contained
in r„N has in the limit for n — ->■ -\- °o a Poisson distribution with mean value
(2 o)fc~! kk~2
a = --—■ . This implies that the threshold distribution function for
/,-!
trees of order k is 1 — e~'\
In proving this vv'e shall count only isolated trees of order k in rn^, i. e.
trees of order k which are isolated subgraphs of TnN. According to Theorem 1.
this makes no essential difference, because if there would be a tree of order
k which is a subgraph but not an isolated subgraph of r N, then rnN would
have a connected subgraph consisting of k + 1 points and the probability
of this is tending to 0 if AT = o {n k I which condition is fulfilled in our
k-2
ease as we suppose N -~ g nk~! .
'Thus we prove
Nln)
Theorem 2a. lj lixn k---^ = (?>() and tk denotes the number of isolated
nk ~!
trees of order k in Prl N(r!) then
)Je~l
(2-1) ]im Pr...V(n)(rft: = }) •-=
n --»-■- co
or j = 0, 1, ..., where
(2.2) X = ^-^\
k\
For the proof we need the following
Lemma 1. Let eriv eni,. . ., Enla be sets of random variables on some
probability space; suppose that eni(l < i ^. ln) takes on only the values 1 and 0. If
)j
(2.3) lim y M(e„,- e„;, . . . £„;,.) = —
n-.-,oo i<;,cij<-..</',■ ra(,i f!
uniformly in r for r = 1, 2, ..., where X > 0 and <Ae summation is extended
over all combinations (ilt i2, ..., «'r) °/ order r of the integers 1, 2, ..., Z„, </jew
?!
i«
(2.4) lim Pl2'cni = ?
(7 = 0,1,
«'. e. the distribution of the sum ~y eni tends for m^4-°° to the Poisson-distri-
bution with mean value /..
584 MISCELLANY

28
BRDOS-RfiSYf
Proof of Lemma 1. Let us put
(2.5)
Clearly
(2.6)
rn(i) = P\2>
2
z ?
l£i'i<'2<... <!>•;£(„
thus it follows from (2.3) that
j = r I r I
(2.7)
(j\ Ar
n - , co j=r \ ' I ' •
(r = 1, 2, . . . )
uniformly in r.
It follows that for any a with | z [ < 1
(2.8)
But
(2.9)
Hm 2\2Pn{i)V
n~-i-°> r_, l,-_r (r
0'= V-
e;~' - 1 .
"V
2X(?) , ^ = ^^(/)(1+^-1.
Thus choosing 2 =x — 1 with 0 < x < 1 it follows that
(2.10)
lim ^>r,(?)*;'= ^-0
for 0 < .r < 1
It follows easily that (2.10) holds for x — 0 too. As a matter of fact
putting Gn(x) = *yPn{j)xi, we have for 0 < x <^ 1
|-Pn(0) - e-»| ^ i G„(x) - e^-')| + [ &*„(*) - ^,,(0) | + | e^-») - e-*|.
As however
ancl similarly
it follows that
Thus we have
\Gn(x) - Prl(H)\ < * 2 Pn(i) < *
|P„(0) - e~* I ^ I Gn(x) - e^"') | + 2 x.
lim sup \Pn(0) - e-;'j ^. 2 a;;
as however x > 0 may be chosen arbitrarily small it follows that
lim P„(0) = e *
585 RANDOM OBJECTS

O-V THE KVOLTJTXOS OV K.VNDOM GRAPHS
29
i. e. that (2.10) holds for x
that
(2.11)
0 too. It follows by a well-known argument
)Je-x
lim P„{j) =
r-
(/ = 0,1,...),
As a matter of fact, as (2.10) is valid for x — 0, (2,11) holds for j =0. If
(2.11) is already proved for j < s — 1 then it follows from (2.10) that
(2.12)
lim yPn(j)xi-
Xh
-xJ-
for 0 < x < 1,
J=s
j--=s
n
By the same argument as used in connection with (2.10) we obtain that
(2.12) holds for x = 0 too. Substituting x = 0 into (2.12) we obtain that (2.11)
holds for j = s too. Thus (2.11) is proved by induction and the assertion of
Lemma 1 follows.
Proof of Theorem 2a. Let Tk") denote the set of all trees of order k which
are subgraphs of the complete graph having the vertices Pt, P2, ..., Pn.
If 8^T^ let the random variable e(8) be equal to 1 if 8 is an isolated subgraph
of rniN; otherwise e(8) shall be equal to 0. We shall show that the conditions
of Lemma 1 are satisfied for the sum 2J E{8) provided that N = N(n) ~
sgT(n)
k~2
k~-\
(2.13)
and 1 is defined by (2,2)
- k
I 2
M(e(S)) =
As a matter of fact we have for any
12 N\
2Nk
1 +0
'N\
More generally if 8lt 82, , . ,, 8r (Sj£ T^) have pairwise no point in common
then clearly we have for each fixed k ;§ 1 and r^ 1 provided that n—>-4-°°,
N —>-f °°
(2.14) M(s(S1)e(S2)...e(Sr))
/22V
C-l)r
2Nrk
l+O
n*1
where the bound of the O term depends only on k. If however the Sj {j
= 1, 2, .,,, r) are not pairwise disjoint, we have
(2.15)
M(s(81)B(S2)...E(Sr)) = 0.
586 MISCELLANY

30
ERDOS—BEXYI
Taking into account that according to a classical formula of Cayley [1]
the number of different trees which can he formed from k labelled points is
equal to kk~2, it follows that
(2.16) 2 M(e(/Sf1) 8(S2
°43r))
]ch-2
k\
2AT
2Nrt
1+0
2N
where the summation on the left hand side is extended over all r-tuples of
trees belonging to the set T("> and the bound of the O-term depends only on k.
Note that (2.16) is valid independently of how N is tending to -|~°o. This
will be needed in the proof of Theorem 3.
Thus we have, uniformly in r
(2.17)
lim
N(n)
M(£(^) 8(8,)
HSr))
r\
for r =1,2, ..
where A is defined by (2.2).
Thus our Lemma 1 can be applied; as tt = £ e{^) Theorem 2 is
proved. str*-"'1
We add some remarks on the formula, resulting from (2.16) for r = 1
(2.1*
M(t„
2^ _^
&*-2
2xV
ifc!
1 + 0
N
kk~2 tK~l e~kt
Let us investigate the functions mH(t) = (k—1, 2, .
k\
),
According to (2.18) nmk
2N\
is asymptotically equal to the average number of trees of
order k in rmN. For a fixed value of k, considered as a function of t, the value
of mk(t) increases for t < and decreases for t > ; thus for a fixed
k k
value of n the average number of trees of order k reaches its maximum for
n
N
the value of this maximum is
l\k-l
Mt
k
e-(fc-0 j.k-2
k\
For large values of k we have evidently
Ml.
Y^nk™
587 RANDOM OBJECTS

ON THE EVOLUTION OP RANDOM GRAPHS
31
It is easy to see that for any t > 0 wc have
mk(t) ^ mk+1(t)
{k = 1,2, .
The functions y = mk(t) are shown on Fig. 1.
It is natural to ask what will happen with the number rk of isolated
trees of order k contained in Tn N if
N(n)
w
k-2
-4-oo. As the Poisson distribution
'- 1 is approaching the normal distribution if 7. ->- + °°, one can guess
I ?! ) v ,
that tk will be approximately normally distributed. This is in tact true, and
is expressed by
~ kk-'(2cp'2c)
Finure la.
y^iee
588 MISCELLANY

32
ERDOS—Rfixri
Theorem 2b. If
(2.19)
but at the same time
N(n)
N(n)
~~'k-2~
nk~l
+
(2.20)
1 i-1
— w log n — w ioglog n
2k 2k
n
then denoting by xk the number of disjoint trees of order k contained as subgraphs
in rnNin) (k = 1, 2, .. .), we have for — °° < x < + °°
(2.21)
where
(2.22)
and
(2.23)
Urn P (7^ ~~ ^n.Mn) _^ „
llm rn,N(n) I - --=^- < X
"-+" I MA>,iV(r,)
0(X)
M
kk~
n,N
k\
2N\
k-l UN
0(¾) = —^ | e 2 du .
Proof of Theorem 2b. Note first that the two conditions (2.19) and
(2.20) are equivalent to the single condition lim MnN(ny= 4- °°, and as
M (rk) ^MnN this means that the assertion of Theorem 2b can be expressed
by saying that the number of isolated trees of order k is asymptotically
normally distributed always if n and N tend to +°° so, that the average number
of such trees is also tending to 4-°°' Let us consider
Mlri
M((2" e(S)Y).
S£T(n)
Now we have evidently, using (2.16)
MW
14-0
2JV-
n*
2
r\
h±lh2[ .. .Ay!/
MJn,N
where Mn_N is defined by (2.22). Now as well known (see [16], p. 176)
(2.24)
r\
\\ A2! ... Ay!
tU)
589 RANDOM OBJECTS

OX THE EVOLUTION OF RANDOM GRAPHS
33
where d-p are the Stirling numbers of the second kind (see e. g. [16], p. 168)
defined by
(2.25)
Thus we obtain
(2.26)
--^^px{X— 1)
MfTt
1+0
-2N
(x-j+1)
2<&mu
Now as well known (see e. g. [16], p. 202)
(2.27) e*^-1) —l=j?jP<tP-
/ = 1 r=;
Thus it follows that
r!
(2.28)
;'=i
r7r
dxr
x=0 /c-0
y± yapm
^1 r! \^i j
J-oo Jfc
We obtain therefrom
M
(2.29) M'
K^n.;
M
l + » /iff
>^r«,v(i-I„,N,
n,N/c=0
&!
1 + 0
r2Arl
Now evidently ^ r~ e~x (k — X)r is the r-th. central moment of the Poisson
distribution with mean value X. It can be however easily verified that the
moments of the Poisson distribution appropriately normalized tend to the
corresponding moments of the normal distribution, i. e. we have for /• = 1, 2,...
(2.30)
Jim
\k—i kl
--^= I xr e 2 dx.
f2 7I
In view of (2.29) this implies the assertion of Theorem 2b.
In the case N (n) = — nlogn + n loglog n + yn + o(n) when
the average number of isolated trees of order k in rn N(n) is again finite, the
following theorem is valid.
Theorem 2c. Let xk denote the number of isolated trees of order k in rn N
(k = 1, 2, . ..). Then if
(2.31)
N(n) = — n log n + n loglog n + yn + o(n)
2k 2k
where — °° < y < + °°, we have
(2.32)
where
(2.33)
lim P
„,Mn> (Tfc = /)
Vt-
(/ = 0,1,
-2fcy
*•*!
590 MISCELLANY

34
BRDOS—RfiNTl
Proof of Theorem 2c. It is easily seen that under the conditions of
Theorem 2c
lim Mn,NM{rk) = A.
n- + »
Similarly from (2.16) it follows that for r = 1, 2, ...
lim J" Mn,Mn) (e(^) e(82) . . . e(^)) = --
s<er
k
and the proof of Theorem 2c is completed by the use of our Lemma 1 exactly
as in the proof of Theorem 2a.
Note that Theorem 2c generalizes the results of the paper [7], where
only the case k = 1 is considered.
§ 3. Cycles
Let us consider now the threshold function of cycles of a given order.
The situation is described by the following
Theorem 3a. Suppose that
(3.1) N(n)^cn where c> 0.
Let yk denote the number of cycles of order k contained in rn N (&= 3, 4, .. .).
Then we have
(3-2) lim P„iMn) (yk = f) = -^- (/ = 0,1,...)
where
(3.3) I = i2-^ .
2 k
Thus the threshold distribution corresponding to the threshold function A(n) — n
(2c)*
for the property that the graph contains a cycle of order k is 1 — e 2k
It is interesting to compare Theorem 3a with the following two theorems:
Theorem 3b. Suppose again that (3.1) holds. Let y% denote the number of
isolated cycles of order k contained in PnN (k = 3, 4, .. .). Then we have
(3-4) lim PniNln){yfl = j)=t!£l (/ = 0ji,...)
where
(2 ce~2c)k
/!
(3.5) (x
2k
Remark. Note that according to Theorem 3b for isolated cycles there
does not exist a threshold in the ordinary sense, as 1 — e~>' reaches its maxi-
-— 1 I n)
mum 1 — e 2ke* for c=— i. e, for N(n) ^- and then again decreases ;
591 RANDOM OBJECTS

OX THE EVOLUTION OF RANDOM GBAPHS
35
thus the probability that FnN contains an isolated cycle of order k never
approaches 1.
Theorem 3c. Let bk denote the number of components of rn N consisting
of k ;§ 3 points and k edges. If (3,1) holds then we have
(3.6)
where
(3.7)
lim Pn.Mn) (du = 1)
(2ce~2c)k
ojj e~ ■'
V,
0' = o,i,
2 k
k2
l+k+~+ ..
2!
kk~
(k — 3)1,
Proof of Theorems 3a., 3b. and 3c. As from k given points one can form
(k — 1) ! cycles of order k we have evidently for fixed k and for N= O(n)
(3.8)
while
(3.9)
M(y,
1 in
2 \k
(k
1)1
M(y*
(k
D\
ri-i
\2,
n — k
2
N-k
(!)
2AT
2i
2.V
21c
As regards Theorem 3c it is known (see [1()] and [15]) that the number
of connected graphs Gk H (i. o. the number of connected graphs consisting
of k labelled vertices and k edges) is exactly
(3.10)
0k= 1(¾ _ i)! ji -|_fr + hl.
kk~3
(fc~3)!|
Now we have clearly
(3.11) M(d,
&,.
n —
k\\
2 )
N -k>
,tn
\\k
Lv
\
1
o V 2N
-e "
71
*
2&
£2 fc*-3
l+M h • • • H
2! (ife-3)!
592
MISCELLANY

36
KRDOS—BEN 11
For large values of k we have (sec [15])
71
(3.12)
and thus
(3.13)
0,.
-l*-'l,
8
12 N i-2-V
M(«*
4&
7£
For N ^— we obtain by some elementary computation using (7) that
for large values of h (such that k = o (re3/4).
(3.14)
M(\.)
4¾
Using (3.8), (3.9) and (3.11) the proofs of Theorems 3a, 31) and 3c follow
the same lines as that of Theorem 2a, using Lemma 1. The details may be
left to the reader.
Similar results can be proved for other types of subgraphs, c. g. complete
subgraphs of a given order. As however these results and their proofs have
the same pattern as those given above we do not dwell on the subject any
longer and pass to investigate global properties of the random graph rn v .
§ 4. The total number of points belonging to trees
We begin by proving
Theorem 4a. If N = o(n) the graph FnN is, with probability lending to
1 for w —>- 4-oo, the union of disjoint trees,
Proof of Theorem 4a. A graph consists of disjoint trees if and only if
there are no cycles in the graph. The number of graphs Gn N which contain
at least one cycle can be enumerated as was shown in § 1 for each value k
of the length of this cycle. In this way, denoting by T the property that the
graph is a union of disjoint trees, and by T the opposite of this property,
i. e. that the graph contains at least one cycle, we have
(4.1)
pniA, (T) gL y
(*
o
'N\
It follows that if N
o(n) we have lim P N(T)
If N is of the same order of magnitude as n i. e. AT -~ en with c .
then the assertion of Theorem 4a is no longer true. Nevertheless if c < 1/2,
1 which proves Theorem 4a.
0.
593 RANDOM OBJECTS

ON THE EVOLUTION OF RA.VDO.M GRAPHS
37
still almost all points (in fact n — 0(1) points) of rnN belong to isolated
trees. There is however a, surprisingly abrupt change in the structure of Tn N
with N ~ en when c surpasses the value— . If c > 1/2 in the average only a
positive fraction of all points of TnN belong to isolated trees, and the value
of this fraction tends to 0 for c -> 4- °° •
Thus we shall prove
Theorem 4b. Let Fn,v denote the number of those points of rnN which
belong to an isolated tree contained in Tn v. Let us suppose that
(4.2 hm —^ = c > 0 .
Then we have
M(In..V(„)) _ |
1 for c ^ 1/2
(4.3) lim - -'—i-^n'L = X(C\ 1
n — for c > —
( 2c 2
where x = x(c) is the only root satisfying 0 < x < 1 of the equation
(4.4) xe~x = 2ce~2c,
which can also be obtained as the sum of a series as follows:
(4.5) x(c)=£ (2ce~2c)k.
k = \ k-
Proof of Theorem 4b. We shall need the well known fact that the inverse
function of the function
(4.6) y = xe~x (0^ x^ 1)
has the power series expansion, convergent for 0 ^, y
(4.7) x= ^ V-.
*T, *!
Lot rk denote the number of isolated trees of order k contained in Pn N. Then
e
clearly
(4.8)
and thus
(4.9)
By (2.18),
(4.10)
if
(4.
2)
holds,
lim
n
Vn,N = JV k T,.
k = \
/( = 1
we have
1 1 l-k~2
(2ce~2c)k.
"-► + » n 2c kl
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38
BRDOS— R&S'YI
Thus we obtain from (4.10) that for c g 1/2
!!-►+» 71 ~"~ 2Cfc = ] k\
As (4.11) holds for any s > 1 w"e obtain
(4.11) lim inf -^^/- ^ — J? — - v~- --'- for any s ^ 1 .
(4.12)
lim inf
n-.+ »
M(F„.V,„,)^ 1 - fc*-'(2ce-*)*
__n,iV(n)_/ ~> _J_ "v
Jfc!
But according to (4.7) for c gi 1/2 we have
■ **-'(2ce-*)*^2c
Thus it follows from (4.12) that for c < 1/2
(4J3) Km inf lIIE^.^"!
lim inf
^ 1.
As however Vn_N(n) ^. n and thus lim sup
M{VntNM
< 1 it follows that
if (4.2) holds and c <L 1/2 we have
(4.14) lim WYji.Nsn)) = ! _
Now let us consider the case c > — . It follows from (2.18) that if (4.2)
holds with c > 1/2 we obtain
(4.15)
n2 n Jt-1
/2A» -^
2N(n). k
+ 0(1)
where the bound of the term 0(1) depends only on c. As however for N(n)
^nc with c > 1/2
2 —
k=n+1 Kl
2 AT(w) — 2 iV(rc)
0
( 1
it follows that
(4.16) M(Fn,Mn))
(N(n)
\n''")
n2 (N(n)\
2 N(n) [ n
+ 0(1)
where x = £ is the only solution with o < x < 1 of the equation
■2N(n) -^("^
we have
(4.17)
n
Thus it follows that if (4.2) holds with c > 1/2
lim
M(F„,Mn)) = *(c)
w 2c
where x(c) is defined by (4.5).
595 RANDOM OBJECTS

ON THE EVOLUTION OF RANDOM GEAPHS
39
The graph of the function x(c) is shown on Fig. la; its meaning is shown
by Fig. lb. The function
for c ^ 1/2
for c > 1/2
is shown on Fig. 2a.
x(c)
2 c
I for cs */2
y'\j^ forc></2
Figure 2a.
Figure 2b.
596 MISCELLANY

40
ehdOs—b&>tyi
Thus the proof of Theorem 4b is complete. Let us remark that in tho
same way as we obtained (4.16) we get that if (4.2) holds with c < 1/2 we have
(4.18)
M(F
n.Nin))
0(1)
where the bound of the 0(1) term depends only on c. (However (4.18) is not
true for c =— as will be shown below.)
2
It follows by the well known inequality of Markov
(4.19)
P(f >a)i£—M(f)
valid for any nonnegative random variable I and any a > M(|), that the
following theorem holds:
Theorem 4c. Let VnN denote the number of those points of rn N which
belong to isolated trees contained in rn N. Then if con tends arbitrarily slowly
to 4-°° for n^>- 4-°o and if (4.2) holds with c < 1/2 we have
(4.20)
lim P(F„,Mn) > n
1 .
The case c > 1/2 is somewhat more involved. We prove
Theorem 4d. Let Vn N denote the number of those points of rnN which
belong to an isolated tree contained in rn N. Let us suppose that (4.2) holds with
olj2. It follows that if a>n tends arbitrarily slowly to -j-°°> we have
(4.21)
where x = x
n,N(n)
2N(n)
N(n)
>v<
n a>,
0
N{n)
is the only solution with 0 < x < 1 of the equation
2N(n)
xe~
2iV|
Proof. We have clearly, as the series >' — (2ce
— h I
-2c\k
is convergent,
D2 (VnN(n))= 0(n). Thus (4.21) follows by the inequality of Chebyshev.
Remark. It follows from (4.21) that we have for any c > 1/2 and any
e > o
(4.22)
lim P
n-*-r o>
n,N(n)
n
2c
< e
1
where x(c) is defined by (4.5).
As regards the case c =1/2 we formulate the theorem which will be
needed latter.
597 RANDOM OBJECTS

ON THE EVOLUTION OF BANDOM GRAPHS
41
Theorem 4e. Let Vn<N(r) denote the number of those points of rnN which
belong to isolated trees of order > r and T„iN(r) the number of isolated trees of
n
order > r contained in -Tn A.. If N(n) -~ - we have for any b> o
(4.23)
and
(4.24)
lira P
lim P
\*as)_yW::e-k
V kT'r k\
Tn,N(n)\r)
■ " Jc"-2 ,
<8
<d
The proof follows the same lines as those of the preceding theorems.
§ 5. The total number of points belonging to cycles
Let us determine first the average number of all cycles in rnN. We
prove that this number remains bounded if N(n) ~ en and c < 1I2 but not
if c = i/2.
Theorem 5a. Let Hn N denote the number of all cycles contained in rn N.
Then we have if N(n) -~ en holds with c < —
2
(5.1)
while we have for c =--
2
(5.2)
lim M(H N(n)) = — log c-c*
n-T" 2 1 — 2 c
4
Proof. Clearly if yk is the number of all cycles of order h contained in
■Tn,N ve have
n
Hn,N = j>" 7k-
k = \
Now (5.1) follows easily, taking into account that (see (3.8))
(5.3) M(yfc)=lh(fc-1)!
u 1 A* /
n
2
N-k.
12N'
n )
2k \n
598 MISCELLANY

42
EH DOS— RENY1
If c = V2 we have by (3.8)
M(r,.1~
2fc
5.4) M(yJ~ — e
l _3,(°
n ! _*' 1
As y — e 2" ~ — logn, it follows that (5.2) holds. Thus Theorem 5a
k^3'2k 4
is proved.
Let us remark that it follows from (5.2) that (4.18) is not true for c = 1Ji.
Similarly as before we can prove corresponding results concerning
the random variable Hn N itself.
We have for instance in the case c = 1/2 for any e > o
(5.5) lim P
Hp.NM _ 1_ I £\
log n 4
This can be proved by the same method as used above: estimating the variance
and using the inequality of Chebyshev.
An other related result, throwing more light on the appearance of cycles
in rn N runs as follows.
Theorem 5b. Let K denote the property that a graph contains at least one
cycle. Then we have if N(n) ~ nc holds with c < 1j2
(5.6) lim PnMn)(K) = 1 - fl^Tce^'.
Thus for c =— it is ,,almost sure" that Fn N(n) contains at least one cycle, while
for c < —the limit for n—>-\- oo of the probability of this is less than 1.
2
Proof. Let us suppose first c < —. gy an obvious sieve (taking into
account that according to Theorem 1 the probability that there will be in rn N(n)
with N(n) -^ nc (c < 1\2) two circles having a point in common is negligibly
small) we obtain
_ - lim M(tf„,A>))
(5.7) lim Pn,N(n)(Z)=e »-+- = fl - 2 c e<+<8.
n—+»
Thus (5.6) follows for c < ^2- As for c —>- V2 the function on the right of (5.6)
tends to 1, it follows that (5.6) holds for c = x/2 too. The function y =
= 1 — yT~^2c ec+c' is shown on Fig. 3.
We prove now the following
Theorem 5c. Let H* N denote the total number of points of rnN which
belong to some cycle. Then we have for N = N(n) ~ en with 0 < c < xl2
(5.8) lim M(H*iN(n))=- 4c*
n —+» 1 — 2 C
599 RANDOM OBJECTS

ON THE EVOLUTION OF RANDOM GRAPHS
43
Figure 3.
Proof of Theorem 5c. As according to Theorem 1 the probability that
two cycles should have a point in common is negligibly small, we have by (5.3)
M(H*.NM)~2 kyk
(2 c)3
4 c3
k-l
2(1 — 2c) 1 —2c
The size of that part of rn N which does not consist of trees is still more
clearly shown by the following
Theorem 5d. Let &n N denote the number of those points of r„_N which
belong to components containing exactly one cycle. Then we have for N = N{n) ~
~ en in case c =f= 1/2
(5.9) lim M(0n>Mn)) = ^ +S (2 ce-2c)*
while for c = 1j2 we have
(5.10) M(#n,N(n)'
1 + ^ + ^ +
1! 2!
+
lck
(*-3)!
rlu
-,2 3
12
where r<x) denotes the gamma-function T(x) = § P-i e-'dt for x > 0.
600 MISCELLANY

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BR DOS—RENYI
Proof of Theorem 5d. (5.9) follows immediately from (3.11); for c= 1/2
we have by (3.14)
li
M('9,vV(n))
h 13
1 'V - •
i /,.= 3 -l
— n
2
2 3
Remark. Note that for c -> '/2
■ - V (2ce-2C)"
2 iS
1 -\ ^...+----
l! (fc-3)!
4(1 — 2 c)2
Thus the average number of points belonging to components containing
exactly one cycle tends to + °° as - for c-*-1^ .
4(1-2 c)2
We now prove
Theorem 5e. For N(n) -~en with 0 < c < L/a ft^ components of rnN(n)
are with probability leveling to 1 for n -> -j-°°, ptf/W ?ws or components
containing exactly one cycle.
Proof. Let ipntN denote tbe number of points of rnjs! belonging to
components which contain more edges than vertices and the number of vertices
of which is less than ]/ log n. We have clearly for N(n) -~ en with c < 1j2
I' in — k\ \
I 2 )
Y 7.-1/ i log2 ^
M(V,„W,))^ >' H'iI2^ ---=0^ J
Thus
P(VnJV(rt) ^ 1) == O
log2
2 /
On tbe other band by Theorem 4c the probability that a component
consisting of more than ]/ log n points should not bo a tree tends to 0. Thus the
assertion of Theorem 5e follows.
§ 6. The number of components
Let us turn now to the investigation of the average number of
components of rn N. It will be seen that the above discussion contains a fairly
complete solution of this question. We prove the following
601 RANDOM OBJECTS

0\ THE UVOLUTION OF RANDOM GRAPHS
45
Theorem 6. If C„ N denotes the number of components of TnN then we have
if N(n) ~ en holds with 0 < c < —
(6.1) M(Cn,Mn)) = n-JVT(n)+0(l)
where the bound of the O-term depends only on c. If N(n)
((3.2)
M(',n,Nfn).
A»+O(log;
have
If N(n) — ere /wMs w/fft c > —w,'e /iawe
2
(6.3)
lim
n — + «
M(£n,Mrf)) = JL
w 2 c
x(c)
x2(c)
2 j
where x = x(c) is the only solution satisfying 0 < x < 1 of the equation xe~x =
= 2ce~2c, i. e.
(6.4) x(c) = >' (2 ce"2')* .
Proof of Theorem 6. Let us consider first the case c < --. Clearly if we
2
add a new edge to a graph, thou cither this edge connects two points
belonging to different components, in which case the number of components is
decreased by 1, or it connects two points belonging to the same component
in which case the number of components does not change but at least one
new cycle is created. Thus2
(6.5)
AT) S HnJi
where Hn N is the total number of cycles in rnN. Thus by Theorem 5a it
follows that (6.1) holds.
Similarly (6.2) follows also from Theorem 5a. Now we consider the case
1
c > —.
2
It is easy to see that for o ;S y <— we have (see e. g. [14])
(6.6)
where
(6.7)
V
x-
2
l-k'1yk
2 In fact according to a well known theorem of the theory of graphs (see [4], p. 29)
being a generalization of Euler's theorem on polyhedra we hnvo ^V — n -f- fn,/v =
= xn,N, where w„n — the ,,cyelomatic number" of the graph rn<N — is equal to
the maximal number of independent cycles, in rn.N (For a definition of independent
cycles sec [4] p. 28).
602 MISCELLANY

46
KUDOS—lUINTI
x can be characterized also as the only solution satisfying (J < x ;S 1 of the
equation xe~x = y.
It follows that if AT(n) -~ nc holds with c -^ 1j2 we have
(6.8) M(C,,N(n)) ==
(2X(n) 4N2(n)
■>X(n) [ n
2n2
-I- 0(1) = n — X(n) + 0(1)
which leads to a second proof of the first part of Theorem 6.
To prove the second part, let us remark first that the number of eoinpo-
71
nents of order greater than A is clearly < —. Thus if "„ \-(A) denotes the
A
number of components of order ^ A of rn v we have clearly
(6.9)
M(;r,„v) = M(,;,(i))-fo
A
The average number of components of fixed order k which contain
, V , k
at least k edges will be clearly according to Theorem 1 of order — , i. e.
I»|
bounded for each fixed value of k. As A can be chosen arbitrarily largo we
obtain from (6.9) that
(6.10) M(;r„v)~ x M(r,)
k = l
According to (2.18) it follows that
(6.11) MC„.V)
n- -
"k\
2X -2--'-'
- e "
n I
and thus, according to (6.6) if X{n) ~c« holds with r. > 1/2 we have
^6.12)
lim
71
1 | X2(C)\
- U"(c) — -
2c 2 I
where .r(c) is defined by (6.4). Thus Theorem 6 is completely proved.
Let us add some remarks. Theorem 6 illustrates also the fundamental
change in the structure of Trl N which takes place if A7 passes - . While the
average number of components of Frl v (as a function of N with n fixed)
71 il
decreases linearly if AT :?S -- this is no longer true for AT > --- ; the average
■2 2
number of components decreases from this point onward more and more
slowly. The graph of
(6.13)
(c) = lim
,\(n)
^('n,N(n))_
71
2c
\x(c)
for
x*(c)
2
0 < c <
for c > 1/2
603 RANDOM OBJECTS

ON THE EVOLUTION OF RANDOM GRAPHS
47
as a function of c is shown by Fig. 4.
From Theorem 6 one can deduce easily that in case N(n) ~ en with
c < 1j2 we have for any sequence con tending arbitrarily slowly to infinity
(6.14) lim P(\CmNM-n+N(n)\<a>n)=l
(6.14) follows easily by remarking that clearly fn N > n — N.
z
1
z- z(c)*
f-c for 0<c i!/2
Figure 4.
For the case N(n) ~ en with c 2? 1f2 one obtains by estimating the
variance of £n N(n) and using the inequality of Chebyshev that for any e > 0
(6.15)
lim P
n
1., x" c
-- F(c) - --
2c 2
< e = 1.
The proof is similar to that of (4.21) and therefore we do not go into details.
§ 7. The size of the greatest tree
If N ~ en with c < 1/2 then as we have seen in § 6 all but a finite
number of points of -TnN 'belong to components which are trees. Thus in this case
the problem of determining the size of the largest component of rnN reduces
to the easier question of determining the greatest tree in Fn N. This question
is answered by the following.
Theorem 7 a. Let An N denote the number of points of the greatest tree which
is a component of f„v, Suppose N = jV(w) ~ en with c =f= %■ Let can be a sequence
604 MISCELLANY

48
HI! DOS—RUSTY I
tending arbitrarily slowly to -j- °°- Then we have
(7.1)
and
(7.2)
where
(7.3)
1 I 5
lim P I An,N(n) ^ - log 71 loglog 11
"-»+» I a I 2
+ <»n
= 0
1( 5 \
lim P ( AnMn) > -- log 71 - - loglog 71 - ft)„ | = 1
n—+» | a 2 I
e~a = 2cex~2c
a«d thus a > 0.)
Proof of Theorem 7a. We have clearly
(7.4) P(^n,JV<n> ^ ^) = P (2 T* ^V\^2 M(T
and thus by (2.18)
(7.5)
i.e. a = 2 c — 1 — log 2 c
It follows that if zx =
we have
(7.6)
P(^nMn) > 2) = O
log n — - loglog 5
ne
+ «r.
P(^jvw ^ 2x) = 0(e-a"»)
This proves (7.1). To prove (".2) we have to estimate the mean and variance
log7i loglogTi —con . We have by (2.18)
M
a '2
_- eam»
2cf2
D2(t2J = 0(M(tzJ)
of tZj where z2 =
(7.7)
and
(7.8)
Clearly
P(An,N(n) > Z2) ^ P(T- ^ 1) = 1 - P(TZ! = 0)
and it follows from (7.7) and (7.8) by the inequality of Chebyshev that
(7.9) P(TZi=0) = O(e-"-).
Thus we obtain
(7.10) P(A,,Mn) > 22) £ 1 - 0(e--") ■
Thus (7.2) is also proved.
Remark. If c < — the greatest tree which is a component of Fn N with
JV ~ cw is — as mentioned above — at the same time the greatest component
605 RANDOM OBJECTS

ON THE EVOLUTION OF RANDOM CRAl'HS
49
of r N, as Fn N contains with probability tending to 1 besides trees only
components containing a single circle and being of moderate size. This follows
evidently from Theorem 4c. As will be seen in what follows (see § 9) for
c > the situation is completely different, as in this case Fn N contains
2
a very large component (in fact of size G(c)n with 67(c) > 0) which is not a
tree. Note that if we put c — — log n we have a. = — log n and — Xogn^k
2 h ha
in conformity with Theorem 2c.
We can prove also the following
Theorem 7b. If N~cn, where c =f= — and e~" = 2ce1~2c then the number
2
of isolated trees of order h =
1
log n loglog n 4-1 resp. of order > h (where
a ' 2 j
I is an arbitrary real number such that h is a positive integer) contained in
rn N has for large n approximately a Poisson distribution with the mean value
, ' a5 2e~"' a? 2 e~al
A = _ resp. a = .
2 c ][2ti 2 cf 2ti(1 — e~a)
Corollary. The probability that FnN^ with N(ri)~nc where c=f= —
does not contain a tree of order > —
a
log n loglog n
-\-1 tends to
exp
if AT
5'2 p-al
a° * e
for n—y -j-°°) where a = 2c — 1 — log 2c.
2c\J2n{\ — e~ ,
The size of the greatest tree which is a component of FnN is fairly large
71
~ — . This could be guessed from the fact that the constant factor in the
expression - - log n loglog n\ of the ,.probable size" of the greatest eompo-
a 2 /
nent of -Tn N figuring in Theorem 7a becomes infinitely large if c = —.
71
For the size of the greatest tree in rnN with N ^- the following
result is valid:
Theorem 7c. If N ~— and AnN denotes again the number of points
2
of the greatest tree contained in Fn N, we have for any sequence a>n tending to
4-°° for n—*- -j-°°
(7.11)
and
lim P(^n,N > w23w„)=0
[7.12)
lim P\A„.N >
= 1
606 MISCELLANY

50
ERDOS—R6>fTI
Proof of Theorem 7c. Wo have by some simple computation using (7)
i [n — k \
(7.13)
M(rft) =
**-«
\N-k+\l nkx-^e-x
k\
Thus it follows that
(7.14)
--0
K)
which proves (7.11).
On the other hand, considering the mean and variance of t* = V* rk,
it follows that
M(t*) > Aw™ where A > 0 and D2(t*) = 0((0¾2)
and (7.12) follows by using again the inequality of Chebyshev. Thus Theorem
7c is proved.
The following theorem can be proved by developing further the above
argument and using Lemma 1.
Theorem 7d. Let r(/i) denote the number of trees of order ^ fin2!3 contained
in r„N( \ where 0 < /i < -\-°° and N(n) ~—. Then we have
2
(7.15)
where
(7.16)
linl Pn,N(n)(T(M) = j)
~7T
(?' = 0,1, ...)
x =
1^12 71
e'x dx
§ 8. When is FnN a planar graph ?
We have seen that the threshold for a subgraph containing k points
and k -\- d edges isn k+d ; thus if N ~ en the probability of the presence
of a subgraph having k points and k -\- d edges in FnN tends to 0 for 71-)-+°°.
for each particular pair of numbers k 2i 4, d >- 1. This however does not
imply that the probability of the presence of a graph of arbitrary order having:
more edges than vertices in FnN with N ~ nc tends also to 0 for n —> +°°.
In fact this is not true for c ^ x/a as is shown by the following
607 RANDOM OBJECTS

ON THE EVOIXTIOX OF RANDOM CRAFHS
51
Theorem 8a. Let xnN{d) denote the number of cycles of Gn N of arbitrary
order which are such that exactly d diagonals of the cycle belong also to Fn N.
Then if N(n)
n + %Vn , ,,/— , i,,
: f- o(yn) where —°° < A < + °°, we have
(8.1)
where
(8.2)
l'm P(Xn,N(n)(d) = 3) =
g'e-i
0 = 0, 1, . . .)
2.6"-rf!
T w
I
%
yM-'eV3.e dy.
Proof of Theorem 8a. We have clearly as the number of diagonals of a
k — gon is equal to —
(8.3) M(^,N(d))=2l(l
(*
1)!
(k(k - 3)'
and thus if .V (n) = *+AJ(™ + 0(Yn)
(8.4)
M(zn,Mn)C0)'
2d+1-dlnd
n
2
-d/
A 1" - ^
It follows from (8.4) that
(8.5) lim M(xnMn)(d))= —j— f y2""1 e V
n-»+ co z • fa a ! J
3 2
dy.
The proof ean be finished by the same method as used in proving Theorem 2a.
Remark. Note that Theorem 8a implies that if N(n) = \- conVn
with 0),,-)-+00 then the probability that TnN(n) contains eyeles with any
prescribed number of diagonals tends to 1, while if N(n) = wn |/ra
the same probability tends to 0. This shows again the fundamental difference
n n
in the structure of T„ N between the eases N < — and N > — . This differ -
2 2
ence can be expressed also in the form of the following
Theorem 8b. Let us suppose that N(n) ~ nc. If c < — the probability
608 MISCELLANY

52 KRDiiS—RfrXYl
that the graph Frl N(n) is planar is tending to 1 while for c > —this probability tends
to 0.
Proof of Theorem 8b. As well known trees and connected graphs
containing exactly one cycle are planar. Thus the first part of Theorem 8b follows
from Theorem 5e. On the other hand if a graph contains a cycle with 3
diagonals such that if these diagonals connect the pairs of points (P:, P]) (i —
= 1, 2, 3) the cyclic order of these points in the cycle is such that each pair
(Pt, P'^ dissects the cycle into two paths which both contain two of the other
points then the graph is not planar. Now it is easy to see that among the
(k(k — 3)^
k \
6
2
I I triples of 3 diameters of a given cycle of order k there are at least
3
triples which have the mentioned property and thus for large values of k
approximately one out of 15 choices of the 3 diagonals will have the mentioned
property. It follows that if N(n) — \-(on }rn with con —>- +°°, the proba-
bility that PnN(„) is not planar tends to 1 for n—>- -(-<». This proves Theorem
8b. We can show that for N(n) = — + ).fn with any real ?. the probability
of rn N(n) not being planar has a positive lower limit, but we cannot calculate
ts value. It may even be 1, though this seems unlikely.
§ 9. On the growth of the greatest component
We prove in this § (see Theorem 9b) that the size of the greatest
component of T N(n) is for N(n) ~ en with c > l/2 with probability tending to 1
approximately G(c)n where
(9.1) 0(^ = 1-^-
2 c
and x(c) is defined by (6.4). (The curve y = 0(c) is shown on Fig. 2b).
Thus by Theorem 6 for AT(?;) -^ en with c > 1/2 almost all points of
rn,N(n) (i' c- a^ but °(n) points) belong either to some small component which
5
is a tree (of size at most 1/a (logn loglogft) -j- 0(1) where a = '2c —1 —log 2c
by Theorem 7a) or to the single "giant" component of the size ~G(c)n.
Thus the situation can be summarized as follows: the largest component
°f r„Mni is of order logw for ~ c < l/„, of order n2'3 for- --^ — and
' ' b n u n 2
of order n for—— ~c > x/2. This double "jump" of the size of the largest
N(n)
component when ^-^ passes the value XL is one of the most striking facts
n
concerning random graphs. We prove first the following
609 RANDOM OBJECTS

OX TUB EVOLUTION' OF RANDOM Cii.U'llS
53
Theorem 9a. Let,WnN(A) denote the set of those points of rn N which belong
to components of size > A, and let H N(A) denote the number'of elements of
the set <3frKN(A). If Nx(n) ~(c — e) n where e>0, c — e ^ l/2 and N2(n) ^cn
then with probability tending to 1 for n —>- 4-°° from the HnN^(A) points
belonging to -^n:Nl(n)(A) more than (1 — b) HnNiM(A) points will be contained
in the same component of rniN,(rl) for any b with 0 < b < 1 provided that
(9.2)
A >
50
e2<32
Proof of Theorem 9a. According to Theorem 2b the number of points
belonging to trees of order ^ A is with probability tending to 1 for n —>- + °°
equal to
£ —- [2(c - fc-)]"-i e-2<c-f>] + o(w) .
\k=\ fc' I
On the other hand, the number of points of Tn Ni(n) belonging to components
of size < A and containing exactly one cycle is according to Theorem 3e
o(n) for c — e ^ x/2 (with jjrobability tending to 1), while it is easy to see, that
the number of points of FniNl(n) belonging to components of size ^ A and
containing more than one cycle is also bounded with probability tending to 1.)
Our last statement follows by using the inequality (4.19) from the fact
that the average number of components of the mentioned type is, as a simple
calculation similar to those carried out in previous §§, shows, of order O
Let Ety denote the event that
n
(9.3)
!#n.wn)M) -nf(A.c-E)\ < rnf(A,c-E)
where t > 0 is an arbitrary small positive number which will be chosen later
and
(9.4)
f(A,c)
A 7./(-1
2 ce
> 0
and let E^ denote the contrary event. It follows from what has been said
that
(9.5)
lim P(E<>») = 0 .
We consider only such rnN^n) for which (9.3) holds.
Now clearly rnN^n) is obtained from rnNiin) by adding .^2(71)-.^(71)^716
new edges at random' to rnN](„). The probability that such a new edge should
(H,
connect two points belonging to ^n Nl(n)(4), is at least
n,N,(n)(^)
2
N2(n)
n
and thus by (9.3) is not less than (1
large and r sufficiently small.
2t) /2 (A, c — e), if n is sufficiently
610 MISCELLANY

54
ERD(5S—RKJ.'YI
As these edges are chosen independently from each other, it follows
by the law of large numbers that denoting by vn the number of those of the
N2(n) —Nx(n) new edges which connect two points of 0^,,^(,1) ancl by Etf)
the event that
(9.6) vn> e(l — 3r)f2(A, c — e)n
and by E$ the contrary event, we have
(9.7) lim P(£<?>) = 0 .
We consider now only such rnNl(n) for which Efi takes place. Now let us
consider the subgraph r*N2(n) of rniN2(n) formed by the points of the set
<%*n,Nx(n>(A) and only of those edges of rnNl^ which connect two such points.
We shall need now the following elementary
r
Lemma 2. Let ax, a2, . . ., ar be positive numbers, "V a; = 1. If max at :g a
j -1
then there can be found a value k (\ ^L k < r ■— 1) such that
; = , 15^
(9.8) and
1 — a * 1 + a
syajg,
2 ;-, 2
1 — a " 1 + a
^ y aj ^ __i—
2 j=/(+i 2
Proof of Lemma 2. Put 8j = S'a, (/ = 1, 2, ..., r). Let )0 denote the
i-i
least integer, for which Sj > x/2. In case 8Jo — x/2 > x/2 — ^0-i choose
k — )0 — 1, while in case 8jo — x/2 < x/2 — SJ-0_1 choose & = ;0. In both
eases we have \ 8k — 1L\ < —° ^1 —which proves our Lemma.
2 2
Let the sizes of the components of r*N^n, be denoted by bv b2, ..., br.
Let E^) denote the event
(9.9) max bj>Hn:NlM(A) (I-8)
and E^ the contrary event. Applying our Lemma with a = 1 — b to the
numbers aj = i it follows that if the event E^ takes place, the
Hn,Ndri)(A)
set <%"niNl(n)(A) can be split in two subsets J%"n and J^„ coa-taining Hn and
H'n points such that H'n + Hnn = HnNi(n)(A) and
(9.10) HniNM(A) ~ < min (H'n, H'n) g max {H'n, H'n) £ HmNM{A) 1 - -)
further no point of -3^"n is connected with a point of -3^^ in r*Nl^ .
It follows that if a point P of the set 3P niNAn){A) belongs to Jt'n (resp.
•Jfn) then all other points of the component of rn>Ni^ to which P belongs are
611 RANDOM OBJECTS

OX THE EVOLUTION OF RANDOM GRAPHS
55
also contained in <y?'n (resp. <3?"n). As the number of components of size > A
of ^njvi(n) *8 clearly < —",Nl "^—- the number of such divisions of the set
W»,.Vi(«)(A)
'Jf n,Ni(n)(A) does not exceed 2A'
If further E^ takes place then every one of the vn new edges
connecting points of -3?n.N^(n\{A) connects either two points of *3?"n or two points
of cJ3T" The possible number of such choices of these edges is clearly
2
+
H'n
2
As by (9.10)
{9.11)
H'
"\ +
2 b2 ,
4
d 2
1
2
, <5>2 , &
1 — b -\ < 1
2 2
it follows that
(9.12)
and thus by (9.3) and (9.6)
(9.13) P(£<?)) < exp nf(A,c- e
Thus if
(9.14) Aed(l — 3r)f(A,c — e) > (1 + T)log4
2
4 ~ 2
then
(9.15)
lim P (#<?>) = 0 .
As however in case c — e > 1/2 we have f(A, c — e) ^. G (c — e) > 0
for any A, while in case c — e — x\2
(9.15a) /
4,
A JWt-
2
F-
>
^k!ek kJZ+iJclek' 2]j A
if A > An
1 50
the inequality (9.13) will be satisfied provided that r < — and A > .
10 s2 <52
Thus Theorem 9a is proved.
Clearly the "giant" component of rnN2^ the existence of which (with
probability tending to 1) has been now proved, contains more than
(l-r){l-d)nf{A,c-B)
612 MISCELLANY

56
Hiinos—i! i;_\rY i
points. By choosing e, t and b sufficiently small and A sufficiently large,
(1 — t) (1 — 6) f(A, c — e) can be brought as near to G(c) as we want. Thus
we have incidentally proved also the following
Theorem 9b. Let QnN denote the size of the greatest component of rnN
If N(n) <~^cn where c > xj2 we have for any r\ > ()
(9.16)
lim P [ q"^{,d - G(c) < >]
where G(c) = 1
x(c)
— and x(c)
ft*-'
V'- —- (2c e-'2c)k is the solution satisfying
2c t~x ft!
0 < x(c) < 1 of the equation x(c) e~x^ =2ce-*c.
Remark. As G(c) —>■ 1 for c—>- + °° it follows as a corollary from Theorem
9b that the size of the largest component will exceed (1 — a)n if c is
sufficiently large where a > 0 is arbitrarily small. This of course could be proved
directly. As a matter of fact, if the greatest component of Fu V(n) with A'(n) -^nc
would not exceed (1 —o.)n (we denote this event by Brl(a, c)) one could by
Lemma 2 divide the set V of the n points Plt . . ., Pn in two subsets V resp.
V" consisting of n' resp. n" points so that no two points belonging to different
subsets are connected and
(9.17)
a, n
< m:
n (n', n") :S max (n', n") < 1 —
u
But the number of such divisions does not exceed 2", and if the n points
are divided in this way, the number of ways AT edges can be chosen so that
only points belonging to the same subset V' resp. V" are connected, is
In .»-
it follows
(9.18)
P(Bn(a, C)) < 2" 1 -
2/
N(n) _ N(ny,
< 2"e 2 .
Thus if a c > Jog4, then
(9.19) lim P(5 ,(«, c)) = 0
lot? 4
which implies that for c > - -and X(n) ^ en we have
(9.20)
lini P(e„„v(n) > (1 «) «) = 1
613 RANDOM OBJECTS

OX THE ETOLlTTrON 01' KAXDOM GKAl'HS
57
We have seen that for N(n) ^ en with c > 1/2 the random graph FnN(n-}
consists with probability tending to 1, neglecting o(n) points, only of isolated
n kk~2
trees (there being approximately— (2c e-2c)k trees of order k) and of
£ C /C .
a single giant component of size ^G(c)n.
Clearly the isolated trees melt one after another into the giant
component, the "danger" of being absorbed by the "giant" being greater for larger
components. As shown by Theorem 2c for N(n) ~ — n log n only isolated
r i r • l -i v N(n)— \l2n\ogn
trees of order <^k survive, while tor ' --
4- °° the whole
graph will with probability tending to 1 be connected.
An interesting question is: what is the "life-time" distribution of an
isolated tree of order k which is present for N(n) ~ en I This question is
answered by the following
Theorem 9c. The probability that an isolated tree of order k which is present
in Fn N ,. where Nx(n) ~ en and c > x/2 should still remain an isolated tree
in -Tn'.v2(n) where N2(n) ~ (c + 0 n (t > 0) is approximately e~2kl; thus the
,,life-time" of a tree of order k has approximately an exponential distribution
with mean value and is independent of the "age" of the tree.
Proof. The probability that no point of the tree in question will be
connected with any other point is
N,(n)
u
This proves Theorem 9e.
in — k\
I 2 I
in,
I2I
i + *A
§ 10. Remarks and some unsolved problems
We studied in detail the evolution of FnN only till N reaches the order
of magnitude 11 log n. (Only Theorem 1 embraces some problems concerning
the range AT(n) ~ na with 1 < a < 2.) We want to deal with the structure
of ^njv(n) f°r N(n) ^cna with a > 1 in greater detail in a fortcoming paper;
here we make in this direction only a few remarks.
First it is easy to see that F ,„-. is really nothing else, than the
complementary graph of FnN^ny Thus each of our results can be reformulated
to give a result on the probable structure of rnN with N being not much
less than ) . For instance, the structure of Fn N will have a second abrupt
change when N passes the value
if N
<
— en with c > ]/2
then the complementary graph of Fn N will contain a connected graph of order
/(c)", while for c < 1/2 this (missing) "giant" will disappear.
614 MISCELLANY

58
ERDOS—Bf'.NYI
To show a less obvious example of this principle of getting result
for N near to
let us consider the maximal number of pairwise independent
points in FnN. (The vertices P and Q of the graph F are called independent
if they are not connected by an edge).
Evidently if a set of k points is independent in FnN^n, then the same
points form a complete subgraph in the complementary graph F'ntN,n ■ As
i,N(n)
n
2
however F„ Nln, has the same structure as F ,„>. it follows by Theorem
"'"(") "'(2) ~ N(")
1, that there will be in FnN^ almost surely no k independent points if
N(n) = o\n
'^-^
i. e. if N(n) —
' k-V\ but there will be in
Fn,N(n) almost surely k independent points if'JV^tt) = I 0J n \[ k-\> where
<an tends arbitrarily slowly to +«>. An other interesting question is: what
ean be said about the degrees of the vertices of FnN. We prove in this direction
the following
Theorem 10. Let DnN,n){Pk) denote the degree of the point Pk in /1^^^
(i. e. the number of points of Fn N(n) which are connected with Pk by an edge). Put
Dn = min DnMn) (Pk) and Dn = max DnMn) (Pk).
N(n)
lim
n -i » n log n
Suppose that
(10.1)
Then we have for any e > 0
(10.2) lim P
We have further for N(n) ^cn for any k
7A<-i <e
1 .
(10.3)
lim P(DnMn) (Pk) = j)
{2 c)'e~2c
(/ = 0,1 ).
Proof. The probability that a given vertex Pk shall be connected by
exactly r others in FnN is
in — l\
r
n
2)
N j
n — l\
2
N—r
2N
r\
615 RANDOM OBJECTS

ON THE EVOLUTION OF BAN DOM GRAPHS
59
thus if N(n) ^cn the degree of a given point has approximately a Poisson
distribution with mean value 2c. The number of points having the degree r
is thus in this case approximately
(2c)r«-
(r = 0,l, ...).
If N(n) = (n log n) o)n with co
a point will be outside the interval
proximately
4-°° then the probability that the degree of
2N(n)
e) and K— (1 + e) is
2
K-2I0gn-<u„i >e-2I0gn-
and thus this probability is o
(■2<an-logn)ke-2a"losn
k\
for any e > 0 .
= 0
Thus the probability that the degrees of not all n points will be between
the limit (1 ± e) 2a>„ log n will be tending to 0. Thus the assertion of Theorem
10 follows.
An interesting question is: what will be the chromatic number of Fn N ?
(The chromatic number Ch(P) of a graph F is the least positive integer h such
that the vertices of the graph can be coloured by h colours so that no two
vertices which are connected by an edge should have the same colour.)
Clearly every tree can be coloured by 2 colours, and thus by Theorem
4a almost surely Ch(FnN) = 2 if N = o(n). As however the chromatic
number of a graph having an equal number of vertices and edges is equal
to 2 or 3 according to whether the only cycle contained in such a graph is
of even or odd order, it follows from Theorem 5e that almost surely Ch (JTnN) ^ 3
for N(n) ~ nc with c < 1j2.
— we have almost surely Ch (rn_NM) ^ 3.
For N(n)
As a matter of fact, in the same way, as we proved Theorem 5b, one
can prove that Fn N(n) contains for N(n) ~— almost surely a cycle of odd
order. It is an open problem how large Ch (rnNM) is for N(n) ~ en witho1^?
A further result on the chromatic number can be deduced from our
above remark on independent vertices. If a graph r has the chromatic number
h, then its points can be divided into h classes, so that no two points of the
71
same class are connected by an edge; as the largest class has at least— points,
h
it follows that if / is the maximal number of independent vertices of F we have
f > —. Xow we have seen that for N(n)
h
f ^ k; it follows that for N(7
71
> .
k
— o
K-i)
almost surely
almost surely Ch {FnN(ri)) >
616 MISCELLANY

60
ERDO.s—UftN'VI
Other open problems are the following : for what order of magnitude
of N(n) has rn^Ni^n) with probability tending to 1 a Hamilton-line (i.e. a path
which passes through all vertices) resp. in case n is even a factor of degree 1
(i.e. a set of disjoint edges which contain all vertices).
An other interesting question is : what is the threshold for the
appearance of a "topological complete graph of order k" i.e. of k points such that
any two of them can be connected by a path and these paths do not
intersect. For k > 4 we do not know the solution of this question. For k = 4
it follows from Theorem 8a that the threshold is -. It is interesting to
2
compare this with an (unpublished) result of (r. Dibac according to which
if N :¾ 2n — 2 then GnN contains certainly a topological complete graph
of order 4.
We hope to return to the above mentioned unsolved questions in an other
paper.
Remark added on May 16, 1960. It should be mentioned that X. V.
Smib>jov (see e. g. MameMamtMecKUU CdopHUK 6(1939) p. 6) has proved a
lemma which is similar to our Lemma 1.
(Received December 28, 1959.)
REFERENCES
[]] Caylev, A.: Collected Mathematical Papers. Cambiidge, 1880-185)7.
[2] Poly'a, G.; "Kombiimtoriseho Anzahlbestimmungen fur Giuppen, Graphen unci
chemis< lie V< rhindungen". Ac/a Mathematica 68 (1937) 145 — 254.
[3] Konk:, D.: Thecrie (lev endlichen und unendlichen Graphen. Leipzig, 193().
[4] Berge, 0.: Thcorie des graphes et ses applications. Paris, Dunod, 1958.
[5] Ford, G. W. — Uhxknbeck, G. E.: "Combinatorial problems in the theory of
graphs, I." Prof. Nat. Acad. Sci. 42 (19,50) L"SA 122—128.
[(>] Gilbert, E. X.: "Enumeration of labelled graphs". Canadian Journal of Math.
8 (1!:57) 405-411.
[7] Krdos, P.— Renyi, A.: "On random graphs, 1". Publicationes Malheinalicae
(Debrecen) 6 (1959) 290-297.
[8] Harary, P.: "Unsolved problems in the enumeration of graphs" In this issue, p. G3.
[9] Krdos, P.: ,,Graph theory and probability-." Canadian Journal ol Math. 11(1959)
34—38.
[10] Austin, T. L. — Fagbn, R. E. — Penney, W. P.—Riordan, J.: "The number of
components in random linear graphs". Annals of Math. Statistics 30 (1 959) 747—754.
[11] TurAn P. i "Egv grafelmeltti szelsoertekfeladatrol".Matematikai t's Fizikai Lapok
48 (1941) 436-452.
[12] TurAn, P.: "On the theory of graphs", Colloquium Mathematician 3 (1954) 19 — 30.
[13] KSvary, T.—Sos, V. T. — Turan, P.: "On a problem of K. Zarenkievicz".
Colloquium Mathcmaticum 3 (1954) 50 — 57.
[14] Renyi, A.: "Some remarks on the theory of trees".Publications of the Math. Inst.
of the Hung. Acad, of Sci. 4 (1959) 73 — 85.
[15] Rknyi, A.: "On connected graphs. I.".Publications of the Math. Inst, of the Hung.
Acad, of Sci. 4 (1959) 385-387.
[16] Jordan, Ch.: Calculus of finite differences. Budapest, 1939.
[17] Erdos, P. —Stone, A. H.: "On the structure of linear graphs". Bull. Amer. Math.
Sec. 52 (1946) 1087-1091.
617 RANDOM OBJECTS

ON THE STRENGTH OF CONNECTEDNESS
OF A RANDOM GRAPH
By
P. ERDOS (Budapest), corresponding member of the Academy,
and A. RENY1 (Budapest), member of the Academy
Let G be a non-oriented graph without parallel edges and without
slings, with vertices Vi, Vk, -.., V„. Let us denote by d(Vk) the valency (or
degree) of a point V,; in G, i.e. the number of edges starting from 14. Let
us put
(1) c(G)= min d(Vk).
If G is an arbitrary non-complete graph, let cp(G) denote the least number
k such that by deleting k appropriately chosen vertices from G (i. e. deleting
the k points in question and all edges starting from these points) the
resulting graph is not connected. If G is a complete graph of order n, we put
cp(G) = n — 1- Let ce(G) denote the least number / such that by deleting /
appropriately chosen edges from G the resulting graph is not connected. We
may measure the strength of connectedness of G by any of the numbers
cP(G), c,.(G) and in a certain sense (if G is known to be connected) also
by c(G). Evidently one has
(2) c(G) ^ c,(G) i= cp(G).
It is known further that any two points of G are connected by at least
cp(G) paths having no point in common, except the two endpoints (theorem
of Menger—Whitney, see [1] and [2]) and by at least ce(G) paths having
no edge in common (theorem of Ford and Fulkerson, see [3]).
We shall denote by vr(G) the number of vertices of G which have the
valency r (/- = 0, 1,2, ...).
As in two previous papers ([4], [5]) we consider the random graph/",,, a-
defined as follows: Let there be given n labelled points Vi, Vi, ..., V„. Let
us choose at random N edges among the l") possible edges connecting
possible choices of these edges
these n points, so that each of the | \2)
N )
should be equiprobable. We denote by l\ ,v the random graph thus obtained.
We shall denote by P(-) the probability of the event in the brackets.
618 MISCELLANY

262
P. ERDOS AND A. R£NYI
The aim of this note is to investigate the strength of connectedness of
the random graph r„,N when n and N both tend to + oo, N=N(n) being
a function of n. As it has been shown in [4], the following theorem holds:
Theorem 1. If we have N(n) = -^-n log n + an + o(n) where a is a real
constant, then the probability of I\ .v((1) being connected tends to exp(—e~-a)
for n—>-+oo.
In this paper we shall prove the following theorem:
1 r
Theorem 2. If we have Af(/i) = —n log n + ^n log log n +a n-\-o(n)
where a is a real constant and r a non-negative integer, then
(3) !im P(c;,(r„.A-(„)) = /-) = l —exp
n->+00
further
(4) lim P(c„(r„ .v(„>) = /-) = 1-exp
and
(5) lim P(c(/Vv(„)) = /-) = 1- exp
/-!
/-!
/-!
Remark. Clearly Theorem 2 can be considered as a generalization
of Theorem 1. As a matter of fact, any of the statements ^,(0) = 0 or
ce(G) = 0 is equivalent to G not being connected and thus for /-=0 (3) and
(4) reduce to the statement of Theorem 1. It has been shown further in [4]
that if N(n)= =■ log n-\-an + o(n) and jH„. .v((l) is not connected, then it
consists almost surely of a connected component and of a few isolated points.
Therefore (5) is for /- = 0 also equivalent to the statement of Theorem 1.
Thus in proving Theorem 2 we may restrict ourselves to the case rgl.
The statement (5) of Theorem 2 gives information about the minimal
valency of points of r„,x. In a forthcoming note we shall deal with the
same question for larger ranges of N (when c(r„,\) tends to infinity with n),
further with the related question about the maximal valency of points of r„, x.
We shall prove further the.following
1 r
Theorem 3. If we have N(n)= -y n logn + ^n \oglogn + an +o(n)
where a is a real constant and r a non-negative integer, then we have
(6) lim P(v,.(I\ a-(,o) = k) = -^- for A = 0, 1, ...
619 RANDOM OBJECTS

ON THE STRENGTH OF CONNECTEDNESS OF A RANDOM GRAPH
263
where 1 = ——; in other words, the distribution of vr(F„,x(n)) tends to a
Poisson distribution.
Proof of Theorems 2 and 3. Let r ^ 1 be an integer and — «■ < « < + <x>.
Let us suppose that
1 r
(7) N(n) = ^ n J°g n + ~2 n ,0§ Io§n + an + °(")■
Let Fn,N be a random graph with the n vertices V\, V-i, ..., V„ and
having A/ edges. Let Pfc(n, A/, r) denote the probability that by removing r
suitably chosen points from rtli y there remain two disjoint graphs, consisting
of k and n—k — r points, respectively. We may suppose k< —^— . First
we have clearly
*<".».-HT7
It follows by some obvious estimations that
(8) £ Pk{n,N{n),r) = 0
log ))
' log log
log)) _r*i-ri V ri
(1-+3) 5 <{< —-—
v ' loe oe« I i '
Now we consider the case /tg(f+3):—, . Let PUn.N.r) denote the
log log n v
probability that by removing r suitably chosen points (the set of which will
be denoted by d) r„, x can be split into two disjoint subgraphs F' and F"
consisting of k and n — k—r points, respectively, but that F„,K can not be
made disconnected by removing only r—1 points. If F,hN has these
properties and if 5 denotes the number of edges of Fn, y connecting a point of
A with a point of F', then we have clearly 5 s /-. Otherwise, by definition,
5 ^ rk. Thus we have
(9) A-(/i, N, r)
620 MISCELLANY

264
P. ERDOS AND A. r£NY]
It follows that
(10) iS I-log /i
From (8) and (10) it follows that for n—>■ + <^
(11) P(cP(rn. xw) = /-)- P(c(r„, A>)) = /-).
As a matter of fact, (8) and (10) imply that if by removing r suitably chosen
points (but not by removing less than r points) r„, a» can be split into two
disjoint subgraphs F' and /""' consisting of k and n — k—r points, respec-
n — /-
tively, where k ^ ~—- , then only the case k=\ has to be considered, the
probability of A: > 1 being negligibly small. It remains to prove (5). This can
be done as follows. First we prove that
(12) lim P(c(r„.A-f,o)=l /--1) = 0.
n -> + 00
For /-=1 this follows already from Theorem 1. Thus we may suppose here
r ^ 2. We have
P(c(r,,,A.)^/--i)^J>K l
and thus
(13) P(c(r„,.v(1!)) ^ /--1) = ollog^
which proves (12).
Now let rr(r„,y) denote the number of vertices of r„iA- which have the
valency r. Then we have clearly by (12)
(14) P (c(r„, .v(I1)) = r) ~ P(iv(r„, A.(1„) =£ 0).
Now evidently
(15) P(K(r„. .V(.o)¥= 0) = £ (-if'1 Si
where
(16) Sj=ZZ ••• 2p(W=^W = ^.-.^)=0'
Evidently, if we stop after taking an even or odd number of terms of the
1
621 RANDOM OBJECTS

ON THE STRENGTH OF CONNECTEDNESS OF A RANDOM GRAPH
265
sum on the right-hand side of (15), we obtain a quantity which is greater
or smaller, respectively, than the left-hand side of (15). Now clearly
P(<TO = /-) =
and thus
(17)
!im Si =
/-!
Now let us consider P(d(Vk) = r, d(Vk,) = r) where k^k.2. If both 14, and
Vk, have valency r, three cases have to be considered: a) either 14, and 14,
are not connected, and there is no point which is connected with both 14,
and Vk,; b) or 14, and 14, are not connected, but there is a point connected
with both; c) 14, and 142 are connected. We denote the probabilities of
the corresponding subcases by Pa(d( 14,)= r, d(142) = r), Pb(d(Vk) = r,
d(Vk) = r) and Pc(d(Vkl) = r, d(Vk2) = r), respectively. We evidently have
In
Pa{d{Vk) = r,d{V^ = r)-
(n-2)\
2) {2n~3)
\ N(n) — 2r
r\1(n — 2r—2)\
and thus
(18)
ZZ P„(d(Vk) = r,d(Vk,) = r)
e--a
On the other hand (denoting by / the number of points which are connected
with both 14, and 14,), we have
Ph{d{Vk) = r,d(Vk,) = r)^
(19)
(/i-2)!
-(2/1-3)1
N{n)—2r j
S /!(/- — l)\(n — 2r + l—2)!
°b?
622 MISCELLANY

266
P. ERDOS AND A. RENYI
Similarly one has
P,.(d(Vky-=r,d(Vkl)=r) =
(20) ((2)-^-3))
= V (^-2)1 I N(n)-2r J M
iTi /!(/•—/—l)l4(/i —2/- + /)1 ( (n\\ U
\W(/i)J
Thus we obtain
1 /V2<*
J)->-+ CO *- V / I
The cases/> 2 can be dealt with similarly. Thus we obtain
l /V-aY
(21) lim 5, = ^ ^-- (7 = 1,2,3,4,...).
11 -> + co J • \ • ■ J
It follows from (16) and (21) that
(22) !im P(,v(/\A(,())^0)--=l-exp(--^-
In view of (2), (11) and (14) Theorem 2 follows.
To prove Theorem 3 it is sufficient to remark that by the well-known
formula of Ch. Jordan
(23) P(.v(/'„ .v,,,o) = *) = 2Vl>'"P!+;k\ Sj+k,
./■■-» V J J
e--2u
and thus by (21), putting / =—--, we obtain for A =0,1,...
(24) lim P(,V(r,,,(l0) = /c) = 'T2-Sr^ = ^r-
Thus Theorem 3 is proved.
Our thanks are due to T. Gallai for his valuable remarks.
(Received 12 October I960)
623 RANDOM OBJECTS

ON THE STRENGTH OF CONNECTEDNESS OF A RANDOM GRAPH
267
References
[1] D. Konig, Theorie der endlichen und unendlichen Graphen (Leipzig, 1936).
[2] C. Berge, Theorie des graphes et ses applications (Paris, 1958).
[3] L. R. Ford and D. R. Fulkerson, Maximal flow through a network, Canadian Journal
of Math., 8 (1956), p. 399.
[4] P. Erdos and A. Renyi, On random graphs. I, Publ. Math. Debrecen, 6 (1959), pp.
290—297.
[5] P. Erdos and A. Renyi, On the evolution of random graphs, Publ. Math. Inst. Hung.
Acad. Sci., 5 (1960), pp. 17—61.
624 MISCELLANY

ON RANDOM MATRICES
by
P. EBD<3S and A. BfiNYI
Introduction
In the present paper we deal with certain random 0 — 1 matrices. Let
^y£(n, N) denote the set of all n by n square matrices among the elements of
which there are exactly N elements (n < N < n2) equal to 1, all the other
elements are equal to 0. The set <_x#(w, N) contains clearly such matrices;
we consider a matrix M chosen at random from the set ^#(w, N), so that each
2^
-1
to be chosen. We ask
element of ^#(n, N) has the same probability
now how large N has to be, for a given large value of n, in order that the
permanent of the random matrix M should be different from zero with
probability Si a whore 0 < a < 1. By other words if M = (e;fc) we want
to evaluate asymptotically the probability P(n, N) of the event that there
exists at least one permutation jv j2, . . . , jn of the numbers 1, 2, . . . , n such
that the product £^. e2j! . . . enjn should be equal to 1. A second way to
formulate the problem is as follows: we shall say that two elements of a matrix
are in independent position if they are not in the same row and not in the
same column. Now our question is to determine the probability that the
random matrix M should contain n elements which are all equal to 1 and are
pairwise in independent position. A third way to state the problem is: what
is the probability of the event that the permanent of the random 0 — 1
matrix M should be positive ?
We prove in § 1 (Theorem 1) that if
(1) N(n) = wlog n -(- en + o(n)
where c is an arbitrary real constant, then
(2) lim P(n,N(n)) = e-2e~c.
This implies that if
NJn) — nlogn
(3) hm ~^~' 5_ = + co ,
n^ t-» n
then
(4) lim P(n, N^n)) = 1 ,
while if
,., ,. A^w) — wlogw
(5) hm ^ ' 2— = — co ,
455
625 RANDOM OBJECTS

456
erdOs-r^nyi
then
(6) lim P(n,N2(n)) =0,
<■+ =
This result can be interpreted also in the following way, in terms of
graph theory. Let rn N be a bichromatic random graph containing n red
and n blue vertices, and N edges which are chosen at random among the n2
possible edges connecting two vertices having different colour (so that each
jn2\
of the possible choices has the same probability). Then P(n, N) is equal
\Lv J
to the probability that the random graph Tn N should contain a factor of
degree 1, i.e. Tn N should have a subgraph which contains all vertices of
Tn N and n disjoint edges, i.e. n edges which have no common endpoint.
Clearly if the permanent of a matrix M consisting of zeros and ones
is positive, then the matrix M does not contain a row or column all elements
of which are equal to 0 (called in what follows for the sake of brevity a 0-row
resp. 0-column), but conversely, MM does not contain a 0-row, nor a 0-column,
it is not sure that its permanent is different from 0. However, from our result
it follows that this is "almost" sure. As a matter of fact, Theorem 1 can be
interpreted as follows: if P(n, N) denotes the probability that perm (M) > 0
and Q(n, N) the probability that M does not contain a 0-row or a 0-column,
then if N = N(n) is chosen so that for n —> °° we should have Q(n, N(n)) —>- 1,
then we have also P(n, N(n))-+ 1.
One can state this result somewhat vaguely also in the following way:
if the permanent of a random matrix with elements 0 and 1 is equal to 0,
then under the conditions of Theorem 1 this in most cases is due to the presence
of a 0-row or a 0-column.
In § 2 we deal with a somewhat simpler variant of the problem, when
the elements e,-,- (1 ^. i ^. n, 1 < )> |S n) of the matrix M are independent
random variables each taking on the values 0 and 1 with probability 1 — p
and p respectively. The results obtained are analogous to those of § 1. In § 3
we add some remarks and mention some unsolved problems.
Besides elementary combinatorial and probabilistic arguments similar
to that used by us in our previous work on random graphs (see [1], [2], [3],
[4], [5]) our main tool in proving our results is the well-known theorem of
D. Konig (see [6]), which is nowadays well known in the theory of linear
programming, according to which if M is an n by n matrix, every element
of which is either 0 or 1, then the minimal number of lines (i.e. rows or columns)
which contain all the Is, is equal to the maximal number of 1-s in independent
position. As a matter of fact, for our purposes we need only the special case
of this theorem, proved already by G. Fbobenius [7], concerning the case
when the maximal number of ones in independent position is equal to n.
§ 1. Random square matrices with a prescribed number of zeros
and ones
Let P(n, N) denote the probability of the event that the random matrix
M (M£ ^#(w, N)\ has a positive permanent. According to the theorem of
Fbobenius—Konig (see [6] and [7]) 1 — P(n, N) is equal to the probability
that there exists a number k such that there can be found k rows and n — k — 1
626 MISCELLANY

ON RANDOM MATRICES
457
columns of M which contain all the ones (0 gL k <I n — 1). If we denote by
Qk (n, N) the probability that there can be found k rows and n — k — 1
columns or k columns and n — k — 1 rows which contain all the ones, and k
is the least number with this property, then clearly
(1.1)
0 ^ 1 - P(n, JV) = 2 Qk(n,N).
fc-0
Now we shall prove that if
(1.2) N(n) = nlogn-}-en-{-o(n)
where c is a real constant, then
(1.3)
further that
(1.4)
PS
lim 2 Qk(n,N{n)) = 0,
lim Q0(n, N(n)) = 1 — e^2e~
Clearly (1.1), (1.3) and (1.4) imply that
(1.5) lim P(n,N(n)) = e-^,
which is the result we want to prove. Thus it remains only to prove (1.3)
and (1.4). Let us consider first (1.4). Clearly 1 — Q0 (n, N (n)) is equal to the
probability of the event that the random matrix M does not contain a 0-row
or a 0-column. Thus we have
(1.6)
where S0 = 1 and
l-Q0{n,N(n))= 2 (- 1)'S.
2n
1=0
(1.7)
/1=0
j n
i — h
((n — h) (n — i + h)
N(n)
further for each I >: 0
(1.8)
N(n)
(1 = 1,2, ..., 2»)
- l)'St^ 1 - Q0(n,N(n)) < 2(- 1)'^.
N.
As clearly for each fixed value of i and for n —v °o; if N(n) is defined by (1.2)
we have
(1.9)
^ = ^7-(1+0(1)),
627 RANDOM OBJECTS

458
EB,D(58-R£XYI
it follows that
(1.10)
Um(l-Q0(n,N(n)))= V(-l)
2'e~
rfhus (1.4) is proved. Now let us prove (1.3).
Let us suppose that M is a matrix such that all the ones of M are
contained in k columns and n — k — 1 rows (k ^ 1), and k is the least number
with this property. Then the matrix M can be partitioned into four matrices
A, B, C, D as shown by Fig. 1, so that D consists only of zeros. Then clearly
each column of C contains at least two ones, because if a column of C would
contain not more than a single 1, then by leaving out this column and adding
the row in which this 1 is contained, we would get a system oi k — 1 columns
and n — k rows which contain all the ones, in contradiction to our supposition
of the minimum property of Tc.
k
n — k
-k- 1
k+ 1
A
C
B
D
Thus it follows that
(1.11) Qk(n,N)<2
Fig. 1.
ln{n — k — 1) +k(k— 1)
I n
\k + lj
k + 1\"
2
N -2k
and thus, that
(1.12)
Qk(n,N(n))£
4log21
I Vn
for k = 1,2,
where A is a positive constant depending only on c. Thus we obtain
(1.13)
W Qk(n,N{n))<1Jn°?^~
^ n v ;;-yre-4iog2w
From (1.13) we obtain (1.3) and this completes the proof of (1.5).
Thus we obtained the following
Theorem 1. Let <_J?{n,N) denote the set of all nbyn square matrices, among
the n2 elements of which N are equal to 1 and the other n2 — N to 0. Let M be
' n2
selected at random from the set ,y£{n, N) so that each of the
N
elements of the
set <_/f{{n, N) has the same probability
N
to be selected. Let P(n, N) denote
628 MISCELLANY

OX RANDOM MATRICES
459
the probability of the event that the permanent of the random matrix M is positive.
Then if
N(n) = n log n -{- en -{- o(n)
where c is any real constant, we have
UmP(w,iV(7j)) = c-2e-c.
§ 2. Random matrices with independent elements
In this § we prove the following theorem which is a variant of Theorem 1.
Theorem 2. Let Mn (p) be a random n by n matrix whose elements £,_,■
(l^^^w; 1 ^. j gL n) are independent random variables such that
(2.1) P(elV=:l) = p and P(e,;■ = 0) = 1 — p .
Let Pn (p) denote the probability of the event that the permanent of the
random matrix Mn (p) is positive. Then we have for
logw + c (1)
(2.2) Pn = ~ +o-
n \nj
(2.3) lim Pn(pn) = e-ar- .
n ->™
Proof of Theorem 2. The proof follows step by step the proof of Theorem
1. We have
(2.4) 0<1-Pn(p)^ 2 QUp)
k-0
where QkiU{p) denotes the probability that there can be found k rows and
n — k — 1 columns, or k columns and n — k — 1 rows of Mn (p) which
contain all the l-s, and k is the least number with this property. In this case
we have
(2.5) l-e0>n(P)=^(-l)''Sf
2n
[-0
where S* = 1 and
s-2u,u-4
P)
in-h(i-h)
Thus we have for each fixed value of i if (2.2) holds
o ip - ic
(2.7) ]imSf =
i!
and therefore
(2.8) lim(l-0o>n(p„)) = c
= e~2e-
629 RANDOM OBJECTS

460
ERD(5S-Rl5NYI
On the other hand we have now for k = 1, 2, . . . ,
n — 1
n
{k + l)
(2.9) Qkyn(p) £ 2
and thus
(2.10) Qk,n(Pn) ^
»2fc(l — »,\(fc+l)(n-fc)
l^f for 4 = 1,2,..
where the constant B depends on c onlv.
Thus
(2.11) Urn ^^,,(^) = 0
and Theorem 2 follows.
w — 1
§ 3. Some further remarks
The results of §§ 1 and 2 could be generalized for rectangular matrices
of size m by n where m < n. In this case the question is: what is the
probability that a random matrix of size m by n consisting of zeros and ones should
contain m elements in independent position, which are all equal to 1 ?
Another possible generalization of our results would be to determine
the probability distribution of the maximal number of ones in independent
position in a random square matrix.
One may ask what can be said about the distribution of the value of
the permanent of a random square matrix, under conditions of Theorems
1 and 2? It is easy to compute in both cases the mean value of the permanent
perm (M); we have evidently under conditions of Theorem 1
E(perm(lf)) = n\
and under conditions of Theorem 2
N(n) — nj
N(n) ,
E(perm(Jfn(pn)))=n!pS.
It is easy to see, that these expressions are of the form e"l0g|0gn + o(")
and thus tend rather rapidly to -\-°°. However one can not draw any
conclusion from this fact, because as is easily seen, the variance of the permament
is still much larger than the square of the mean value. An interesting related
problem is of course to evaluate under the conditions of Theorem 1 and 2 the
probability of the determinant of the random matrix being different from 0.
Another problem arises in connection with the graph-theoretical
interpretation of the questions discussed in the present paper: To compute the
probability that a random graph having n vertices and N edges should contain
a factor of the first degree? We hope to return to these problems in another
paper.
(Received November 11, 1963)
630 MISCELLANY

OX' KAXDOM MATRICES 461
BEFEBENCES
[1] Erd6s, P.—Benyi, A.: "On random graphs, I." Publicationes Mathematical
(Debrecen) 6 (1959) 290—297.
[2] Erd6s, P.—Benyi, A.: "On the evolution of random graphs." Publ. of the Math.
Inst, of the Hung. Acad. Sci. 5 (1960) 17—61.
[3] Erd6s, P.—Renyi, A.: "On the evolution of random graphs." International Stat.
Inst., 32. Session, Tokyo, 1960, 119. 1—5.
[4] Erd6s, P.—Benyi, A.: "On the strength of connectedness of a random graph."
Acta Math. Acad. Sci. Hung. 12 (1961) 261—267.
[5] Erd6s, P.—Benyi, A.: "Asymmetric graphs." Acta Math. Acad. Sci. Hung, (in print).
[6] K5nig,D.: "Graphok 6s matrixok." Matematikai es FizikaiLapok 38 (1931) 116—119.
[7] Frobenius, G.: "Uber zerlegbare Determinanten." Sitzungsberichte der Berliner
Akademie 1917, S. 274—277.
631 RANDOM OBJECTS

Studia Scientiarum Mathematicarum Hungarica 3 (1968) 459—464.
ON RANDOM MATRICES II
by
P. ERDOS and A. RENYI
§ 0. Introduction
This paper is a continuation of our paper [1]. Let Jl{n) denote the set of all
n by n zero-one matrices; let us denote the elements of a matrix M„£Jb?(n) by Sjk
(1 sjsj; 1 =?A'-g/7). Let p denote an arbitrary permutation p — (p1, p2) ..., pn)
of the integers (1, 2, ..., n) and /7„ the set of all n\ such permutations. Let us put
for each p £ /7„
(0-1) e(/?) = £lpi-e2p2 ... enpn.
Thus the permanent perm (Af„) of M„ can be written in the form
(0. 2) perm (M„) = 2 e(p)
p£//„
Thus each e,(p)(p£Iln) is a term of the expansion of perm (Mn).
Let us call two permutations p' = (p\, ..., p'„) and /?" = (/?]', ...,/^)
(p' dlln, p"€lln) disjoint if p'^pl for fc= 1,2, ...,«. Let now define (for each
A/„ £J?{nj) v = v(M„) as the largest number of pairwise disjoint permutations
p(1), ..., //v> such that 8(/7(0) = l (/=1,2, ..., v). Clearly
(0. 3) perm (Af„) is v(M„)
thus v(M„)^l is equivalent to perm(M„)>0.
Let us denote by Jl(n, N) the set of those m by n zero-one matrices, among
the n2 elements of which exactly N elements are equal to 1 and the remaining n2 —N
to 0 (0<A^-<m2). Let us choose at random a matrix M„s from the set Jt(n, N)
( 2N
with uniform distribution, i.e. so that each of the
elements of J/{n, N) has the
same probability ,
l
to be chosen.
Let us denote by P(n, N, r) the probability of the event
v(M„tN)mr (r=l,2, ...).
Clearly P(n, N, 1) is the probability of the event perm (M„ A,)>0.
In [1] we have shown that if
(0.4) Nl(n) = n log n + en + o(n)
where c is any fixed real number, one has
(0.5) lim P(«, Nx{n), 1) = e~2'-c.
632 MISCELLANY

460
P. ERDOS AND A. RLNY1
This implies that if oj(n) tends arbitrarily slowly to + <=° for n -*• + <=° and
(0. 6) N*ftn) = n log n + <x>(n)n
then
(0.7) limP(«,iVt(«), 1)=1.
II—» oo
In the present paper we shall extend this result, and prove the following
Theorem 1. For any fixed natural number r, if
(0. 8) N?(n) = n log n + (r—l)n log log n + nco(n)
where <x>(n) tends arbitrarily slowly to + <=° for »-► + «>, we have
(0.9) lim P(n,N*(n),r)=l.
Clearly (0. 7) is the special case r=\ of (0. 9). (0. 5) can be generalized in a
similar way (see Theorem 2). Evidently, the interesting case is when co{n) tends
slower to + <=° than log log n.
The method of the proof of Theorem 1 and 2 follows the same pattern as that
in [1].
In § 2 we formulate — similarly as in [1] — an analogous result for random
zero-one matrices with independent elements, while in § 3 we add some remarks
and mention some related open problems.
§ 1. Random matrices with a prescribed number of zeros and ones
We prove in this § Theorem 1. We suppose ca2 as the theorem was proved
for r= 1 in [1].
Suppose that M is an n by n zero-one matrix belonging to the set Jl(n, N*(n))
where N*(n) is denned by (0. 8), and suppose that v(M)^r— 1.
Clearly we can delete from each row and column of such a matrix r— 1 suitably
selected ones so that the permanent of the remaining matrix M' should be equal
to 0. As regards the matrix M' we distinguish two cases: either the deletion can be
made so that M' contains a row or a column which consists of zeros only, or not.
Let us denote by <2i(«, r) the probability of the first case, and by Q2(n, r) the
probability of the second case. Clearly if a row (column) of M' consists of zeros only,
the corresponding row (column) of M contains at most r— 1 ones. Conversely,
if M contains such a row or column, then clearly v(Af)Sr-l. Thus Q^{n, r) is
equal to the probability of the event that in M there is at least one row or column
which contains at most r—\ ones. Thus we have
(1.1) QAn,r)^2nZ
j'=o
-"n
U)
[Nr(n) j) = 0(e_ra(n)) = o(1)
(NJn))
633 RANDOM OBJECTS

ON RANDOM MATRICES, 11
461
Let us pass now to the second case. Let k be the least number such that one can
find in M' either k columns and n — k — l rows, or k rows and /7 — k— 1 columns,
which contain all the ones of M'\ according to the theorem of Frobenius (see [2]
[nil
and [3]) as perm (Af') = 0, such a k exists, and l^/c^f —-— because the case
fc = 0 has already been taken into account (this was our first case). We may suppose
that all ones of M' are covered by k columns and n — k—l rows (the probability
of the other case when the ones of M' are covered by k rows and n — k—l columns
being the same by symmetry). It follows — as in [1] — that M' contains a submatrix
C consisting of k+ 1 rows and k columns, such that each column of C contains
at least two ones. Let C be the corresponding submatrix of M. It follows that
(1.2)
Q2(n,r)^2 Z Qk
where qk
'
l^sk^
\n- ll
I 2 J
is the probability of the event that M contains a k + 1
by k submatrix C such that each column of C contain at least two ones, and the
submatrix D of M formed by the same rows as C and by those columns which do not
intersect C, contains at most r—\ ones in each row. Evidently
0-3) qk*
n
k
n
k+l.
k+l
2
k (k+ 1)0--1)
2
j=0
(k + l)(n- k))(n(n- k-l)+ k(k-l))
J ){ N*-2k-j J
2 '
It follows from (1. 2) and by an asymptotic evaluation of the expression at the
right hand side of (1. 3) that
(1.4)
As
(1.5)
Q2(n,r)=o(l).
1 - P(«, NXn), r) = 6,(«, r)+ Q2(n, r)
it follows in view of (1.1) and (1. 4) that (0. 9) holds. Thus Theorem 1 is proved.
By the same method we can prove the following result, which generalizes
(0. 5) for ri=2.
Theorem 2. If
(1. 6) N£n) = n log/7 + (r— \)n log log 77 + en + 0(/7)
where rSl is an integer and c is any real number, we have
(1.7)
lim ?{n,Nr{n),r) = e ^1)1 .
634 MISCELLANY

462
P. ERDOS AND A. RENYI
§ 2. Random zero-one matrices with independent elements
Similarly as in [1] let us consider now random n by n matrices A/= (¾)
(1 ^=i,j = n) such that the e;j- are independent random variables which take on the
values 1 and 0 with probabilities pn and (1 — /?„). It can be shown that the following
result is valid:
(2.1)
Theorem 3. For any fixed natural number r, put
log « + (/•— 1) log log n + co(n)
Pn
where co(n) tends arbitrarily slowly to + <=° and let M be an n by n random matrix
the elements of which are independent random variables, taking on the values 1 and 0
with probability pn and 1-/7,, respectively. Then the probability of v(M)^r tends
to 1 for «^+<=°.
Note that the special case r = 1 of Theorem 3 is contained in Theorem 2 of our
previous paper [1].
As the idea of the proof is essentially the same as that of (0. 9), and the
computation even somewhat simpler, we omit the proof of Theorem 3. Theorem 3 can be
sharpened in the same way as Theorem 2 sharpens Theorem I.
Let us put
(3.1)
§ 3. Remarks a«d open problems
H(n,k)= min (perm(M„)).
M„6.«(n)
Clearly fi(n, 1)=1 and /<(«, 2) = 2; however,for /c^3 the question concerning
the value of fi(n, k) is open. We have clearly fi(k, k) = k\ and
(3.2)
fi(k,k-l) = k\
1
2\
+
(- If
k\
but the value of /<(«, k) for n^k + 2 is not known. Clearly for determining fi(n, k)
it is sufficient to consider those matrices Mn which contain exactly k ones in each
row and in each column. As each such matrix is the sum of k disjoint permutation
matrices, i.e. for such a matrix we have v(M„) = /c, thus the problem of determining
fi(n, k) is the same as the problem raised by Ryser (see [7], p. 77) concerning the
minimum of the permanent of n by n zero-one matrices having exactly k ones in
each row and each column. Of course for particular values of n and k one can
determine /<(«, k) (e.g. /i(5, 3) = 12), but what would be of real interest is the
asymptotic behaviour of fi(n, k) for fixed k^3 and n-* + <=°.
Let us put
(3.3)
lim inf ]/^{n, k) = /½.
635 RANDOM OBJECTS

ON RANDOM MATRICES, II
463
It seems likely that fik>l for /c^3. One reason for this conjecture is that if the
conjecture of Van der Waerden is true, we have
k"n\
(3.4) /,(«,*) S—~
(kY
77"
e
k k
i.e. jufc^— >1 for /c^3. We guess that /ik is even larger than — .
If in particular Mn is the matrix defined by Sj j = £j J + i =£j j_i = 1 (we put
Ej,m = Bj,m-n f°r w > «) and 8j7 = 0 if |/—_/| =2, then it can be easily shown that
perm (Mn) — Ln + 2 where Ln is the 77-th Lucas number, i.e. the 77-th term of the
Fibonacci-type sequence
(3.5) 1,3,4,7,11,18,...
and
,* *\ r Vr >'5 +J 3
(3. 6) hm \Ln = > - .
«->=» z e
As regards /1(77, fc), at present it is known only that
(3.7) lim /1(77,3)= +».
This was conjectured by Marshall Hall and proved by R. Sinkhorn [8]. As
a matter of fact, Sinkhorn proved /1(77, 3) = 77 for all /7 =r 3. Of course (3. 7) implies
lim u{n,k) = +«- for fc = 4, 5, ... too.
/1-* + 00
An interesting open problem is the following: evaluate asymptotically
P(t7, 77 log 77 + (/--1 )77 log log 77, r) if r is not constant, but increases together with n.
There is a striking analogy between Theorem 1 and the following well known
result (see e.g. [4]): If N*(n) balls are placed at random into 77 urns, and N*(n) is
given by (0. 8) (with <u(n)-»- + <=°) then the probability of each urn containing at
least r balls, tends to I for 77 -► + =. The relation between this problem and that
of § 1 is made clear by the following remark. If we interpret the rows (columns)
of M as urns and the ones as balls, then there are 77 urns, and each of the N*{n)
,,balls" falls with the same probability I/77 in any of the ,,urns".
In another paper ([5]) we have proved the following theorem (Theorem 1 of [5]):
a random graph T(t7, N) with 77 vertices where 77 is even and N = ^ n log 77 +77 co(n)
edges where <u(w) ■*+» for 77 -*• + <=°, contains a factor of degree one with probability
tending to 1 for n— +<*>.
Theorem 1 of the present paper suggests the following problem: does a random
graph r(n, N) where 77 is even and
yV = - 77 log 77 -\ — 77 log log 77 + 0)(77)77
where 01(/7) ■* + », contain at least r disjoint factors of degree one with probability
tending to 1 for 77^=°? To prove this, besides the method of [5] the results of [6]
have to be used.
636 MISCELLANY

464 P. ERDOS AND A. RENYI: ON RANDOM MATRICES, II
REFERENCES
[1] Erd6s, P. and Renyi A.: On random matrices, Magyar Tad. Akad. Mat. Kutatd Int. Kdzl
8 (1964) 455—461.
[2] Frobenius, G.: Uber zerlegbare Determinantcn, Sitzungsberichle der Berliner Akademie, 1917,
274—277.
[3] Konig, D.: Graphok es matrixok, Mat. Fiz. Lapok 38 (1931) 116—119.
[41 Erdos, P. and Renyi, A.: On a classical problem of probability theory, Magyar Tud. Akad.
Mat. Kutato Int. Kdzl. 6 (1961) 215—220.
[51 Erdos, P. and Renyi, A.: On the existence of a factor of degree one of a connected random
graphs, Acta Math. Acad. Sci. Hungar. 17 (1966) 359—368.
[6] Erdos, P. and Renyi, A.: On the strength of connectedness of random graphs, Acta Math.
Acad. Sci. Hungar. 12 (1961) 261—267.
[7] Ryser, H. J.: Combinatorial mathematics, Carus Math. Monographs, No. 14. Wiley, 1965.
[8] Sinkhorn, R.: Concerning a conjecture of Marshall Hall (in print).
Mathematical Institute of the Hungarian Academy of Sciences, Budapest
(Received March 12, 1968.)
637 RANDOM OBJECTS

Chapter 15
Latin Squares
The contents of [120] are described by the title.
The methods of [314] are completely number-theoretic. However, the
result is of such importance as to warrant its inclusion in this volume. Let
N{n) denote the number of mutually orthogonal Latin squares of order n.
Euler conjectured that if n = 2(mod 4) then N(n) = 1. This was disproved by
Bose, Parker, and Shrikhande. In fact, N(n) = 1 only for n = 1, 2, 6. Chowla,
Erdos, and Straus show that N(n) actually approaches infinity and does so at
a "respectable" rate, N(n) > en"9'. Richard Wilson has recently improved
thi.s bound to N(n) > cnini.
The topics of this section, and many related ones, are covered in a book
by Denes and Keedwell on Latin Squares that is currently in press
(Hungarian Academy of Sciences, Academic Press, joint publishers).
Papers in Chapter 15
[120] (with I. Kaplansky) The asymptotic number of Latin rectangles
[314] (with S. Chowla and E. G. Straus) On the maximal number of pairwise
Orthogonal Latin squares of a given order
[413] (with A. Ginzburg) On a combinatorial problem in Latin squares
639 LATIN SQUARES

THE ASYMPTOTIC NUMBER OF LATIN RECTANGLES.*
By Paul Eedos and Irvixg Kaplaxsky
1. Introduction. The problem of enumerating n by k Latin rectangles
was solved formally by MacMahon [4] using his operational methods. For
fc = 3, more explicit solutions have been given in [1], [2], [3], and [5].
While further exact enumeration seems difficult, it is an easy heuristic
conjecture that the number of n by lc Latin rectangles is asymptotic to
(»!)fcexp (—kC2). Because of an error. Jacob [2] was led to deny this
conjecture for k = 3; but Kerawala [3] rectified the error and then verified the
conjecture to a high degree -of approximation. The first proof for & = 3
appears to have been given by Eiordan [5].
In this paper we shall prove the conjecture not only for lc fixed (as
n—» co ) but for fc < (log«)3/2~e. As indicated below, a considerably shorter
proof could be given for the former ease. The additional detail is perhaps
justified by (1) the interest attached to an approach to Latin squares (lc = «),
(2) the emergence of further terms of an asymptotic series (4), (3) the fact
that (log ft)3/2 appears to be a "natural boundary" of the method. (We
believe however that the actual break occurs at fc = ft1/3.)
2. Notation. An n by lc Latin rectangle L is an array of n rows and fc
columns, with the integers 1, • • • , n in each row and all distinct integers in
each column. Let N be the number of ways of adding a (lc -\- l)-st row to L
so as to make the augmented array a Latin rectangle. We use the sieve method
(method of inclusion and exclusion) to obtain an expression for X. From n !,
the total number of possible choices for the (fc-(- l)-st row. we take away
those having a clash with L in a given column—summed over all choices of
that column, then reinstate those having clashes in two given columns, etc.
The result can be written
(1) N = Jt(—)rAr(n — r)l
r=0
where Ar is the number of ways of choosing r distinct integers in L. no two
in the same column. In particular A0 = 1, At = nlc. To estimate the higher
values of Ar we apply t-he sieve method again. The total number of ways of
* Received November 30, 1945.
230
641 LATIN SQUARES

TUB ASYMPTOTIC NUMBER OF LATIN EECTAXGLES. 231
selecting r elements of L, not necessarily distinct integers but with no two in
the same column, is nCrhr. This over-estimates A,', we have to take away those
selections which include a specified pair of l's, 2's, • • ■ , or n's, then reinstate
those which include two pairs, etc. We may write the result
(2) Ar = Z(—)'B(r,s).
s
Here B(r,s) is precisely defined as follows. Take any s of the n tC„ pairs of
l's, • • • , n's which can be formed in L. Suppose that this.selection involves in
•all y elements; y may be as large as 2s, or as small as the integer for which
j,C2 = s. Find the number of ways of adjoining r — y further elements, so as
to form a set of r elements with no two in the same column. The result of
summing over all choices of s pairs is, by definition, B(r,s). We note in
particular that
(3) J5(r,0) =„C',fcr
(4) B(r, 1) =w*C2 „-2CV-2 r-2.
The B's may be analyzed further as follows. Let F(s, t) be the number
of ways of choosing s pail's of l's, • • • , n's, which use up t elements in all,
and for which no two of the t elements lie in the same column. The number
of ways of expanding this Selection of t elements to r elements, with no two
in the same column, is n-tCr-t ^r~t- Hence
(5) B(r, s) = 2 F(s, t) n.tCt.t lC-K
t
It is to be observed that extreme limits for the summation in (5) are given by
t ^ 2s and s ^ (C2 or, more generously, Vs ^= t.
These quantities F(s, t) are the ultimate building blocks from which the
exact value of N is constructed. We shall discuss them further in 4. For the
present the following crude inequality will suffice:
(6) ^F(sJ) < n'/2(fc2i)'2.
The proof of (6) is as follows. The left hand side is just the number of ways
of choosing a set of (any number of) pairs which involve in all precisely t
elements. In such a choice at most \t/2~] distinct integers are permissible,
and these may be taken in less than nt/2 ways. In all we have at most
642 MISCELLANY

232
PAUL BRDOS AND IRVING KAPLANSKY.
tC2 < t2 pairs to dispose of in the selection. For each of these t2 pairs we
have icCzt/2 < h2t possibilities and hence for all of them at most (fc2£)'J
choices. This establishes (6).
The various quantities defined in this section will be used without further
explanation in the remainder of the paper.
3. Proof of the main result. We first prove
Theorem 1. If k < log n)3/2~e, then for sufficiently large n
(7) \NJc/n[ — l | < n~c
where c is a positive constant depending only on e.
Proof. Define A{r, x) by
a>-l
(8) A{r,x)=^{—YB{r,s),
»=i
where x = [(logn)1"^]. Then by the sieve's well known property of being
alternately in excess and defect we have
(9) \Ar — B(r, 0) — A(r, x)\ ^B(r, x).
In (1) make the substitution
Ar={Ar — B(r,0)—A(r,x)} + B(r, 0) +A(r,x)
and use (3) and (9). We find
(10) \N — 2 (—YnCrTcr{n — r) ! |^| G\+H,
where
(11) 0 = 2 (—)rA{r,x){n — r)\,
r=0
(12) H = ^B(r,x)(n — r) I.
We proceed to study 0. With the use of (8) and (5), and an interchange
of summation signs, (11) becomes
643 LATIN SQUARES

THE ASYMPTOTIC NUMBER OF LATIN RECTANGLES. 233
G/H.\ =2 { — )'^F{S, t) 2 ( — yn-tCr-tb'-'/Wr
s-l t r-t
where (n)r = n(n— 1) ' ' ' (n — r-f-1) is the Jordan factorial notation.
The change of variable r = t -\- u transforms the final sum into
(—)*/(«)'2 (—*)"/"!= (-)^^-5)/((1).
where 6 is the remainder after n — t terms of the series for e~k. Then
(13) |ff |e*/n!^2 ZF(s,t)(l + 8eJ<)/(n)t.
S=l t
As noted above, the limits for I lie between V s and 2s. Hence t ^ %x < 2 log n.
From this we readily deduce
(14) l/(n)t<c1n-t,
(15) 0e* < c2,
where Ci, c2 are absolute constants. From (6), (13), (14), and (15) we
obtain
| ff [ e*/n\ < c, 2 {lc-ty°/nt/2
with c3 = d(l +c2). In the fraction under the summation sign, the
logarithms of numerator and denominator are respectively of the orders t2 log log n
and tlogn. Since i < 2(logH)1~£, it follows that for large n
where Ci is a positive constant depending only on £. Hence
(1G) | 67 | ek/n ! < 2xc3n-c* < irc-\
We next turn our attention to the term II given by (12). From (5)
and an interchange of orders of summation,
IIJn ! = 2 F(x, t) 2 n-tCr-t fc'-y(n),..
( r=t
The final sum is the product of l/(?i)(.by a portion of the series for e*.
Hence
644 MISCELLANY

234 PAUL ERDOS AND IRVING KAPLANSKY.
H/n\ < ek1lF(x,t)/(n)t < ^2 {k2ty'/n*'2
t t
by (6) and (14). The fraction to be estimated is the same as above but
the summation now starts at V.T 2: c6(log m) (1~e)/2. It follows that t log n
5; c6(log n)3/2~c/2, and we are able to swallow up a further term e2k whose
logarithm is less than 2 (log n)3/a-E. Hence for large n
e2k(lc2t)t2/ii^2 < n~c->
and
(17) Hek/n ! < 2xr,>rc' < n-08.
Combining (16), (17), and (10), we obtain (7), for the sum on the left of
(10) may run to infinity at a cost of 0(n~°). This concludes the proof.
(We may note that for the case where le is fixed as n—■» oo, the proof
could be abridged as follows. We take x = 1; then the term G disappears,
and an estimate of H is easily obtained from (4).)
From Theorem 1 we readily derive our main result:
Theorem 2. Let f{n, Tc) be the number of n by h Latin rectangles and
suppose fc< (log n)3/2~e. Then
(18) f(n, fc)(«!)"*exp (,/7,) -» 1 as n-»oc.
Proof. From Theorem 1 it follows that f(n,i-\-\) lies between the
limits f(n,i)n] e~*(l ± ?rc). Taking the product from i = 1 to fe — 1, we
find that f(n, k) lies between the limits
(tt!)fcexp (-,,-(7,)(1 ± Ji"0)*.
Since (1 -\-rrc)k and (1—n~c)*—»1 as n—»oo, we obtain (18).
4. Further terms of the asymptotic series. A more careful argument
reveals that the error term in (7) is actually of the order of fc2?!,"1. By
detaching the term B{r, 1) as well as B{r, 0) in (2), we can reduce the error to
the order of fc4«r2. Continuing in this fashion, we may compute successive
terms of an asymptotic series. The existence of such a series was conjectured
by Jacob [2,337].
We shall merely sketch the results. Applying (1), (2), and (5) as we
did in 3, we find
xV/n! = 2 (—yZF(s,t)(e-*-e)/(n)t.
s t
645 LATIN SQUARES

THE ASYMPTOTIC NUMBER OF LATIN RECTANGLES.
235
The term 6 may be dropped and we have
V (n)s (n)a (n)*
Thus all that is required is evaluation of the F's. That F(l, 2) = nkC2 was
already implicitly noted in (4). For F(2,3) we observe that not more than
one integer may be Used, that there are then nkC3 choices for the three
elements, and 3 choices for the two pairs within them. Hence F(2,3) = 3n^C3.
Similarly F(2,4:) includes the term 3n^C4, corresponding to the choice of
only one integer. If two different integers are taken, there are ah initio
iiC2(kC2)2 choices; but we must eliminate selections which include two
elements in the same column. An application of the sieve process to this last
difficulty yields
F{2,±)=3nkCi + nC2{liC2y- — nkC2{k — iy + X,
where X is the number of instances in which integers i, j both occur in two
different columns. It is noteworthy that this is the first term which depends
upon the particular Latin rectangle to which a (k + l)-st row is being added.
A simple argument shows that X ^ nkC2(k— 1), so that X/(n)4 is of
order n"3 or less, as are all the later terms of (19). Hence we have, correct
up to n~2:
(20) Nf/ni = 1 — ?^ + 2p^ + *C>W*y+. ■ ■
(n)2 (n)3 (n)4
= l—ftC2/n+*C2(& +4) (34-7)/12^+- ■ ■.
By taking the product of the terms (20) from 1 to k— 1, we obtain the
asymptotic series for /(n, fc), the number of Latin rectangles:
(21) /(n,fc)(n!)-*exp (fcC2)
= 1—fcC3/n+fcC3(fc3 —3fc2+ 8fc —30)/12¾2+• • -.
For k = 3, the right side of (21) becomes 1 — 1/n — l/2n2 + • • • . In the
(.able below we compare this with the exact value given by Kerawala in [3].
n
5
10
15
20
25
1 — 1/n — l/2n2
.78
.895
.93111
.94875
.9592
Exact value
of (21)
.76995
.89560
.93126
.94881
-.95923
646 MISCELLANY

236
PAUL EEDOS AND IRVING KAPLANSKY.
In attempting to push the asymptotic series still further, we run into
the difficulty that terms like X, i. e., terms dependent upon the preceding
Latin rectangle, begin to play a role in (20). However, it may be that in
(21) at least the term in n~3 can be obtained without consideration of X, for
heuristically it seems likely that the "expectation" of X is o(n).
In conclusion we remark that the form of (21) strongly suggests that at
about k = n1/a the expression ceases to be valid. We are unable to prove this
rigorously.
Stanford University
and
University of Chicago.
BIBLIOGRAPHY
1. L. Dulmage, Problem E 650, American Mathematical Monthly, vol. ol (1944),
pp. 580-587.
2. S. M. Jacob, " The enumeration of the Latin rectangle of depth three . . .",
Proceedings of the London Mathematical Society, vol. 31 (1930), pp. 329-354.
3. S. M. Kerawala, "The enumeration of the Latin rectangle of depth three by
means of a difference equation," Bulletin of the Calcutta Mathematical Society, vol. 33
(1941), pp. 119-127.
4. P. A. MacMahon, Combinatory Analysis, vol. I, Cambridge, 1915, pp. 246-263.
5. J. Riordan, "Three-line Latin rectangles," American Mathematical Monthly,
vol. 51 (1944), pp. 450-452.
647 LATIN SQUARES

ON THE MAXIMAL NUMBER OF PAIRWISE
ORTHOGONAL LATIN SQUARES OF A GIVEN ORDER
S. CHOWLA, P. ERDOS, and K. G. STRAKS
1. Introduction. In the preceding paper Bose, Shrikhande, and Parker
give their important discovery of the disproof of Euler's conjecture on Latin
squares. In this paper we show that their results can be strengthened to imply-
that N(n), the maximal number of pairwise orthogonal Latin squares of order
n, tends to infinity with n. In fact there exists a positive constant c, such
that AT(») > nc for all sufficiently large n.
Our proof involves no new combinatorial insights, but is based entirely on
a number-theoretical investigation of the following inequality due to Bose
and Shrikhande.
Theorem A. If k < N(m) + 1 and 1 < u < m then N{km + u)
> mln{N(k), N(k + 1), 1 + X(m), 1 + N(u)} - 1.
The only other results on Latin squares which we need are due to H. F.
MacNeish.
Theorem B. (1) N(ab) > min{A/(a), AT(6)}.
(2) N(g) = q — 1 if q is the power of a prime.
In § 2 we give a proof of the fact that N(n) tends to infinity, using only
the most elementary tools. In § 3 we use Brun's method to obtain
quantitative results on the lower bound of N{n). Finally, in § 4, we discuss the
theoretical limitations on the results that can be derived from Theorem A.
2. Proof that
lim N(n) = °° .
Let x be an arbitrarily large positive integer. Let
(1) k + 1 = I] PT (P Prime).
Then by Theorem B we have
(2) N(k + 1) > 2X - 1 > x
and
Received August 8, 1959. This paper was written while the authors were members of the
Number Theory Institute (Summer, 1959) in Boulder, Colorado. The authors wish to
acknowledge with gratitude the opportunity for collaboration given them by this Institute.
204
648 MISCELLANY

I'AIRWISK ORTHOGONAL LATIN SOrAKES
205
(3) X(k) > x
since all prime factors of k are greater than x. Now set
(4) m, = kk Yl <t (° prime).
Note that while mi is defined in terms ot n, it has an upper bound which
depends on x alone. If n is sufficiently large then the interval (n/(k + l)mi,
(n — l)/kmi) contains a number mi such that
(5) m-i = 1 (mod kl).
Thus the least prime factor ot mi is greater than k.
If we set m = mim2 then from Theorem B and equations.(4), (5) we obtain
(6) \(m) > mm{.Y(»Zi), -V(w2)} > mhi{2h - 1, k\.
> k.
Thus the first condition of Theorem A is satisfied. Finally we set u = n — km.
Since we had chosen n/(k + l)mi < mi < (» — \)jkrri\ we have km + 1
< n < (k + l)m, so that
(7) 1 < u < m
which satisfies the second condition ot Theorem A. From (1), (4), and (5)
we see that n and km are incongruent module any prime less than x and
therefore u has no divisors less than x. Thus
(8) .V (u) > x.
Combining (2), (3), ((5), and (8) we obtain from Theorem A
(9) .V(«) > x - 1
for arbitrary x and sufficiently large w.
3. Numerical estimates on the lower bound of N(n). In addition to
Theorems A and B we need a result of Brim's sieve method. We shall use
the following theorem due to H. Rademacher (1).
Theorem C. Let P(D; x;pi, . . . , pr) denote the number of positive integers,
y, no greater than x which lie in an arithmetic progression A + tD(t = 0, 1, . . . ,)
whereQ < A < Dand (A, D) = 1 and so that y ^ at (mod pt),y ^ 6; (mod pi)
(i = 1,..., r).
If pi < . . . < pT are primes with pi > 7, then
w» w > DTi'Tr c'"■'""
where C and C are positive constants.
We shall also need the following simple fact.
649 LATIN SQUARES

20() S. CIIOWLA, 1>. ICKDOS, AND !■".. G. STRAUS
Lemma D. 27/e number of integers, y, no greater than x which are divisible by
a prime factor p of n so that p > nc, is no greater than x/cnc.
Proof. Obviously there are at most x/p numbers y divisible by p and
therefore the number in question is no greater than
x £ :< x ^2 -~ < x —■- = x/cn
„\,i p fin n n c
p>nc
since there are less than 1/r prime factors of u which exceed »''.
Case I. n is even. Pick k so that
(10) k=-l (mod 2[5r'°g2K]), k = 1 (mod 15);
k ^ 0 or -1 (mod p) for p prime, 7 < p < n'w: k < »"10.
We note that this restricts k to an arithmetic progression with difference
Z>=l5.2[",0,,"]<Cl«1'".
Thus by Theorem C there are at least
Cm1'10 ™ i
m\ Z-JL /"•'-jib'so
cxn log n (1/90)
81/910 /, 2 ni 79/900 ^ 81/910 /, 2
= c? n /log n — L n > c3n /log »
choices of &.
According to Lemma D the number of natural numbers below nino which
have a prime factor greater than n]'->0 in common with n does not exceed
90»s/90. Since 81/910 > 8/90, it follows from (11), and the fact that k has
no factors less than w,/9n, that we can choose k so that
(12) (k,n) = 1.
From (10) and Theorem B it follows that
(13) X(k)>n"*° -l>\nll'n,
N(k+ i) >min [\nl"\nllW\ - 1 > \nllt\
for n sufficiently large.
We now set » = »i + n^k where 0 < »i < k and let u = »i + «,&, where
we pick Hi subject to the following conditions.
(14) Ui ^ »i (mod 2);
«i ^ —njk (mod p), p \ k\ .
, , . . ( /> prime, 3 < p < &;
Mi p »2 (mod /)) '
r - 159/200
Note that the incongruence (mod 2) implies that ?« is odd.
650 MISCELLANY

PAIRWISK nRTIIOGONAt. I.AT1N SOUARK.S l'()7
The incongruences modulo 2, 3, and ~> can be satisfied by restricting iii to
a progression with difference 30. In order to apply Theorem C we need
(wi, 30) = 1. If (u1: 30) > 1 we write u1 = w,' . (ult 30) with (m/, 30) = 1 ;
then according to Theorem C, the number of such choices of k/ is at least
f. 159/200
I 1 - , <-• n f-l Z.TH/10 . 159/200 /, 2 /-v 79/100 A
(lo) -^-.——,- — C k ' > rtn /log n — L n ' > 0
30 log" /fe
for w sufficiently large.
From (14) we see that w is not divisible by any prime less than k and
prime to k. If u were divisible by a prime p which divides k, then »i, and
hence », would be divisible by p, in contradiction to (12). Hence from (13)
we obtain
(Hi) N{u) > k > X(k) > |w1/,J1 tor sufficiently large «.
Also, if we set m = (n — u)/k, then
(17) m > n/nino -(1+ »lb'J/200) > hn9'10
> w1'10 +(1 + m159'300) > u > 1
tor sufficiently large «. Finally, according to (12) and (14), all prime factors
ot m exceed k so that
(18) X(m) > k > X(k) > lnlln.
According to (17) and (18) our choice of k, u, m satisfies the conditions
of Theorem A. Thus by (13), (Hi), and (18) we have
(19) X(n) > lnll'n for all .sufficiently large even n.
Case II. n odd. Instead of applying Theorem (' to k we apply it to k + 1
with equation (10) replaced by
(10') k + I = 1 (mod 2Ul J) k + 1 = 2 (mod 15);
k + 1 ^ 0 or 1 (mod p) for 7 < p < »1/iK1;
-¾ + 1 < n"1".
The rest of the argument on k proceeds as before and equation (12) remains
unchanged, while (13) becomes
(13') X(k) > min|iw,/<J1, «1/,J0j - 1 > |»I/91,
X(k + 1) > »1/90 - 1 > |w1/,J1.
The choice of u is modified so that (14) is replaced by
(14') ui ?^ n« (mod 2); nx ■&. — nl/k (mod p)
tor all primes 3 < p < k which do not divide k\ while Uj ^ »2 (mod p) tor
all primes 3 < p < &; z*t < nu"Jl-'"\
651 LATIN SQUARES

208 s. c'howla, r. erdos, and e. g. STRAUS
It then follows from (13') and (14') that both n and m = (n — u)/k are
odd, and the remainder of the argument proceeds exactly as before to yield
the following.
Theorem. There exists a number »o so that for all n > n0 we have
N{n) > i»"91.
4. Remarks. The exponent 1/91 in our result is far from best possible. We
have not used the best available sieve method, nor have we even squeezed
the last drop out of the sieve method quoted. It seems, however, reasonable
to defer such efforts in the hope that other theorems of the type of Theorem A
can be developed, which may eliminate the twofold use of the double sieve of
Theorem C. This would be accomplished, for example, if either the occurrences
of both N(k) and N(k + 1) or the inequality N (m) + 1 > k could be
eliminated.
Theorem A can never lead to N(n) > »"2 since we must have n > mk and
N(m) + 1 > k so that k < m < n112 and N(k) < nxi\
On the other hand, our result seems to eliminate the possibility of a
reasonable modification of MacNeish's conjecture which would express N(n) in
terms of prime power divisors of n; since for any positive c there are infinitely
many n for which even the greatest prime power divisor is less than nc.
Reference
1. H. Rademacher, Beitrdge zur Viggo Brunschen Methode in der Zahlentheorie, Abbh. oVlath.
Sem. Hamburg, 3 (1924), 12-30.
Number Theory Institute (1959), Boulder, Colorado
University of Colorado
University of California, Los Angeles
652 MISCELLANY

ON A COMBINATORIAL PROBLEM IN LATIN SQUARES
P. ERDOS and A. GINZBURG1
1. Denote by Sn an arbitrary latin square with n elements [alt a2, ..., an).
A row and column of this square, intersecting on the main diagonal (i.e.
diagonal beginning at the left lower corner) will be called corresponding.
After striking out n — c arbitrary rows and corresponding columns,
a square Tc with c x c entries remains. Sueh a square will be called a principal
minor. It is clearly determined by denoting its c elements belonging to the
main diagonal.
Denote by Tc^^ ...,-, (ilt i2, . . ., iq = 1, 2, . . ., n all different) the number
of columns in Tc containing the elements a,,, a,-s, . . ., a,-, simultaneously.
Let h^ be the minimum of k;if ;2>. _ mi iq. We shall consider the following problem:
Assuming that n and ¥® are two given positive integers, what is the
minimal c (denoted by b), such that from an arbitrary Sn at least one Tc ean be
obtained with the prescribed &(?).
The problem is solved by a method used already in [1] and [2].
The question for the case of &(2) arises in connection with so called
generalized normal multiplication tables of groups (and other systems) [3],
[4], [5]. Such tables are complete (i.e. the product of any two group elements
appears explicitly in them) if and only if Ic^ Si 1. E.g. the following
10
8
5
4
2
1
0
9
7
4
3
1
0
14
8
6
3
2
0
14
13
C 5
4
1
0
13
12
11
3
0
14
12
2
0
12
11
9
0
13
10
9
7
11 8 , 6
10 i 7 5
is a generalized normal multiplication table of the group Z15 (the cyclic group of
order 15). The multiplication is performed according to the rule
9ij Qjk = gtk>
' Technion, Israel Institute of Technology, Haifa.
407
653 LATIN SQUARES

408
eedOs-gixzburg
where g-,j is the element placed at the intersection of the i-ih column and
■j-th row.
It can be directly inspected that in the above table &(2) = 1 and it is
really complete. It can also be shown that for n = 15 and k^ = 1, b 2^ 7.
2. Prom the definition of b follows directly
(1)
b > k^
Assuming that the main diagonal is occupied by one element av one improves
(1) to
In- 1
6-1)
1
b > &<«>
The following theorem gives an upper bound for b when W^ = 1.
Theorem. In any given n by n latin square there can be found a principal
minor of order not more than
Cn"+[ (log71)^+1 .
(G a sufficiently large absolute constant) containing every q-tuple
K> a*,. • • • - a;,) (*'i. H
iq — 1, 2, . . ., n all different) in some column.
Proof. We shall show that if 2t — [Cre'+I (logw)*+I] elements are chosen
at random on the main diagonal, then all but o
2t
of the principal minors so
obtained will contain every g-tuple in some column.
For this it will suffice to show that the number of principal minors in which
a given g-tuple (ax, a2, . . ., aq) does not oeeur in any of its columns is o
We shall now estimate this number.
n
2t)
\ ni
First we choose t elements at random. This can be done in wavs.
UJ "
Denote the chosen columns by ?,, i2, ..., it. In every is (1 5i s ^ t) the
elements av a2, . . ., aq occur in the rows denoted correspondingly by $\
j2s\ ..., fg\ When choosing the remaining t elements on the diagonal we have
to take care that none of the t g-tuples (f^\ ffi, ..., ffi) occurs amongst
the t chosen elements, for otherwise (av a2, ..., aq) would occur in a column
of our minor.
It is easy to show that there are at least of these g-tuples
q2 - q + 1
which are disjoint (this follows from the fact that there are at most q g-tuples
which contain the same element). Denote the number of disjoint g-tuples
by u. The number of ^-tuples not containing any of the u g-tuples equals by a
simple sieve process:
(2)
n — t
t
n — t — q
t-q
n — t — 2q
t — 2q
654 MISCELLANY

OX A COMBINATORIAL PROBLEM IN LATIN SQUARES
409
It can be shown that the sum (2) is o
[Cni+[ (logre)*+I]
where G is a sufficiently large absolute constant. Our proof of this fact uses
standard probabilistic arguments and is inelegant and therefore we supress
it. (A proof due to Prof. N. G. de Bbuijn is given in Addendum).
Now the total number of ways of choosing 2t elements on the diagonal
so that no column of the obtained principal minor should contain the fixed
g-tuple (av a2, . . . , aq) is less than
Since there are o-tuples we obtain for all but o choices principal
minors of order 2t with every <?-tuple in some column.
This completes the proof.
For 5=1 there is an explicit formula for the sum (2) (see [6] p. 316)'
In this case
n — t — v
t — v
Now
n — 2t
t
n
\2t
t and J? I— 1)"
n\ (n — 2t)\ (n — 2t)\ (2t)\ t) t\
(n — t)\t\ {n -3t)\t\ n\ (2t)\
n — 2t\
t I
(n — 2t) (n — 2t — 1) . . . (n — 3t + 1)
(n — t) (n — t — 1)
(n — 2t+ 1)
t
n — 2t + 1 I
<
n — t
i-LUi
For 2t = [Cn* (log n)*]
= o
if C > 2.
3. At present we can not decide if this theorem is close to being best
possible (from (1) it follows that in any case b > w'"+1). It seems though
that it will not be easy to improve it with the method of this paper. Following
a suggestion of Prof. H. Hanani we can show that for any p prime or a power
of a prime a quadratic table of p2 + p + 1 by p2 + p + 1 can be constructed
containing p3 -|- p2 + p +1 elements, such that any pair of these elements
occurs at least in one column of the table and no element occurs more than
once in one column or one row of it. This can be done as follows: it is well
known that from p2 + p + 1 elements p2 + p + 1 p + 1-tuples can be formed
with every pair of elements in one (and only one) p + 1-tuple. Now replace
655 LATIN SQUARES

410
BRDOS - GINZBURG
every one of the above p2 + p + 1 elements by p new elements and a ,,zero".
The total number of the obtained elements will now be p3+ p2 + p+ 1 and
they are divided in p2 + p + 1 p2 + p + 1-tuples. Every pair of the new
elements occur clearly in one of the p2 + p + 1-tuples. (Some of the pairs
occur in p -)- 1 such p2 + p + 1-tuples.) It is easy to see that the replacing
can be performed in such a way, that no element occurs twice in one row of
the obtained quadratic table. This table can now be extended to a latin square,
since it fulfils the condition of Ryseb [7].
4. A number of unsolved problems arise in connection with the above
one:
1) To find bounds for b in case when fc(?) > 1.
2) Given three positive integers n, fc(?), d. What is the minimal c such
that from an arbitrary Sn at least one Tc can be obtained (if any) with
lax (k;u
- #«> < d .
3) Given an arbitrary Sn. What is the c of the maximal minor (not
necessarily principal) in which all elements are different.
Addendum
The following proof is due to Prof. N. G. de Bbuijn.
m\ lu) tm — g^ lu\ Im — 2q)
t) \\)\t — q) UJU — 2g,
l r., . ._ ., _r. i
2
=— r (l+x)m3f'1-
2ni J
(l + x)"
dz,
where the integration is performed along a circle around 0. There is a saddle
point near x = —, so we take the radius of the circle equal to — .
t t
r i Tfi \ ~ q~\u
The contribution of the saddle point is about 1 — 1 —J times what
it would be if u = 0. If u = 0 it has the value , so under the assumption
t = C^+i (logm)«+! ,
u ■-
it,
■2+1
< 0 < 1
u and q are integers, q = 0(1), we find for £
't\i
So indeed it is o m~q
exp
m
m
(m|exp(-0C?+1logm).
if Cj is large enough.
(Received July 13, 1963)
656 MISCELLANY

ON A COMBINATORIAL PROBLEM IN LATIN SQUABES 411
BIBLIOGRAPHY
[1] Erd6s, P.: "Graph theory and probabiUty." I.: Can. J. Math. 11 (1959) 34—38.
II.: Can. J. Math. 13 (1961) 346—352.
[2] Erd6s, P.: "Some remarks on the theory of graphs." Bull. Amer. Math. Soc. 53
(1947) 292—294.
[3] Tamari, D.: "Les images homomorphes des groupoides de Brandt et l'immersion
des semi-groupes." Comptes rendus 229 (1949) 1291—93.
[4] Tamari, D.: "Representations isomorphes par de systemes de relations. Systemes
associatifs." Comptes rendus 232 (1951) 1332—34.
[5] Gdstzburg, A.: "Systemes multiplicatifs de relations. Boucles quasiassociatives."
Comptes rendus 250 (1960) 1413—16.
[6] Netto, E.: Lehrbuch der Kombinatorik. II. Auflage. Chelsea P. C. N. Y.
[7] Ryser, H. J.: "A combinatorial theorem with an application to latin rectangles."
Proc. Amer. Math. Soc. 2 (1951) 550—552.
657 LATIN SQUARES

Chapter 16
Geometry
Many of the problems in this section are of interest to the amateur
mathematician. In [106] it is shown that if an infinite set of points in the plane have all
their distances integral then they must be collinear. In most of the other
papers the set of distances achieved by n points in Euclidean space is
considered. For example (see [121]), in the plane, what is the maximum number of
unit distances among n distinct points? The bounds known are not very close.
In [109], which this editor failed to include, is the following beautiful
result. "If A, B, C are three points not in line (in a plane) and
k = [max (AB, BC)], then there are at most A(k + 1)2 points P such that
PA — PB and PB — PC are integral. For \PA — PB\ is at most AB and
therefore assumes one of the values 0, 1, . . . , k, that is, P lies on one of the
k + 1 hyperbolas. Similarly P lies on one of the (k + 1) hyperbolas
determined by B and C. These (distinct) hyperbolas intersect in at most A(k +1)2
points." This implies the results of [106].
A good general reference is the book Combinatorial Geometry in the Plane by
Debrunner, Hadwiger, and Klee. A paper by ErdSs on geometry problems
will appear in the Proceedings of the Teheran Mathematics Conference
(March 1972). Another paper by ErdSs, "On a problem of Greenbaum,"
will appear in the Canad. Math. Bull, in 1973 or 1974.
The results of [121] were improved by L. Moser, "On the different
distances determined by n points," Amer. Math. Monthly, 59(1952)85-91.
Papers in Chapter 16
[106] (with N. H. Anning) Integral distances
[121] On sets of distances of n points
[175] (with P. Bateman) Geometrical extrema suggested by a lemma of
Besicovitch
[325] On sets of distances of n points in Euclidean space
[vi] (with G. Szekeres) On some extremum problems in elementary
geometry
[576] (with G. Purdy) Some extremal problems in geometry
659 GEOMETRY

INTEGRAL DISTANCES
NORMAN H. ANNING AND PAUL ERDOS
In the present note we are going to prove the following result:
For any n we can find n points in the plane not all on a line such that
their distances are all integral, but it is impossible to find infinitely many
points with integral distances {not all on a line).1
Proof. Consider the circle of diameter 1, x2+y2=l/4. Let
pi, p2, ■ ■ • be the sequence of primes of the form 4&+1. It is well
known that
2 2 2
pi = a, + bi, Ui ^ 0, hi ^ 0,
is solvable. Consider the point (on the circle x2 + y2 = 1/4) whose
distance from (— 1/2, 0) is bi/pi. Denote this point by (x;, yt). Consider
the sequence of points (- 1/2, 0), (1/2, 0), (x«, yt), i= 1, 2, • • • . We
shall show that any two distances are rational. Suppose this has been
shown for all i <j. We then prove that the distance from (x/, y,-) to
(xi, yi) is rational. Consider the 4 concyclic points (— 1/2, 0), (1/2, 0),
(x,-, yi), (x,-, yi); 5 distances are clearly rational, and then by Ptolemy's
theorem the distance from (x,-, y,-) to (x,-, y,) is also rational. This
completes the proof. Thus of course by enlarging the radius of the
circle we can obtain n points with integral distances.
It is very likely that these points are dense in the circle x2+y2= 1/4,
but this we can not prove. It is easy to obtain a set which is dense on
x2+y2=l/4 such that all the distances are rational. Consider the
Received by the editors February 20, 1945.
1 Anning gave 24 points on a circle with integral distances. Amer. Math. Monthly
vol. 22 (1915) p. 321. Recently several authors considered this question in the
Mathematical Gazette.
661 GEOMETRY

5945l
INTEGRAL DISTANCES
599
point xi whose distance from (—1/2, 0) is 3/5; the distance from
(0, 1/2) is of course 4/5. Denote (-1/2, 0) by Ph (1/2, 0) by Pi,
and let a be the angle PJPiXi. a is known to be an irrational multiple
of ir. Let Xi be the point for which the angle PiP^Xi equals ia; the
points Xi are known to be dense on the circle X2-\-y2 = 1/2, and all
distances between x,- and x,- are rational because if sin a and cos a are
rational, clearly sin ia and cos ia are also rational.
To give another configuration of n points with integral distances,
let m2 be an odd number with d divisors, and put
2 2 2
m = Xi — ji.
This equation has clearly d solutions. Consider now the points
(m, 0), (0, yi) * = 1, 2, • • ■ .
It is immediate that all the distances are integral.
These configurations are all of very special nature. Several years
ago Ulam asked whether it is possible to find a dense set in the plane
such that all the distances are rational. We do not know the answer.
Now we prove that we cannot have infinitely many points
Pi, Pi, • • • in the plane not all on a line with all the distances PiP;
being integral.
First we show that no line L can contain infinitely many points
Qi, Qi, • • • . Let P be a point not on L, Qi and Q,- two points very
far away from P and very far from each other. Put d(PQt)=a,
diQiQi) = b, d(PQj) = c (d(A, B) denotes the distance from A to B.)
(1) c^ a+ b - 1.
Let QiR be perpendicular to PQj. We have
a < d(PR) + (d(QtR))*/d(PR), b < d{QiR) + {d{QiR)Y/d{QjR).
Thus from (1)
(.d(QiR)y( + ——\ > l
KV \d(PR) d(Q;R)J
which is clearly false for a and b sufficiently large. (d(QiR) is clearly
less than the distance of P from L.) This completes the proof.
There clearly exists a direction PiX such that in every angular
neighborhood of PiX there are infinitely many P,-.
Let P2 be a point not on the line PiX.
Denote the angle XPiPi by a, 0<a<ir. Evidently the P< cannot
form a bounded set. Let Q be one of the P,- sufficiently far away from
662 MISCELLANY

600
N. H. ANNING AND PAUL ERDOS
Pi, where the angle QPiX equals e (e sufficiently small). Denote
d(Pu Pi) = a, d(Pu Q)=b, <f(P2, Q) = c. We evidently have
c1 = a2 + b2 — lab cos (a — e).
a, b, c all are integers. From this we shall show that if b and c are
sufficiently large, e sufficiently small, then
(2) c = b — a cos a.
Put
c = b — a cos a + 5, 5 > 0.
Then
{f> — a cos a + 5)2 = 62 — 2a£ cos a + a2 cos2 a + 25(6 — a cos a)
+ 52 > a2 + 52 - 2aZ> cos (a - e)
if 6 is sufficiently large and e sufficiently small. Similarly we dispose
of the case 5<0. Thus (2) is proved.
From (2) we have
a2 + 62 — lab cos (a — e) = b2 — lab cos a + a2 cos2 a
or
a2 sin2 a
cos (a — e) — cos a = •
26
Thus we clearly obtain
t < d/b.
Thus clearly all the points Q< have distance less than c2 from the line
PiX. Let Qi, Q2, Q3 be three such points not on a line, where diQiQj)
are large. Let QiQ3 be the largest side of the triangle QiQiQ*. Let QzR
be perpendicular to Q1Q3. We have as before
d(QG,) £ d(Qu Q*) + d(Q&) - 1;
also
d(QiQt) ~ diQJl) < e, d(Q&3) - d(Q3R) < e
an evident contradiction; this completes the proof.
By a similar argument we can show that we cannot have infinitely
many points in w-dimensional space not all on a line, with all the
distances being integral.
University of Michigan
663 GEOMETRY

ON SETS OF DISTANCES OF n POINTS
P. ERDOS, Stanford University
1. The function f(n). Let [P„ ] be the class of all planar subsets P„ of n
points and denote by f{n) the minimum number of different distances
determined by its n points for P„ an element of {P„}. Clearly, /(3) = 1 (with the three
points forming the vertices of an equilateral triangle) /(4) = 2, /(5) = 2. The
following theorem establishes rough bounds for arbitrary n. Though I have sought
to improve this result for many years, I have not been able to do so.
Theorem 1. The minimum number fin) of distances determined by n points
of a plane satisfies the inequalities
{n - 3/4) »'* - 1/2 ^ f(n) S cn/(log m)1'2.
Proof. Let Pi be an arbitrary vertex of the least convex polygon determined
by the n points, and denote by K the number of different distances occurring
among the distances PiP; (i = 2, 3, • • • , n). If N is the maximum number of
times the same distance occurs, then clearly KN^n — 1.
If r is a distance that occurs N times then there are iV" points on the circle
with center Pi and radius r, which all lie on the same semi-circle (since Pi is a
vertex of the least convex polygon). Denoting these points by Qi, Qi, • ■ • , Qn,
we have QiQ^<QiQi< • ' • <QiQn, and these N—l distances are pairwise
distinct. Thus fin) Si max (N — 1, (n — l)/N), which is a minimum when N(N—i.)
= n — \. This yields the first part of the theorem.
Considering now the points (x, y) with integer coordinates for O^x, y^n11*,
we obtain at least n points Pi which pairwise have distances of the form
(w2+i>2)1/2, O^w^m1'2, 0^i;^m1/2. Now it is well-known that the number of
different integers not exceeding In which are of the form u2-\-vi is less than
cw/(log m)1/2, and the proof is complete.*
For n points in ^-dimensional space the same method yields Cin1!k <f(n)
<CtnVk.
2. Some conjectures concerning f(n). Let us assume that our n points form
a convex polygon. Then I conjecture that/(m) Si [n/2], with the equality sign
valid when the n points are vertices of a regular w-gon. I am unfortunately
unable to prove this. The following conjecture is stronger: In every convex polygon
there is at least one vertex with the property that no three vertices of the
polygon are equally distant from it. If this is the case, then clearly we would obtain
[n/2] different distances by considering all the distances from such a vertex.
A still stronger conjecture is that on every convex curve there exists a point
P such that every circle with center P intersects the curve in at most 2 points.
3. The function g(n; r). Denoting by g{n; r) the maximum number of times
a given distance r can occur among n points of a plane we establish
* Landau, Verteilung der Primzahlen, vol. 2.
248
664 MISCELLANY

ON SETS OF DISTANCES OF 71 POINTS
249
Theorem 2. n^'110*1"*" <g(n;r) <m3'2.
Proof. Assuming that there are x,- points at distance r from Pit clearly
g(n;r) = max 3^2?_iX;. We suppose that Xi^x^^ ■ ■ • j^x„. Now the a;,- points
at distance r from P, can contain at most two points with distance r from P,.
Hence
(1) E (*< -2i+2)£n for j = 1, 2, ■ ■ ■ , n.
Put [n1li]=a, nlli-a = e, 0 = e<l. We have from (1)
xi+ x2+ ■ ■ • + xa = n+ 2( ) = 2» - 26M1'2 + «2 - «1/2 + «
(2) \ 2 /
<2n - 2m1^
for m=4. Thus
1
(3) xa < — (2m - 2m1'2) = 2»"2.
a
Hence from (2) and (3)
n
X) *.' < 2m - 2«m1/2 + (» - a)2w1/2 = 2»3'2
•-1
or
gin; r) < «3'2.
By again considering the set of points (.r, y), O^S.r, y^a we easily obtain
(using well known theorems about the number of solutions of u2-{-v- = m)*
g(n) > nHc'l0«Iog *
which completes the proof.
It seems likely that g(n) <n1+t.
4. Maximum and minimum distances. If r is the diameter of the points Pi,
it is well known that r can occur only n times, j This follows almost immediately
from the fact that if PiP2 = r and P3Pt = r the lines PiP2 and P3P4 must intersect,
for otherwise a simple argument shows that the diameter of PiPiPsPt would be
greater than r. Connect P,- with P, if and only if their distance is r. We
distinguish two cases. In Case 1, every P,- is connected with at most two other P's.
In this case the number of lines, i.e., of pairs of points at distance r is clearly =m.
* See e.g. P. Erdos, London Math. Soc. Journal, 1937, vol. 12, p. 133. The proof would depend
on the prime number theorem for primes of the form 4¾+ 1 (or on some weaker elementary result
concerning the distribution of primes of the form 4fe + l).
\ Jahresbericht der Deutschen Math. Vereinigung, vol. 43, 1934, p. 114.
665 GEOMETRY

250
ON SETS OF DISTANCES OF n POINTS
If Pi would be connected with three vertices say P2, Ps, P* where P1P3 is
between P1P2 and P1P4 then P3 can not be connected with any other P„ since
P3P1 would have to intersect both P1P2 and P1P4 (the angle P2P1P4 is of course
^7r/3), and thus be greater than r. Now we can just omit P3 and since both the
number of points and the number of distances are reduced by 1, the proof can
be completed by induction.
It would be interesting to have an analogous result for n points in k
dimensional space. Vazsonyi* conjectured that in three-dimensional space the
maximum distance can not occur more than In— 2 times.
If one could prove that in ^-dimensional space the maximum distance can
not occur more than kn times, the following conjecture of Borsuk would be
established : Each k-dimensional subset of diameter 1 can be decomposed into k-\-1
summands each having diameter <1.
Let now r' denote the minimal distance between any two P's. First it is easy
to see that r' can not occur more often than 3n times. This is immediately
clear from the fact that since r' was the minimal distance between any two P's,
there can be no more than 6 P's at distance r' from any given P.
Connect P,- with P, if and only if their distance is r'. A simple argument
shows that no two such lines P1P2 and P3P4 can intersect (otherwise there would
be two P's at distance <r'). Thus the graph we obtain is planar, and from
Euler's theorem it follows that the number of edges of such a graph is not
greater than 3m —6. Thus we have proved the following
Theorem 3. Let the maximum and minimum distances determined by n points
in a plane be denoted by r and r', respectively. Then r can occur at most n times and
r' at most 3m —6 times.
It is easy to give n points where the maximum distance occurs exactly n
times. By more complicated arguments we can prove that the minimal distance
r' can occur not more than 3m — cm1/2 times, where c is a constant. On the other
hand the example of the triangular lattice shows that r' can occur 3m — cim1'3
times. I did not succeed in determining exactly how often r' can occur.
One could try to generalize Theorem 3 to higher dimensions. But already
the case of three-dimensional space presents great difficulties. It would be of
some interest to determine the maximum number of points on the unit sphere
of k dimensions such that the distance of any two is ^1.
* Oral communication.
666 MISCELLANY

GEOMETRICAL EXTREMA SUGGESTED BY A LEMMA
OF BESICOVITCH
PAUL BATEMAN, The Institute for Advanced Study, and PAUL ERDOS,
Syracuse University
1. Introduction. In [l] Besicovitch needed as a lemma a result of the
following type.
Theorem 1. Given a set T of coplanar circles, the center of no one of them being
in the interior of another, and U the circle {or a circle) of Y whose radius does not
exceed the radius of any other circle of T, then the number of circles meeting U does
not exceed 18.
Besicovitch proved the weaker theorem obtained from this one by replacing
18 by 21. In this paper we shall prove Theorem 1 as it stands. The number 18
cannot be replaced by a smaller number, as is shown by the example in which
all of the circles have radius 1 and the centers are at the following points in a
polar coordinate system: the origin, the points (1, h -60°) where h = 0, 1, ---,5,
and the points (2 cos 15°, (26+1)-15°) where k=0, 1,---,11.
We prove Theorem 1 by establishing its equivalence with Theorem 2 and
then proving the latter.
Theorem 2. It is impossible to have 20 points in* a circle of radius 2 such that
one of the points is at the center and all of the mutual distances are at least 1.
Naturally one can ask the general question: For any positive integer »,
what is the radius r(n) of the smallest! circle containing n points one of which
is at the center and all the mutual distances between which are at least 1?
Thus Theorem 2 says that r(20)>2. A related question is the following: Of all
sets of n points in the plane such that the mutual distances are all at least 1
(with no restriction on the arrangement of the points) what set has the minimum
diameter J Z>(w)? The following theorem answers this question for n = 7.
Theorem 3. A set of seven points in the plane whose mutual distances are all at
least 1 has diameter at least 2, with this value being attained only by the set of
points consisting of the vertices and circumcenter of a regular hexagon of side-
length 1.
The asymptotic behavior of r(n) and D(n) is well-known§; in fact, for large n
the regular hexagonal lattice gives about the best results, so that D(n)~2r(n)
~(12/7r2)1'4M1'2 as n—> *. However we are interested here in small values of n.
* The term "in" is supposed to include the boundary.
f It is not difficult to see that the greatest lower bound is attained here.
I The diameter of a set of points is the least upper bound of all the mutual distances. For a
finite set this is simply the greatest mutual distance. Also note that the diameter of a polygon is
equal to the diameter of the point-set consisting of the vertices.
§See [3], [4], [5], [6].
306
667 GEOMETRY

1951] GEOMETRIC EXTREMA SUGGESTED BY A LEMMA OF BESICOVITCH 307
For each n the values of r(n) and D(n) can be calculated to any desired degree
of accuracy by constructing a sufficiently fine mesh, but the proofs of our
theorems will show that to accomplish this practically is not easy.
The first few values of r{n) are as follows: r(2)=r(3) = • • • = r(7) = l,
r(8) = i cosec (180°/7) = 1.15 • • • , r(9) =¼ cosec 22?5 = 1.30 • • • , r(10) =\
cosec 20° = 1.46 ••• , r(l 1) = ¾ cosec 18° = 1.61 ••• , r(12) = 1.68 ••• , and
r(13) = V3 = 1.73 • • • . The evaluation of r(8) through r(ll) will come out in
the proof of Theorem 2; the evaluation of r(12) and r(13) can be effected by
similar methods. Further, the example before the statement of Theorem 2 shows
that r(19) ^ 2 cos 15° = 1.93 ••• . Also r(20) > 2.
The first few values of D{n) are also easy to find: D(2) =D(3) = 1, .0(4)
= V2 = 1.41 • • • , .0(5) =H1+V5) = 1.61 ..., D(6) = 2 sin 72° = 1.90
Further, Theorem 3 gives .0(7) =2. For n = 3, 4, 5 the vertices of the regular
w-gon of side-length 1 give the minimum diameter; for n = 6 the best result is
given by the set of points consisting of the vertices and circumcenter of a regular
pentagon of circumradius 1.
2. Proof that Theorem 1 is equivalent to Theorem 2. To show that Theorem
1 implies Theorem 2 we proceed as follows. Suppose we have k points in a circle
of radius 2 such that one of the points is at the center and all the mutual
distances are at least 1. If we construct a circle of unit radius about each point
and then apply Theorem 1 to this set of k circles, with U as the circle around
the central point, we get k^ 19.
In showing that Theorem 2 implies Theorem 1 (and, in fact, in proving
Theorem 2) we use a polar coordinate system with radial distance p and
amplitude d. The set of points such that p f^t we denote by C{t). Now we may assume
that the U of Theorem 1 is the unit circle of our polar coordinate system. Thus
we have a set A of k — 1 circles of radius at least 1 each one of which meets the
the unit circle U, the center of no one of these k circles (including U) lying in
the interior of another. Then it suffices to show that we can construct a set A*
of k— 1 points in C(2) whose mutual distances and distances from the origin
are all at least 1. We do this by choosing a point for A* corresponding to each
circle D of A in the following way: if the center of D lies in C(2), we pick the
center; if the center X of D lies outside of C(2), we pick the point R lying on
the circle p = 2 and having the same amplitude as X. As a result of this
correspondence the circle of radius 1 about a point Q of A* is contained in the
corresponding circle of A; hence Q is at distance at least 1 from the points of A*
which were originally centers of circles of A. Thus it remains only to prove that
if two circles of A have centers X and Y both of which lie outside of C(2), then
the corresponding points R and 5 of A* have mutual distance at least 1. Let
OX = x, OY = y, angle XOY=\p. Then by the properties of A we have XY2
^max {(x-1)2, (y-1)2}; that is,
x1 + y1, — 2xy cos ^ ^ max {(x — l)2, (y — 1)2}.
Now if we suppose that y^x we have
668 MISCELLANY

308 GEOMETRIC EXTREMA SUGGESTED BY A LEMMA OF BESICOVITCH [May
x2 + y2 - (y - 1)2 1 x2 - 1
cos 4> = = 1
2xy x 2xy
1 x2 - 1 1 / 1V 7
^ — + = 1 (l ) ^-'
x 2x2 2 \ x/ 8
since x>2. Therefore i?52 = 8 —8 cos i/^l. Thus the points chosen for A*
have the desired properties and our proof of equivalence is finished.
3. Proof of Theorem 2. The proof is based upon the following lemmas, of
which the first is basic and the others are simple corollaries thereof.
Lemma 1. Let r and R be such that 0<R— 1 ^r^R and suppose that we have
two points P and Q which lie in the annulus r^p^R and which have mutual
distance at least 1. Then the minimum cj>(r, R) of the angle POQ has the following
values
R2 + r2 - 1 1
4>(r, R) = arccos > ifR-l<r<R ;
2Rr J ~ ~ R
/ 1\ 1 1
S(r, R) = arccos [ 1 ■) = 2 arcsin ■ > if R < r < R.
^- \ 2R2/ nn J
2R R
In particular
4>{\, 1.10) > 54?0 ¢(1.10, 2) > 16?8
¢(1, 1.15) > 51?5 ¢(1.15, 2) > 20?0
¢(1, 1.20) > 49?2 ¢(1.20, 2) > 22?3
¢(1, 1.25) > 47? 1 ¢(1.25, 2) > 24? 1
¢(1, 1.30) > 45?2 ¢(1.30, 2) > 25?5
¢(1, 1.45) > 40?3 ¢(1.45, 2) > 28?3
¢(1, 1.60) > 36?4 ¢(1.60, 2) > 28?9
Proof. It suffices to consider the case in which OQ = R and PQ= 1. If we put
OP=p, then our problem is to find the minimum of /(p) = angle POQ
= arccos {(i?2+p2— l)/(2i?p)} forpin the interval r^p^i?. By differential calculus
we see that although/(p) may have an interior maximum for p = (i?2—1)1/2,
it cannot have an interior minimum. If we compare /(r)=arccos {(i?2+r2
-\)/{2Rr)} and f(R)= arccos {1-l/(2i?2)}, we see that when R-l^r^R
— \/R the minimum is achieved for OQ = R, PQ=l, and OP = r, while when
R-l/R^r^R the minimum is achieved for OQ = R, PQ=\, and OP = R.
We shall find it convenient in what follows to use the term admissible
points to refer to a set of points in l^p^2 whose mutual distances are all at
least 1.
669 GEOMETRY

1951] GEOMETRIC EXTREMA SUGGESTED BY A LEMMA OF BESICOVITCH 309
Lemma 2. There are at most 12 admissible points in the annulus 1.45 ^p g 2.
Proof. This follows from the fact that 130(1.45, 2)>360°. (Note that the
constant 1.45 could be replaced by any number r such that 4>(r, 2)>360°/13,
for example by 1.402.)
Lemma 3. It. is impossible to have 7 admissible points in C(1.15), 8 admissible
points in C(1.30), 9 admissible points in C(1.45), or 10 admissible points inC{\.60).
Proof. In fact, 70(1, 1.15)>360°, 80(1, 1.30) >360°, 90(1, 1.45) >360°,
100(1, 1.60) >360°. (Note that the constants 1.15, 1.30, 1.45, 1.60 could be
replaced by any values less than f cosec (180°/7), f cosec 22?5, f cosec 20°,
^ cosec 18°, respectively. This remark, along with the fact that the vertices of
the regular &-gon of side length 1 with center at the origin do constitute a set
of k admissible points in C{% cosec (180°/&)}, provides the evaluation of r(8),
r(9), r(10), r(ll)).
Lemma 4. It is impossible to have 7 admissible points in C(1.30) such that 6
of them lie in C(1.10).
Proof. If we had 7 admissible points in C(1.30), 6 of which lay in C(1.10),
then of the 7 angles subtended at O by pairs of consecutive points (considered
in order of amplitude) 5 would be each at least 0(1, 1.10) and the other two
would be each at least 0(1, 1.30). But 50(1, 1.10) + 20(1, 1.30) >360°.4.
Lemma 5. It is impossible to have 8 admissible points in C(1.45) such that 7 of
them lie in C(1.25). It is impossible to have 8 admissible points in C(1.45) such that
6 of them lie in C(1.15).
Proof. The proof is similar to that of Lemma 4. The first part follows from
the fact that 60(1, 1.25) + 20(1, 1.45)>363?2; the second from the fact that
40(1, 1.15)+40(1, 1.45)>367?2.
Now we come to the proof of Theorem 2. We suppose that we have 19
admissible points and seek to get a contradiction. By Lemma 2 there are at
most 12 admissible points outside C(1.45) and by Lemma 3 at most 8
admissible points in C(1.45). Thus there are two cases to consider, according to
whether we have 7 admissible points in C(1.45) and 12 admissible points
outside, or 8 admissible points in C(1.45) and 11 admissible points outside. The
first case we subdivide further, according to whether one of the 7 points lies
outside of C(1.30) or not.
Case la: 12 admissible points Blt • • • , B12 outside of C(1.45), 7 admissible
points Ai, • • • , At in C(1.45), one of the Ai, say Ak, outside of C(1.30). All the
angles BiOBi+1 exceed 0(1.45, 2)>28?3. (We mean to include the angle BnOBi,
of course; similarly in the sequel.) But for any i the angle AkOBi exceeds
0(1.30, 2)>25?5 and so one of the angles BiOBi+1 exceeds 20(1.30, 2)>51?0.
However 11 (28?3) + 51?0 = 362?3, a contradiction.
670 MISCELLANY

310 GEOMETRIC EXTREMA SUGGESTED BY A LEMMA OF BESICOVITCH [May
Case lb: 12 admissible points Bu • • • , Bn outside of C(1.45), 7 admissible
points Ai, • • • , Ai in C(1.30). By Lemma 3 one of the A{, say Ak, lies outside
of C(1.15) and by Lemma 4 another one of the Ait say Am, lies outside of
C(1.10). For each i the angle AkOB{ exceeds 0(1.15, 2)>20?0 and the angle
AmOBi exceeds 0(1.10, 2)>16?8. Hence an angular sector of more than 40?0
and another sector of more than 33?6 are ruled out as possible locations for
points Bi. These sectors cannot overlap, since the angle AkOAm is at least
0(1, 1.30) >45?2. If one of the angles BiOBi+1, say BnOBn+1, includes both these
sectors, then B„OBn+i exceeds 40?0+33?6 = 73?6 and we have a contradiction,
since 110(1.45, 2) + 73?6>384?9. Otherwise we see that out of the 12 angles
BiOBi+i one exceeds 40?0 and another exceeds 33?6. Since at most 9 admissible
points can lie in C(1.60), at least 10 of the 12 points Bt lie outside C(1.60).
Hence at least 8 of the 12 angles BiOBi+1 exceed 0(1.60, 2)>28?9. But 40?0
+ 33?6+6(28?9)+ 40(1.45, 2)>360?2, a contradiction.
Case II: 11 admissible points Bi, • • • , Bn outside C(1.45), 8 admissible
point Au • ■ • , As in C(1.45). By Lemma 3 one of the Ait say Ak lies Outside
C(1.30); by Lemma 5 a second one of the A{, say Am, lies outside of C(1.25) and
a third one of the A{, say A„, lies outside of C(1.15).
Now for each i the angle B{OAk exceeds 0(1.30, 2) >25?5, the angle BtOAm
exceeds 0(1.25, 2)>24?1, and the angle B{OAn exceeds 0(1.15, 2)>20?0. Hence
three angular sectors of more than 51?0. 48?2, and 40?0, respectively, are ruled
out as possible locations for the points Bi. If one of the angles BiOBi+i, say
BpOBP+1, includes two of these sectors, then BpOBp+i exceeds 20?0+24°l +
40?3 = 84°4, since AkOAm, AmOAn, and AnOAk are each at least 0(1, 1.45)
>40?3; this gives a contradiction, since 100(1.45, 2) + 84?4>367?4.
On the other hand if no angle BiOBi+i includes two of the proscribed sectors,
then we see that out of the 12 angles BiOBi+i one exceeds 51?0, another exceeds
48?2, another exceeds 40?0, and each of the 8 remaining exceeds 0(1.45, 2)
>28?3. But 51?0+48?2+40?0+8(28?3)=365?6, a contradiction.
4. Proof of Theorem 3. We require the following lemmas, of which the first
three are well-known.
Lemma 6. The area of a triangle does not exceed \\/3 times the square of the
longest side.
Lemma 7. The product of the diagonals of a quadrilateral is at least twice the
area.
Lemma 8. If a convex polygon has perimeter not less than 2tt, its diameter
exceeds 2.
Lemma 9. If there is a point in a triangle whose distance from each vertex is at
least 1, then some side of the triangle has length at least V3.
Proof. One of the sides of the triangle must subtend an angle of 120° or more
at the point in question.
671 GEOMETRY

1951] GEOMETRIC EXTREMA SUGGESTED BY A LEMMA OF BESICOVITCH 311
Lemma 10. If O <^<\iv and ABCD is a convex quadrilateral with •$.ABC
= !*"+£, <BCD^lir-l^, and \^AB, BC, CD^2, then AD>2.
Proof. PutAB = x, BC = y, CD=z, $.BCD =0. Then
AD2 = {y - a; cos (fir + £) - z cos#}2 + {x sin (fir + £) - ssintf}2
= X2 + y2 + s2 - 2yz cos 5 + 2sz cos (0 + fir + £) - 2*;y cos (fir + £).
Considering this expression as a function of x, y, z, 6 Over the domain l^x, y,
2 = 2, fir — §£fg0 ^7r, we see from the positivity of the partial derivatives that the
smallest value of AD2 occurs for x = y = z= 1, 0 = §ir — §£. It suffices therefore to
prove that
/(£) = 3 - 2 cos (fir - f£) + 2 cos (^ + If) - 2 cos (fir + £)
exceeds 4 for 0<£±£fir; but this follows from the fact that/(0) =4, /(fir)=5
-4 cos (4ir/9), and/"(£)<0 for 0^£^fir.
Lemma 11. If ABCDE is a convex pentagon such that AB, BC, CD, DE^ 1,
AE^ V3, and <£C>120°, then either the diagonal AD>2 or the diagonal BE>2.
Proof. If $.A >90°, then BE>2; il <££>90°, then .4I>>2. Suppose then
that neither angle adjacent to AE exceeds 90°. Then if <£C=120°+£, we see
that either $.B^ 120°- |£ or ^.D^ 120°- |£, for otherwise the angle sum of the
pentagon would be less than 90° + 90°+(120°+£) + 2(120°-|£) =540°. Hence
by Lemma 10 either AD>2 or BE>2.
Now we turn to the proof of Theorem 3. Suppose then that we have seven
points in the plane such that the mutual distances are all at least 1. In proving
that the diameter of the set of seven points is at least 2, we consider five cases,
according as the convex hull of the seven points is a triangle, quadrilateral,
pentagon, hexagon, or heptagon. As stated in the theorem we shall find that the
diameter actually exceeds 2 throughout, except for the subcase of the hexagonal
case in which the angles of the hexagon are all 120°, the sides all have length 1,
and the seventh point is at the center.
The triangular case. Suppose that a circle of radius 5 is drawn about each
of the seven points as center. Then no two of these circles can properly intersect.
If one side of the triangle is intersected by the circles around two of the four
inner points, then that side has length at least 3^/3/2 by the Pythagorean
theorem. If no side of the triangle is intersected by two of the circles about
inner points, then the area of the triangle is greater than the area of three
circles of radius ^, namely 37r/4>V3; thus by Lemma 6 at least one side of
the triangle has length greater than 2.
The quadrilateral case. Again we construct a circle of radius 5 about each of
the seven points. If one side of the quadrilateral is intersected by the circles
about two of the three inner points, then that side has length at least 3y/3/2 > 2.
Suppose then that no side is intersected by more than one of the circles about
672 MISCELLANY

312 GEOMETRIC EXTREMA SUGGESTED BY A LEMMA OF BESICOVITCH [May
the inner points. If just one or two sides are intersected by inner circles, then
the area of the quadrilateral is greater than the area of three circles of radius -j,
namely 3tt/4>2, and thus by Lemma 7 at least one of the diagonals of the
quadrilateral has length greater than 2.
Now notice that if a side of the quadrilateral is intersected by the circle
around one of the inner points, then that side has length at least \/3 by the
Pythagorean theorem. If three sides are intersected by inner circles and the
quadrilateral is a rectangle, then all sides of the rectangle have length at least
V3 and hence the diagonals have length at least \/6. If three sides are
intersected by the circles about inner points and one of the angles of the quadrilateral
exceeds 90°, then at least one of the sides adjoining that angle has length V3 or
greater, while-the other side adjoining it has length at least 1; hence (by the
law of cosines) the diagonal spanning this angle has length greater than 2.
The pentagonal case. Again if the circles of radius f about the two inner
points intersect the same side of the pentagon, then that side has length at least
3\/3/2. On the other hand if these two circles intersect two different sides of
the pentagon, then both these sides have lengths at least V3, the pentagon has
perimeter at least 3-\-2y/3>2ir, and thus by Lemma 8 the diameter of the
pentagon exceeds 2.
Thus we assume that at least one of the inner points, say K, has a distance
from the perimeter of the pentagon greater than ^. Then the other inner point,
say L, must lie in one of the triangles formed by K and consecutive vertices of
the pentagon, say the triangle KAB. By Lemma 9 one of the sides of the
triangle KAB has length at least \/3. If either AK or BK has length y/3 or more,
the diameter of the pentagon exceeds \/3-\-%>2. We suppose then that AB
^ V3. Since one of the angles ALK and BLK is at least 90°, either AKt y/2
or BK = y/2. Suppose AK = y/2. Then prolong AK until it meets the pentagon
again at a point X. If AX>2, we are finished. If AX^2, then KX^2- y/2,
the side on which X lies has length at least 2 {1 - (2- V2)2} 1,2> 1.62, the
pentagon has perimeter greater than 3 + 1.73+1.62 = 6.35>2ir, and thus the
pentagon has diameter more than 2 by Lemma 8.
The heptagonal case. This case is settled immediately by Lemma 8. Actually
by pushing our methods further it is possible to prove that of all convex
heptagons with sides of length at least 1 the minimum diameter occurs for the regular
heptagon of side-length 1, in which case it is 1 + 2 cos (2tt/7) = 2.24 • • • .
The hexagonal case. Dismissing the case in which a side of the hexagon has
length greater than 2, we separate the proof into cases according to the number
and arrangement of those angles of the hexagon which exceed 120°. First of all
we note that if two adjacent angles of the hexagon exceed 120°, the diagonal
spanning these two angles has length greater than 2. This case occurs, for
example, if there are four or five angles of the hexagon exceeding 120°.
If exactly one angle of the hexagon exceeds 120°, say is 120°+£, then one of
the angles adjacent to it is at least 120° — ^£; the proof is completed in this case
by Lemma 10. If exactly two angles of the hexagon exceed 120°, say have the
673 GEOMETRY

1951] GEOMETRIC EXTREMA SUGGESTED BY A LEMMA OF BESICOVITCH 313
values 120°+a and 120^+/3, and these two angles are non-adjacent, then
somewhere in the hexagon we have a pair of adjacent angles of which one is
120°+£, £>0, and the other is at least 120°—f£; for otherwise the pentagon
would have an angle sum less than (120°+a) + (120°+/3)+ (120° -fa)
+ (120°-f/3) + (120°-f max {a, /3 } ) + 120° = 720°. Again the proof is
completed by Lemma 10.
In case that exactly three of the angles of the hexagon exceed 120° and no
two of these are consecutive, let A, B, C, D, E, F be the vertices of the hexagon
and let A, C, E be those at which the angles exceed 120°. First we remark that
if one of the sides of the hexagon has length greater than 5(Vl3 — 1), we have a
diagonal of length greater than 2 by the law of cosines; thus we may assume all
sides of the hexagon to have length not exceeding 2(Vl3 —1) < V3. The three
diagonals A C, CE, EA divide the hexagon up into four triangles. One of these
four triangles contains the seventh point and hence one of the three diagonals
A C, CE, EA has length at least \/3 by Lemma 9. Suppose EA ^ V3. Then by
applying Lemma 11 to the pentagon ABCDE we find a diagonal of length
greater than 2.
The only case left is that in which no angle of the hexagon exceeds 120°,
i.e., all angles are 120°. We easily see that every diagonal spanning two angles
has length at least 2 and that the diameter is exactly 2 if and only if all the
sides of the hexagon have length 1 (with the seventh point at the center
naturally).
5. Further questions. Another function which we could consider is the
diameter d(n) of the smallest circle containing n points whose mutual distances are
all at least 1, without the restriction that one point be at the center. Obviously
D(n) ^d(n) — 2r(w) and the same general remarks that were made about D(n)
and r{n) could be made about d(n). The first few values of d(n) are ^(2) = 1,
d(3) =2/V3 = l. 15 ••• ,^(4) = V2 = 1.41 •■• , d(5) =cosec 36° = 1.70--- ,
d(6)=2,d(7)=2.
Naturally we can consider the analogue of d(n) for other figures than the
circle, for example, the side-length t{n) of the smallest equilateral triangle
containing n points whose mutual distances are all at least 1. An interesting
unsolved question about t{n) is whether or not t{\k{ k-\-1} +1) >k— 1 for k a
positive integer. Obviously /(jfe{fe+l}) ^fe—1, since the regular hexagonal
lattice gives ^k {k +1} admissible points in a triangle of side-length fe — 1.
In the proof that D(7) = 2 we remarked that of all convex heptagons whose
sides have length 1 or more the minimum diameter is assumed by the regular
heptagon of side-length 1. An analogous statement can be made for triangles,
quadrilaterals, and pentagons. However for hexagons the minimum diameter is
assumed by the equilateral hexagon of side-length 1 whose angles are alternately
90° and 150°. The situation for »-gons with w>7 is an open question.
Another interesting problem is whether or not r(w+l)>r(») for w^7 and
whether D{n+\) >D{n) for n^.3. It follows from Theorem 3 that D(8)>D(7)
674 MISCELLANY

314 GEOMETRIC EXTREMA SUGGESTED BY A LEMMA OF BESICOVITCH [May
= 2, since the diameter of a set of seven points (with mutual distances 1 or
more) is greater than 2 except for one special configuration (for which we can
make a separate argument).
Of course we can consider the analogues of r(n) and D(n) in k dimensions,
say rk(n) and Dk{n). Clearly Dk(n) = 1 for n^^+1. It would be interesting to
have a good estimate for Dk(k-\-2) from below. It is probably true'that Dk(k+2)
^ V4/3, but this seems difficult to prove. On the other side it is easy to give an
example to show that Dk(k-\-2) ^ \/4k/(3k— 2). As for rk(n), we do not even
know the smallest value of n such that rk{n)>\, except for k = 2. For n—»°o
there are asymptotic relations of the form (cf. [2])
%Dk(n) ~ rk{n) ~ cknllk,
but Ck is not known for k > 2.
6. Added in Proof: After this paper was submitted for publication, a proof
of our Theorems 1 and 2 was published by E. F. Reifenberg, Math. Gaz., vol.
32, 1948, pp. 290-292. His proof is in general similar to ours, although
considerably different in detail.
The problem mentioned briefly in the third paragraph of §5 has been
discussed recently in a paper by S. Vincze, Acta Univ. Szeged. (Sect. Sci. Math.)
vol. 12, part A, 1950, pp. 136-142. For n not a power of 2 he shows that the
minimum diameter possible for convex n-gons whose sides have length at least
1 is 5 cosec(7r/2w). In particular for even n having at least one odd prime factor
the regular »-gon of side length 1 does not give the minimal diameter. Vincze
points out that a closely related problem was considered earlier by K. Rein-
hardt, Jber. Deutsch. Math. Verein., vol. 31, 1922, pp. 251-270.
References
1. A. S. Besicovitch, A general form of the covering principle and relative differentiation of
additive functions, Proc. Cambridge Philos. Soc., vol. 41, 1945, pp. 103-110.
2. R. A. Rankin, On the closest packing of spheres in n dimensions, Ann. of Math. (2), vol. 48,
1947, pp. 1062-1081.
3. B. Segre and K. Mahler, On the densest packing of circles, This Monthly, vol. 51, 1944,
pp. 261-270.
4. A. Thue, Uber die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene,
Christiana Videnskabs-Selskabets Skrifter 1910, no. 1.
5. L. Fejes (T6th), Extremal distributions of points in the plane, on the surface of the sphere
and in space, Acta Sci. Math. Nat. (Univ. Francisco-Josephina, Kolozsvar) no. 23, 1944, iv + 55 pp.
6. L. Fejes (T6th), Uber einem geometrischen Satz, Math. Zeit., vol. 46, 1940, pp. 83-85.
675 GEOMETRY

ON SETS OF DISTANCES OF n POINTS IN EUCLIDEAN SPACE
by
P. ERDOS
Let [P(rP] be the class of all subsets P(n^> of the k dimensional space
consisting of n distinct points and having diameter 1. Denote by gk{n, r) the maximum
number of times a given distance r can occur among n points of a set PW. Put
Gk{n) = max gk{n, r), gk(n) = gk(n, 1)
r
(i. e. gk(n) denotes the maximum number of times the diameter can occur
as a distance among n points of k dimensional space and Gk(n) denotes the
maximum number of times the same distance can occur between n suitably
chosen points in k dimensional space). It is well known [1] that g2{n) =n
and I [2] proved that
,1 rc/loglogn
< G2(n) < n3
Furtherl conjectured that G2(n) < nln'E for every e > 0 if n > n0(e). Vazsonyi
conjectured that g3{n) = 2n — 2 and this was proved simultaneously and
independently by Gbunbaum [3], Heppes [4] and Stbaszewicz [5] (all
using similar methods). I am going to prove
(2)
^•w43 < G3(n) < c2-n5l3
Perhaps G3(n) < ti4<3+£ holds for all n > n(s).
One could have expected that Gk(n) = o(n2) andgk(n) < ck- n for every k.
In 1955 Lekz showed that this is not so. In fact Lenz showed that (Lenz's
result is unpublished)
9i{n) >
and consider the following
The proof of Lenz is very simple. Put s -
n points in four-dimensional space:
(x„ i/,., 0, 0), 1 < i < s, (0, 0, Xj, yj), s+ 1 < j £n
where 0 < x,, xjy yh yj < —r- , xj + yf = — , x2j + y2j = — . Clearly all the
165
676 MISCELLANY

166
s (n — s) = •- distances between the points (a.-,, yn 0. 0) and (0, 0, x]} y)
is 1 (and 1 is the diameter of the set (xt, ,?/,, 0, 0); (0, 0, Xj, yy).
By a slight modification of this method Lenz in fact proved that
71
<74(rc) > h c3n for a certain c3 > 0. Lexz then asked: what is the limit
4
of gk(n)jn'2 as n -> °°. In this note I am going to prove the following
Theorem. For every k > 4
]
k
2
lini gk(n)'n2 = lini6/,.(71) n2 = —
n—a: ' n-=° ' 2
Clearly gk(n) g Gk(n) and gk(n) g gk+1(n). Gk(n) g Gk+1(n). Thus to
prove our Theorem it will suffice to show that for every I >- 2
(4)
and
lini g2i (w)>2 ^
]
21
lini 6r2,., (ii)/n2 <
■11
The proof of (4) is trivial generalization of the proof of Lexz. For each
t, 1 < t < I denote by Ij the group of
points whose first 2£ — 2
coordinates are 0 the it — 1-th and 2t-th coordinates are x,. yt, 1 <L i g
x„ yt > 0, x] + y\
and the remaining il — 2t coordinates are (>. Clearly
for any tl =f= t2 the distance between any two points of Itl and /,, is 1 and the
set U It has diameter 1. Thus
9u{n) ~2L
I
I
O(ii)
which clearly implies (4).
Next we prove (5). If (5) is not true then there exists an e > 0 so that
for a certain I ;> 2 and infinitely many ns
G
2/--1
(71,) >
21
£1 111
A(ns
In other words there exists a set />^' + 1) in 2^+1 dimensional space
and a distance r which occurs among at least A(?is) pairs of points of P^2/"*1).
Connect any two points of P& : '> whose distance is r. Thus we obtain a graph
677 GEOMETRY

ON SET? Ob' DISTANCES OP n POINTS I.N BUOL1DRAN SPACE lg7
of ns vertices and A(ns) edges. By a theorem of A. H. Stone and myself1
[6] this graph contains for sufficiently large ns = ns(e) a subgraph of 3 (I + 1)
vertices .r(/> 1 ^. i ^ 3, l^^^Z+1 so that any two vertices x(/i> and
x(/2> are connected by an edge if tx =j= t2 (in other words the distance between
x('>> and x<'2> is r if t1 =f=t2). But then a simple geometrical argument shows
that the I + 1 planes (a;^>, x^'>, x!j'>), 1 g ( 5S Z + 1 must be mutually
perpendicular, which implies that the dimension of the space spanned by the
x(P is at least 21 -|- 2. This contradiction proves (5) and thus the proof of
our Theorem is complete.
By a sharpening which I recently obtained of the result of Stone and
myself I can prove
(6) cfc(n)</-- .±_U* + 0(n2-«*
1
2
2
1
~k
2
where eA. —> 0 as /fc ->- °°, I do not know how close (6) is to the true order of
magnitude of Gn(n). Perhaps the result of Lenz
(') Gk(n) > I - - -~ \ n* + ckn
gives the right order of magnitude.
Now we are going to prove (2), First wc prove the upper estimate. Let
xx, x2, , , ,, xn be n points in three dimensional space, assume that there are
a, points at distance r from x,. Clearly to any three points x;i, x;2, Xj3 there
can be at most two points x, at distance r. Thus since the total number of
triplets (xJit xJz. x;J is a simple argument gives
, = i 13 I 13 )
or
(8) 2«kct»3-
I-1
n n
If "V a? is given "Va, is maximal if all the a, are equal. Thus (8) implies
(=-i i-i
n
^ ai < c2n° *
i = }
which proves the upper bound in (2).
1 The theorem in question states as follows; To every e, K>2 and I there exists
an «0 (f, r, I) so that if n > n0 (s, r, I) and Gn is a graph of n vertices and more than
n2 — — „ +e edges then Gn contains rl vertices x'P l<.i <l, 1 <, t < r so
{2 I (r— 1) J
that for every t-^ ^ t2, a;; 2> and a;,,2' are connected by an edge for every 1 <, ilt i2 s£ I.
678 MISCELLANY

i68 URDUS
To prove the lower bound in (2) consider the points (x, y, z) of integer
coordinates 0 52. x, y, z 5i [n1 *]. Clearly the number of these points is less
than n but is greater than n (1 — e). The square of the distance between
two of these points is of the form
(9) u2 + v2 + w2, 0 ^. u, v, w < n1'*
The numbers (9) are all less than or equal 3re2 3 and since there are more than
nil e)\
such distances, clearly for some r the same distance must occur
2 J
at least l/Tw4'3 times, which completes the proof of (2), From deep number
theoretic results it follows that for suitable r the same distance occurs more
than c5w43 loglog n times and this is the best lower bound I can get for G3(n)
at the present time.
(Received December 18, 1959,
REFERENCES
[1] Aufgabc 167, Jahre.sbericht der Deutschen Math. Vereinigung 43 (1934) 114,
[2] Ebdos, P.: ,,On sets of distances of re points," Amer. Math, Monthly 53 (194(1)
248-250,
[3] Grunbaum, B, : "A proof of Vtfzsonyi's eoujoeturo," Bull. Research Council of
Israel 6A (1956) 77-78,
[4] Heppes, A, ; „Beweis einei' Vcrmutung von A, Vay.sonvi," Acta Math. A ad. Sci,
Hung. 7 (1957) 463-4()6,
[5] Stkaszewicz, S, ; "Sur un problemc geometriquc de P, Erdos," Bull. Acad. Poll,
S.*i. CI III. 5 (1957) 39—40,
[6] EhdOs, P, and Stone, A, H. : "On ihc structure of linear graphs," Bull. Amer.
Math. Soc. 52 (1946) 1087-1091.
679 GEOMETRY

ON SOME EXTREMUM PROBLEMS IN ELEMENTARY
GEOMETRY
By
P, ERDOS anil G. SZEKERES*
Mathematical Institute, Eiitvos Lorand University, Budapest and University Adelaide
(Received May 20, 1960)
Dedicated to the memory of Professor L, Fejer
1. Let S denote a set of points in the plane, N(S) the number of points in
6'. More than 25 years ago we have proved [2] the following conjecture of
Esther Klein —Szekeres;
There exists a positive integer f (n) with the property that if N (S) > f (n)
then S contains a subset P with N (P) = n such that the points of P form a
convex n-gon.
Moreover we have shown that if /0 (n) is the smallest such integer then
2 a 4\
t0(n) < and conjectured that /0 (»)=.-2" 2 for every n>3J We are
n — 2;
unable to prove or disprove this conjecture, but in § 2 we shall construct a set
of 2"" - points which contains no convex n-gon. Thus
2"^Mn)<|2'^49l
I n — 2
A second problem which we shall consider is the following: It was proved
by Szekeres [3] that
(i) In any configuration of N = 2" -j-1 points in the plane there are three
points which form an angle (< n) greater than (I — I jn 4- I//?N'2)3t,
(ii) There exist configurations of 2" points in the plane such that each
angle formed by these points is less than (1 — I In) n + f with e > 0 arbitrarily
small.
The first statement shows that for sufficiently small e > 0 there are no
configurations of 2" -f 1 points which would have the property (ii). Hence in
a certain sense this is a best possible result; but it does not determine the exact
limiting value of the maximum angle for any given N(S).
Let a (m) denote the greatest positive number with the property that in
every configuration of in points in the plane there is an angle /3 with
(1) ft^a(m).
* Tliis paper was written while P. E)rdos was visitini* at the University of Adelaide.
1 The conjecture is trivial for n -- 3; it was proved by Miss Klein, for n — 4 and by
E, Makai and P. TurAn for n - ri.
680 MISCELLANY

54
P, ERDOS AND G, SZEKERES
From (i) and (ii) above it follows that « (;«) exists for everv m > 3 and that for
2"<m<2"+1,
(2) [1 - 1/« + 1/« (2" + 1)*] * < a (//7) < [1 - 1/(/7 + I)] n.
Two questions arise in this connection;
1, What is the exact value of a (m).
2. Can the inequality (1) be replaced by
(3) (i > a (m)
For the first few values of rn one can easily verify that
113 2
a (3) = — n, a (4) = — ■ n, a (5) = — n, a (6) = a (7) = a (8) = — re,
6 Z 3 3
and that the strict inequality (3) is true for m = 7 and 8. For 3 <, m < 6 the
regular m -gons represent configurations in which the maximum angle is equal
to a (m); but we know of no other cases in which the equality sign would be
necessary in (1),
In § 4 we shall prove
Theorem 1. Every plane configuration of 2" points (n £= 3) contains an angle
greater than (1 — \jn)n.
The theorem shows in conjunction with (ii) above that for n > 3, a (2") =
= (1 — 1//7) iz and that the strict inequality (3) holds for these values of m.
The problem is thus completely settled for m = 2", n > 2.
It is not impossible that « (m) = (1 — \\n)n for 2"_1 < m < 2", n > 4,
and that (3) holds for every m > 6. However, we can only prove that for
0 < k < 2"-\ a (2" — k) > (1 — 1/n) n — knj2 (2" — k) (Theorem 2).
Finally we mention the following conjecture of P. ErdOs ; Given 2"+ 1 points
in /7-space, there is an angle determined by these points which is greater than
— ■ x. For /7 = 2 the statement is trivial, for n = 3 it was proved by N, H.
Kuiper nad A, H, Boerdijk,2 For n > 3 the answer is not known; and it seems
that the method of § 4 is not applicable to this problem. **
2. In this section we construct a set of 2"-2 points in the plane which
contains no convex n-gon. For representation we use the Cartesian (x, y) plane.
All sets to be considered are such that no three points of the set are collinear,
A sequence of points
(x,„ y„), v = 0,1, ..., k, x0<x1< ... <xl:
is said to be convex, of length k, if
r»-*-_i, <?>+!-* f0rv=l, ..., k-l;
xr — X,, - ! xn+1 — x,,
2 Unpublished.
** (Added in proof: This conjecture was recently proved by Danzer and Grunbaum
a surprisingly simplex way,)
681 GEOMETRY

ON SOME EXTREMUM PROBLEMS
concave, of length k, if the same is true with the inequality sign reversed. It
lli -i- i 2
was shown in [2] that a set of more than
points must contain either
/e also
points
a concave sequence of length k or a convex sequence of length I. We have also
stated, without proof, that there exists a set Su of f (k, I) =
[ k — 1
which contains no concave sequence of length k and no convex sequence of
length I. We shall first construct an explicit example of such a set. Skl consists
of points
[*, &/(*))> x
k + l —2
k— I
where gu (x) is defined inductively as follows:
1) = ^,(1) = 0
(a)
g:, W
gu W
where
c., = Max !
If k > I, /> I, then
ii, -i,/
for 1 <; x
k+ I — 3i
k— 1
(k + l — 3\
k—\ I
+ ckl for
||A- + /-
2) tlk + l-
I
tk + l-
k + l
k —
x <
lk-r-1-
k + / — 3|
k — 2 I
Clearly the construction is such that ghl (x) is monotone increasing and every
slope in Skl is positive.
Now if A denotes the set of the first f (k, I — 1) =
[k+l
points of
Skl and B the set of the last f (k — 1,1) points then a concave sequence in Sk!
which contains two points in A cannot contain a point in B and a convex
sequence in Ski which contains two points in B cannot contain a point in A. For
/fc _i_ i 3M
and the maximum slope in B
k— I II
the maximum slope in A is<gfr_,„1
i (k+l — 3
ik~l,l
2
so that B is entirely above any line connecting points
of A and A is entirely below any line connecting points of B, by the definition
of ckl. Hence the maximum concave sequence in Sld has length k — 1 and the
maximum convex sequence has length I— I.
682 MISCELLANY

56
P. ERDOS AND G. SZEKERES
To construct a set S of 2"~2points which contains no convex n-gon, we
proceed as follows: Let
a,. = 2 Max
1 I ii« —2i
" k \gl ■ „ ill
Ik —
A
A" I I
(A --- 1, ,,,, n—2), and define Sk, A =-- 1, , . ., n — 1 as follows;
oel Oj = Oi n —li
•S/;-l-l — Shr,
Then
^ (n — i) a,,— ^ a,-\, A= 1,
ii i i '
n-i
s = U s,
lias the required property.
The number of points in S is
" ,1 /» —2i „
- !A—ll
so we only have to show that every convex polygon in S has less than n sides.
Note that Sx consists of the point (1,0) alone and (x, y) 6 Slit k > 1 implies
x ^ 0, y < 0, Also
i ,n 9 i
'A &K- 1. /l- /.-1
k
(n - A) a, +
in —2.
/z — A + —
2
'/;— gA, n-/,-
|/7 — 2|
Ik—ll
<
(n — k)a
n — 2) , I
i n — k
h 'A—1 2
(A= 1,
so that the slope of any line connecting Sk and S,l + i is less than— I , \n — A + ■
and greater than— I / n + A— - . Therefore the slope of any line connecting
S„ and S„ 1 < A < / < n — 1 is less than — 1 \n — A
and greater
than
\n — l +
683 GEOMETRY

ON SOME EXTREMUM PROBLEMS
57
Suppose now that P,, (r = I, , . ,, r) is a non-empty subset of Ski, 1 < A, "_
r
•c,,,<fcr^n—I and such that P= Uniforms a convex polygon. Since
the slope of lines within each Sk, is positive, P, for 1 < J < r consists of a single
point and Px must form a concave sequence, Pr a convex sequence. But then
the total number of points in P is at most
fci + (kr — k1— 1) -J- (/i —A"r) = n—\.
3. The proof of Theorem 1 requires a refinement of the method used in
[3], and in this section we set up the necessary graph-theoretical apparatus.
We denote by C<N) the complete graph of order JV, i. e. a graph with A'
vertices in which any two vertices are joined by an edge. If G is a graph, then
S (G) shall denote the set of vertices of G. If A is a subset of S (G), then G\A
denotes the restriction of G to A. An even (odd) circuit of G is a closed circuit
containing an even (odd) number of edges.
A partition of G is a decomposition G = Gt + ,. , -J- G„ into subgraphs
Gj with the following property: Each G,- consists of all vertices and some edges
of G such that each edge of G appears in one and only one G,- (G,* may not
contain any edge at all). We call a partition 0 = 0-^+ ,,, -\- G„ even if it has the
property that no G,- contains an odd circuit.
Lemma 1, If a graph G contains no odd circuit then its vertices can be divided
in two classes, A and B, such that every edge of G has one end point in A and one
in B.
This is well-known and very simple to prove, e. g, [I], p 170,
Lemma 2, If C-N> = G± + ,,, H- G„ is an even partition of the complete
graph C(V> into n parts then JV < 2n.
This Lemma was proved in [3]; for the sake of completeness we repeat
the argument. Since Gx contains no odd circuit we can divide S (C(N>) in classes
A and B, containing N1 and JV2 vertices respectively, such that each edge of G±
connects a point of A with a point of B. But then Gx + ... -i- G„ induces
an even partition G'> -f- ,,, ->- G„ of G' = C(N> | A and since G' is a complete
graph of order JV,, we conclude by induction that JV, < 2n_1. Similarly N., < 2"_1
hence N = JV, -•-- JV2 < 2",
To prove Theorem I we shall need more precise information about the
structure of even partitions of C(N>, particularly in the limiting case of JV = 2",
The following Lemmas have some interest of their own; they are formulated
in greater generality than actually needed for our present purpose.
Lemma 3, Let N = 2" and C(N> = Gt + ,,, -f G„ an even partition of
C(N> into n parts. Then the total number of edges emanating from a fixed vertex
p 6 S (C<N>) in Gy,4- ... + G;;, where 1 < /, < /'., <"...< /, < n, is at least
2'— I and at most 2n—2"-'',
Clearly the order in which the components G,- are written is immaterial,
therefore we can assume in the proof that /„ = v, v = I, ,, ., i. We also note
that the first statement follows from the second one by applying the latter to
the complementary partition G,;i --,,, - Gn and by noting that the number
of edges from p in G, -'■ ...■'■ G„ is 2" -- 1, We shall prove the second state-
684 MISCELLANY

58
P, ERDOS AND G, SZEKERES
ment in the form that there are at least 2"~l vertices in S (On)) (including p
itself) which are not joined with p in Gt + ... + Gt. The statement is trivial
for n = 1; we may therefore assume the Lemma for n— 1.
By assumption, G± contains no odd circuit. Therefore by Lemma 1 we
can divide its vertices into two classes, A and B, such that no two points
of A (or of B) are joined in Gv Both classes A and B contain 2n~1 vertices;
for otherwise one of them, say A, would contain more than 2"_1 vertices and
C(N>S A would have an even partition G'z 4- ... -|- G'n into n — 1 parts, contrary
to Lemma 2.
Let A be the class containing p. Hence there are at least 2n~1 vertices with
which p is not joined in Gv This proves the Lemma for i = I. Suppose i > I
and consider the partition G2 + . . . + G'n of C<N>| A, induced by G1 + G2 + ...
-f- Gn. Since the order of C(N>J A is 2"_1, we find by the induction hypothesis
that there are at least 2"_1~<'-1' = 2n~' vertices in A to which p is not joined
in Gg H •..+ G,'. But p is not joined with any vertex of A in Glt therefore it
is not joined with at least 2n~' vertices in Gx + , ,. + G,.
In the special case of i = I we obtain
Lemma 3.1. Let N = 2" and C<N> = G1 + ... + Gn an even partition.
Then every p is an endpoint of at least one edge in every G,-.
If N < 2", then Lemma 3.1 is no longer true, but the number of vertices
for which it fails cannot exceed 2" — N. More precisely we shall prove
Lemma 4. Let N = 2" — k, 0 <, k < 2", and C<N> = G-, + ... + Gn
an even partition of C-N) into n parts. Denote by v (p), p e S = S (C(N>) the number
of graphs G,. in which there is no edge from p. Then
2 (2*(p>— \)<k,
P(S
Proof, For n = I the statement is trivial, therefore assume the Lemma
for n— 1. Let qlt ,.., qs be the "exceptional" vertices in Gx from which there
are no edges in Gj and denote by Q the union of vertices qt, i = \, ..., '}.
{Q may be empty). By Lemma 1, S is the union of disjoint subsets A, B and Q
such that every edge in Gx has one endpoint in A and one in B. Denote by Ax
the union of A and Q, by B1 the union of B and Q. Let a and b be the number
of vertices in A and B respectively. Then a + 6-f/ = 2" — k and a -f / < 2"_1,
b + / < 2"-1 by Lemma 2, applied to C<N> )^ and On)| ^. Write a = 2"'1 —
— / — fcj, 6 = 2"-1 — / — k2 so that k1 > 0, k2 > 0 and 2n — ] — kx — k,=
= 2n — k,
(4) k = / + fci + k2.
By applying the induction hypothesis to C{N'>\A1 and to the partition
G!z --'■ ... + G^ induced by G2 + ... + Gn, we find
2 (2"<p>— 1)+^ (2*)-1— 1) < /q
for some /u, (p) > v (p). Therefore a fortiori
2 (2'(p> -1)+2 (2KP) ' — 1) ^ A"i
piA piQ
685 GEOMETRY

ON SOME EXTREMUM PROBLEMS
59
and similarly
V (2'C) — 1) + V (2'W-1— 1) < A'2.
Hence
2- (2K/» - 1) - / < k, + k2=k-j
p <s
by (4), which proves the Lemma.
4. Before proving Theorem I we introduce some further notations and
definitions. To represent points in the Euclidean plane E we shall sometimes use
the complex plane which will also be denoted by E. If qlt p, </<, are points in E,
not on one line and in counterclockwise orientation, the angle (< n) formed
by the lines pq1 and pql will be denoted by A(q1pq.i).
A set of points S in E is said to have the property Pn if the angle formed by
any three points of S is not greater than (I — 1/n) n. We shall briefly say that
S is Pn or not Pn according as it has or has not this property.
A direction a in £ is a vector from 0 to eix on the unit circle. An n-partition
of E with respect to the direction a. is a decomposition of £ — {0} into sectors
7,, k = 1, ..., 2/7 where 7,; consists of all points
z = re'^+'i), r > 0, (k—l)—^.cp<k —.
n n
With every set of points S = {plt ..., pN} and every ^-partition of E with
respect to some direction a we associate a partition C(N> = Gx -f . .. + G„
of C(N> in n parts according to the following rule: pM pv are joined in G,- if and
only if the vector from p,, to p„ is in one of the sectors 7,-, Tn + i.
The following Lemma was proved in [3].
Lemma r>. If the set S is Pn then the partition C(N> = G± + ... + Gn associated
with any given n-partition of E is necessarily even.
We shall also need
Lemma 6. If Pi p, . .. pn are consecutive vertices of a regular n-gon P and q
is a point distinct from the centre and inside P then there is a pair of vertices
(p„ pj) such that A(ptq pj) >(1 — 1/n) jr.
The proof is quite elementary; if p( is a vertex nearest to q and if q is in
the triangle p,- ps p^ then at least one of the angles A (p} q p,), A (p, q p;-_v a)
is ,> ( 1— 1//7)71.3
Lemma 5 and Lemma 2 give immediately the result that a set of 2" -| 1
points in the plane cannot be Pn. Our purpose, however, is to prove Theorem 1
which can be stated as follows:
Theorem l*. A set of 2" points in the plane is not Pn.
Proof. Let S be a set of N = 2" (/7 > 2) points in the plane, plt p2, ..., p„
the vertices of the least convex polygon of S, in cyclic order and
counterclockwise orientation. We can assume that no three vertices of S are collinear,
otherwise S is obviously not P,.. We distinguish several cases.
3 See Problem 4086, Amer. Math. Monthly, 54 (1947), p. 1 17. Solution by C. R. Phelps.
686 MISCELLANY

GO
P. ERDOS AND G. SZEKERES
Suppose first that there is an angle A (/),-i/>,■/), .-1) ■-_ (1 — ly'O n. Let a
be the direction pt p, . 1 and <x («) the n-partition of E with respect to a. Then
there are no points of S in the sectors 7\ and Tn,. 1 corresponding to o-(u),
hence there is no edge from the vertex pt in the component Gn of the associated
partition C<N:> -= Gt -l ... J- G„. We conclude from Lemma 3.1 that G1 -- . ..
- G„ is not an even partition, hence by Lemma 5, S is not Pn.
Henceforth we assume that all angles A {p, .-ythPi i) are equal to (1—1 //z) n
so that the least convex polygon has 2n vertices. For convenience let these
vertices be (in cyclic order and counterclockwise orientation) pxq^ p,q, . . . pnqrV
and denote by a,- the side length ptqt and by b, the side q,p, rl.
Suppose that all angles A (pt._ 1 pi pi + 1) and A (qt 1qiqt + 1) are equal to
(1 — 2 In)n. Then an elementary argument shows that the triangles pt ^q, 1 p .
cl,-\Pil1h PiQiPi-i etc are similar, hence
ai bi a, , ,
which implies a1 = a, -.= ... --= an, bt = b2 =-- . .. = b„. We conclude that
lh Pj- • • Pn is a regular n-gon.
Now let q be any point of S inside the least convex polygon. If q is inside
the triangle /?,</, p,+i then clearly A(piqpi+1) >(1 — ljn)zr. Therefore we
may assume that each </ inside the least convex polygon is already inside p1 . .. p,,.
Since /z > 3, there are at least two such points, hence we may assume that q
is not the centre. But then by Lemma 6, A(pfq pj) > (1 — 1 >n) zx for suitable
/' and /.
The last remaining case to be considered is when not all angles
A (p, .1pi pi ,-A are equal; then for some i, A (p,-_i />,• Pi + A < (1 —2[n) n. Since
A (P, -l 9/-1 Pi) = ^ (P/ ?/ Pi + i) ~ (1 — 1/^)^1 we may assume that there are
no points of S inside the triangles Pi-Xqt xpt and ptq,pH 1. But then if a is the
direction of pt pl^1 and <x («) the corresponding n-partition of E,
0^) = 0^-. ..tG„ the partition of C<N> associated with cr (a), then inG^-- G„
there are only two edges running from p,, namely p,-ql:i and /?,</,. By
Lemma 3 (with i = 2, /\ = n — 1, /2 = n), C(N> ^ G, -=- . . . 4- G„ is not an
even partition and by Lemma 5, S is not Pn.
Finally we prove
Theorem 2. In a plane configuration of N == 2" — A- points (0 <T k <' 2"~')
Mere is an angle > (1 — \jn — A/2N)tt.
Proof. Suppose that all angles formed by the points of S are <. (1 — 1 n) n -
b — b' where b == A ir/N and b' > 0. Let p 6 S and
f=--<f(P) - A{qlPq2) < (1 — 1/„)„__ i_ d'.ftCS, ?2 6 S
the maximum angle at p. If there are several such angles, make an arbitrary
but fixed choice.
687 GEOMETRY

ON SOME EXTREMUM PROBLEMS
61
Let a -- a (p) be the direction of pqx so that there are no lines pq, q^S
in the sectors
z=±re'^ °K 0 < G< (1 — l/n)jr— ~d — 6'.
If S has N = 2" — k points, then there are N pairs of directions + a (p).
Hence there is a direction ft such that at least k-'~ I directions f„« (p„), ;> = 0, 1,
..., k, f,.—. ± I are in the interval
ft <E„a(pv)<f3-\ b + ~8'.
By a slight displacement of ft we can achieve that all directions in S should
3
be different from the direction ft -\ b.
4
3
3
Consider now the ^-partition cr\ft-\ b of E with respect to ft -|—- b
I 4 4
and construct the associated partition C(N> = G1 -[- ...-!- Gn with the following
3 I
modification; Every pq with direction between ft -i b and ft + <5 ^ <5' shall
1 JT
be joined in G„ instead of G1 and every p </ with direction between ft -J— 5 + -—
2 n
3 :r
and /8 H <5 + "-shall be joined in G., instead of Gx. With this modification
4 n
it is still true that the partition C(N> ■= Gx -f ... + Gn is even if S has no angle
greater or equal to (I — 1/n) zz b — b'. But it is easy to see that the vertices
3
p, are not joined with any vertex of C(N> in G^ For if ft -\ b < s„ a (p,.) <
-.ft + b + -5' then the edges emanating fromp,, in the sectors 7\, Tn+1are count -
3
ed to Grl; and if ft < ?,. « (p„) < /8 -f — 5, the only possible edges from p,
in the sectors 7\, 7"„ + 1 are in directions between ft J, b + - - and ft -\ — 6-\- —
2 n 4 «
and these are counted to G\. Thus we have a contradiction with Lemma 4 and
the Theorem is proved.
Note added in proof. In the case af k = 1, Theorem 2 can be improved as
follows.
Theorem 3. Every plane configuration of 2" — I points (n > 2) contains
an angle not less than (I — ljn)n.
The theorem shows in particular that a (2" — I) = (I — \jn)n, but we
cannot decide whether the strict inequality (3) is valid for m = 2"— I.
688 MISCELLANY

62
P. ERDOS AND G. SZEKERES; ON SOME EXTREMUM PROBLEMS
Proof. Suppose N (S) = 2" — 1 and all angles in S are less than (I — \jn)zi.
Let q be an interior point of S, that is one which is not on the least convex
polygon. Let
A{qiqqi) = {\~ l/n)™—3, d>0
be the largest angle at q. If /3 is the direction of q q^, a = /3 + — d, <r («) the
corresponding ^-partition of E, C(N) = G\ + ... + G„ the partition of C{N>
associated with <r (a), then clearly there are no edges from q in G^ We conclude
from Lemma 4 and Lemma 5 that from all other vertices of S there is at least
one edge in every G,- (/ = 1, ..., n). We show that this leads to a contradiction.
Let P ~ (¾ ..., pm) denote the least convex polygon of S. Since each
angle in P is less than (1 — 1 jn) n, we have m < 2 n — 1. Therefore if 7\, ..., T2n
are the sectors of cr (a) there is a pt such that if p,-_i/5, is in TA._1 then ptpt a
is not in Tk. But then there is no edge from pt in Gk, as easily seen by
elementary geometry.
References
[11 Konig, D., Theorie der encllichen und unendlichen Graphen, (Leipzig, 1936).
[2] Erdos, P. and Szekeres, G., A combinatorial problem in geometry, Compositio Math.,
2(1935), 463—470.
[3] Szekeres, G., On an extreinum problem in the plane, Amer. Journal of Math., 63 (1941),
203—210.
689 GEOMETRY

Reprinted from Journal of Combinatorial Theory Vol. 10, No. 3, May 197 I
All Rights Reserved by Academic Press, New York and London Printed in Belgium
Some Extremal Problems in Geometry
Paul Erdos and George Purdy
Department of Mathematics, The University of Calgary, Alberto, Canada
Communicated by the Editor-in-Chief
Received January 7, 1969
1. Introduction
Let there be given n points Ai ,..., Xn in ^-dimensional Euclidean space
Ek . Denote by d(X\ , X,) the distance between X, and X, . Let A(X1,..., Xn)
be the number of distinct values of d(X{, X,), 1 < ; < j < n. Put
fk(n) = min A(X1,..., X„), where the minimum is assumed over all
possible choices of A\ ,..., Xn . Denote by gk(n) the maximum number of
solutions of d(X{, X,) = a, 1 < ; <j < n, where the maximum is to be
taken over all possible choices of a and n distinct points A'i ,..., Xn . The
estimation of fk(n) and gk(n) are difficult problems even for k = 2. It is
known that [1,2]:
C[H2'3 <fi(n) < c2«/Vlog n, (1)
cind
nl-[r,.-0oglogfl)] <g2(n) <Ci„M (2)
where the c's denote positive absolute constants.
It seems that in (1) the upper bound and in (2) the lower bound is close
to the right order of magnitude, but we cannot even show /2(n) > "1"'
or g>(n) < «1+e.
If k > 4 the study of gk(n) becomes somewhat simpler [3],
A. Oppenheim asked us the question of investigating the number of
triangles chosen from n points in the plane which have the same non-zero
area. In this note we investigate this question and its generalizations.
246
690 MISCFXLANY

SOME EXTREMAL PROBLEMS IN GEOMETRY
247
2. Notations
Let X0 , X1,..., Xn be n distinct points in A:-dimensional space Ek ,
A > 0, r > 2.
We define glP(n; X1,..., Xn ; A) (n > /• + 1, & Js r) to be the number of
/•-dimensional simplices of the form Xi ■■■ Xt having volume A. We let
g^in; X, ,..., Xn) = max ^'(n; ^ ,..., Xn ; J)
and
g[r)(n) = max gW(n; X, ,..., Xn).
Let X0 be a fixed point and define
Gt\n; X, ,..., Xn ; A)(n > r; k > r)
to be the number of /--dimensional simplices of the form X0Xi ■■• A^
having volume /1. We let
G?(n; X0 ,..., Afn) = max G(kr\n; X0 ,..., Xn ; J)
and
G£r)(n) = max Gkr\n; X0,..., Xn).
Clearly g<» ^ nGkr\n - 1) < nG^(n).
We see that g^X") = gsO1) in the notation of the introduction.
We extend fk(ri) to fk\n) and Fkn(n) in a similar way:
Let fk\n; X1,..., A-,,) be the number of distinct volumes occurring
among all the /'-dimensional simplices Xi '" Xt , and let f^\ii) =
min fk\n; X1,..., Xn) where the minimum is taken over all possible
choices of A\ ,..., Xn , except where X1,..., Xn all lie in an (r —
^-dimensional subspace (not necessarily through the origin).
Similarly, if X0 is a fixed point, let Fkr\n; X0,..., Xn) be the number of
distinct volumes occurring among the r-dimensional simplices A'0A'! ■■■ Xi ,
and let Flk(n) = min Fkr\n; Xa,..., Xn) where the minimum is taken over
X0 ,..., Xn not lying is an (r — l)-dimensional subspace.
Clearly we have the following: f[r\n) < nF[!\n - 1) < nF[!\n),
fk\n)=Mn) in the notation of Section 1. ^"(n) < g[!\n),
G£Un) < Gkr\n), /&(„) ^ fkr\n), and F&(n) > Fkr\n) (k > r).
691 GEOMETRY

248
ERDOS AND PURDY
3.
Oppenheim pointed out that the generalized construction of Lenz (see,
e.g., [3]) gives us lower bounds for g and G. To illustrate, we show that
G<2)(2n) > «2.
Let (Xi, yt) (1 < i < n) be distinct pair of real numbers such that
xt2 + yt* = 1. Let Xi = (0, 0, xi ,yt,), F,- = (X{, yt, 0, 0), (1 < i < n),
X0 — (0, 0, 0, 0). Then the n2 triangles X0Xi Y} are congruent and therefore
have the same area.
The same method shows that G$(kri) > nk and gM+2,(kn + 1) ^ w*+1-
It seems to us that
^U") = {k + 1)fc+1 (1 + 0(1)),
i.e., that Oppenheim's example is asymptotically best possible.
It also seems that
and we have proved this for k = 2, but we do not include the proof here.
Theorem 1. Gf'(n) < 4n3/2 and therefore g(22)(n) < 4n5/2.
Proof. Suppose that, for some least n, G^\n) > 4n3/2. Then n > 4.
Let
G<2)(n; A-0,..., *„ ; J) = m. > 4«3 2, Zl > 0.
Let G be the graph whose vertices are X1,..., X„ and whose edges are all
the XjXj such that the triangle XaXiX} has area A. Then every vertex
A'i of G is adjacent to at least [4\/h] other vertices, since, otherwise,
removing Xi would reduce the number of triangles by at most 4vV and
we would have
Gf\n - 1) > 4n3/2 - Wn > 4(n - 1)3/2
contradicting the minimal choice of n. If 1 ^/^ n, therefore, there are
at least [4Vn] points Xj such that the triangle XaXtX} has area A. These
points lie on two lines parallel to XQXj . One of these linear sets of points,
692 MISCELLANY

SOME EXTREMAL PROBLEMS IN GEOMETRY
249
say S, , contains at least £[4\/n] points. Consider the points Xt on the
first [\/n] lines S/l <j < [V~n]). Then
n >
[Vrt] [Vn]
U St > X I Si I - I _ I S, n S,
1 1 lsSisSJsS[Vrt]
which is false for n ^ 4.
Theorem 2.
g22)(n) > en2 log log n (« > n0).
Proof. Let n 55 "o > where n0 will be chosen later. Let a = [\/log n]
and let X1,..., A"„, (m < n) be the integral points (x, y) where 1 < x < nja
and 1 < y < a. It is enough to show that g^\m; X1 ,..., Xrn ; |a!) ^
en2 log log n for n ^ nQ. Let (¾ , yx) and (x2 , >'2) be integral^ points
satisfying
x, < x2 < a!,
a
Ji < J2 < «•
We may choose nQ large enough so that (nja) — a! > (nj2a) for n ^ «0.
Let ¢/= (x2 — x1, y.2 — y-^. The ¢/ + 1 points (x3, y3) given by
x3 = x1 + -j (x2 - xj + 01(^- jj-1,
A:
>3 = yi + -j O2 - A) (0 < fc < d),
are clearly among the points X1,..., X„, . Also
l < y\ < ys < y2 < a,
1 ^ x1 < x3 < x2 + a! < «/a.
(3)
(4)
693 GEOMETRY

250
erdOs and purdy
The area of the triangle (xf, yt) (1 < /' < 3) is easily seen to be \a\,
and condition (4) ensures that no unordered triple XtXjXk is represented
more than once in the form (xt , y{) (1 < ;' ^ 3).
Let 0 < d < V'a. We choose (x1, yj and (x2 , y2) so that
(x2 — xx , y2 — yj = d, i.e., x2~ xx = fid and y2 — y1 = vd, (fx, v) = 1,
i.e.,
For each (/x, v), (xt, j'J, ¢/, this determines (x2 , y2). It is well known
and easy to prove by elementary number theory that in a rectangle of
sides t1 and t2 the number of points with coprime coordinates is
(1 + o(l)) —5 tj2 as tx —»■ co and ?2 —»■ oo.
The point (¾ , ^) can be chosen in
@[^G-a!)]>cnways
and thus the number of choices of {xx, yj, (x2 , y2) is greater than cn2jd2.
Now on the line (xx, yj)(x2 , y2) there are d + 1 lattice points given by (3).
Thus there are d + 1 choices for (x3, y3). Thus the number of triangles
(x1, y^(x2 , y2)(x3, y3), (x2 — xx, y2 — yx) = d having area \a\ is more
than c{n2jd). Summing for d we get the result.
Theorem 3. G^\n) < en2-1/3 and therefore g32\n) < en3-1/3.
Proof. Suppose that G3*\n) > c«2-1/3, for some n. Then for some
A > 0 and Xa,..., Xn in E3, G<2)(n; X0,..., Xn,A)> en2"1'3. Let G be the
graph whose vertices are X1,..., Xn and whose edges are XtXj such that
the triangle X0X{Xj has area A. By a theorem of Sos, Turan, and Kovari [4],
there exist Y1, Y2 , Y3 and Zx,..., Zk such that Y{ and Z3 are joined for
1 < ;' < 3, 1 <j < &, provided that c is sufficiently large, depending
only on k. Hence three cylinders with axes X0Y1, X0X2 , X0Y3 all contain
Z1,..., Zk on their surfaces. But by elementary geometry this is impossible
when k is greater than some absolute constant.
Somewhat similar methods work in higher dimensions. Using a theorem
on generalized graphs proved in [5, Theorem 1] it can be shown that, e.g.,
g^\n) 5¾ en3-' for some e, 0 < e < 1, and also that G^\n) < cknk~'K
694 MISCELLANY

SOME EXTREMAL PROBLEMS IN GEOMETRY 251
We may obtain a trivial upper bound for/^'(n). Consider the points
(x, y, z) with integer coordinates 0 < x, y, z =¾ n1/3. There are at least n,
and if XYZ are three such points, the area A of triangle XYZ is not at most
Since 4A2 is an integer, we see that
/£s,(n)<g)V.
The same method yields
/£r,00<(*)2n2r'*
(the result for/f'(") implies gf> > en11/5).
4.
Finally we would like to mention a few related combinatorial problems:
Let there be given n points in the plane. How many quadruplats can one
form so that not all the six distances should be different ? It is not difficult
to show that one can give n points so that there should be en3 log n
quadruplets with not all the distances distinct, but that one cannot have en1 2
such quadruplets. It seems that the maximum is less than /73 + t but we
could not prove this.
A well known theorem of £Pannwitz states that in a plane set of n points
of diameter 1 the maximum distance can occur at most n times and n is
best possible. Similarly we can ask: Let there be given n points in the plane.
How many triangles can one have which have the maximal (or minimal
[non-zero]) area ? Unfortunately we have only trivial results. The maximum
area can occur at most en2 times and it can occur en times.
Many more questions could be asked, here we state only a few of
them. Let there be given w points in A:-dimensional space. What is the
largest set of pairwise congruent (similar) triangles2 ? What is the largest
set of equilateral, (isosceles) triangles2 ?
695 GEOMETRY

252
ERDOS AND PURDY
References
1. P. Erdos, On Sets of Distances of n Points, Amer. Math. Monthly 53 (1946), 248-250.
2. L. Moser, On the Different Distances Determined by n points, Amer. Math. Monthly
59(1952), 85-91.
3. P. Erdos, On Some Applications of Graph Theory to Geometry, Canad. J. Math.
19 (1967), 968-971. See also P. Erdos, On Sets of Distances of n Points in Euclidean
Space, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 165-169.
4. T. Kovari, V. T. S6s, and P. Turan, On a Problem of K. Zarankiewicz, Colloq.
Math. 3(1954), 50-57.
5. P. Erdos, On Extremal Problems of Graphs and Generalized Graphs, Israel J.
Math. 2(1964), 183-190.
696 MISCELLANY

Bibliography of Paul Erdos
This bibliography, except for the last few entries, was compiled by the
Hungarian Academy of Sciences. The additional papers are designated by roman
numerals ([i], . . . , [vi]).
The asterisked papers are reprinted in this volume.
1932
[I] Egy Kilrschdk-fele elemi szdmelmeleti tetel dltaldnostidsa, Mat. Fiz. Lapok 39
(1932)1-8.
[2] Beweis eines Satzes von Tschebyschef, Acta Litt. Sci. Reg. Univ. Hungar. Fr.-
Jos. Sect. Sci. Math. 5(1932)194-198.
1934
[3] Bizonyos sz&mtani sorok torzsszdmairol, Bolcseszdoktori ertekezes. Sarospatak,
1934, 1-20.
[4] On the density of the abundant numbers, J. London Math. Soc. 9(1934)278-
282.
[5] A theorem of Sylvester and Schur, J. London Math. Soc. 9(1934)282-288.
[6] (with P. Turan) On a problem in the elementary theory of numbers, Amer. Math.
Monthly 41(1934)608-611.
[7] (with Gy. Szekeres) Uber die Anzahl der Abelschen Gruppen gegebener Ordnung
und uber ein verwandtes zahlentheoretisches Problem, Acta Litt. Sci. Reg. Univ.
Hungar. Fr.-Jos. Sect. Sci. Math. 7(1934)94-103.
1935
[8] On primitive abundant numbers, J. London Math. Soc. 10(1935)49-58.
[9] On the density of some sequences of numbers, J. London Math. Soc. 10(1935)
120-125.
[10] Note on sequences of integers no one of which is divisible by any other, J. London
Math. Soc. 10(1935)126-128.
[II] Note on consecutive abundant numbers, J. London Math. Soc. 10(1935)118—
121.
697 BIBLIOGRAPHY OF PAUL ERDOS

[12] The representation of an integer as the sum of the square of a prime and of a square-
free integer, J. London Math. Soc. 10(1935)243-245.
[13] On the difference of consecutive primes, Quart. J. Math. Oxford Ser. 6(1935)
124-128.
[14] On the normal number of prime factors of p — 1 and some related problems
concerning Euler's 4> function, Quart. J. Math. Oxford Ser. 6(1935)205-213.
[15] (with P. Turan) Ein zahlentheoretischer Satz, Bull. Inst. Math. Mec. Univ.
Kouybycheff Tomsk 1(1935)101-103.
[16] Uber die Vereinfachung eines Landauschen Satzes, Bull. Inst. Math. Mec. Univ.
Kouybycheff Tomsk 1(1935)144-147.
[17] Uber die Primzahlen gewisser arithmetischer Reihen, Math. Z. 39(1935)473-
491.
* [18] (with Gy. Szekeres) A combinatorial problem in geometry, Compositio Math.
2(1935)463-470.
1936
[19] A generalization of a theorem of Besicovitch, J. London Math. Soc. 11(1936)
92-98.
[20] On the representation of an integer as the sum of k k-th powers, J. London Math.
Soc. 11(1936)133-136.
[21] (with P. Turan) On some sequences of integers, J. London Math. Soc. 11
(1936)261-264.
[22] On the integers which are the quotient of a product of three primes, Quart. J.
Math. Oxford Ser. 7(1936)16-19.
[23] On the integers which are the quotient of a product of two primes, Quart. J. Math.
Oxford Ser. 7(1936)227-229.
[24] On a problem of Chowla and some related problems, Proc. Cambridge Philos.
Soc. 32(1936)530-540.
[25] On the arithmetical density of the sum of two sequences one of which forms a basis
for the integers, Acta Arith. 1(1936)197-200.
[26] (with H. Davenport) On sequences of positive integers, Acta Arith. 2(1936)
147-151.
698 BIBLIOGRAPHY OF PAUL ERDOS

[27] (with T. Grunwald and E. Weiszfeld) V'egtelen grdfok Euler-vonalairol. Mat.
Fiz. Lapok 1936, 129-141.
[28] Note on some additive properties of integers, Publ. de Congres International
des Math., Oslo, 1936, 1-2.
[29] (with E. Feldheim) Analyse mathematique.—Sur le mode de convergence pour
rinterpolation de Lagrange, C. R. Acad. Sci. Paris 203(1936)913-915.
1937
[30] On the density of some sequences of numbers, II, J. London Math. Soc. 12
(1937)7-11.
[31] On the sum and difference of squares of primes, I, J. London Math. Soc. 12
(1937)133-136.
[32] On the sum and difference of squares of primes, II, J. London Math. Soc. 12
(1937)168-171.
[33] Note on the number of prime divisors of integers, J. London Math. Soc. 12
(1937)308-314.
[34] (with J. Gillis) Note on the transfinite diameter, J. London Math. Soc. 12
(1937)185-192.
[35] On the easier Waring problem for power of primes, I, Proc. Cambridge Philos.
Soc. 33(1937)6-12.
[36] (with P. Turan) On interpolation, I. Quadrature and mean convergence in the
Lagrange interpolation. Ann. of Math. 38(1937)142-155.
[37] (with V. Jarnik) Eine Bemerkung ilber lineare Kongruenzen, Acta Arith. 2
(1937)214-220.
[38] (with R. Oblath) Uber diophantische Gleichungen der Form n\ = x" ±y" und
nl ± ml = xv, Acta Litt. Sci. Reg. Univ. Hungar. Fr.-Jos. Sect. Sci.
Math. 8(1937)241-255.
[39] (with G. Grunwald) Uber die arithmetische Alittelwerte der Lagrangeschen
Interpolationspolynome, Studia Math. 7(1937)82-95.
1938
[40] (with Chao Ko) Note on the Euclidean algorithm, J. London Math. Soc.
13(1938)3-8.
699 BIBLIOGRAPHY OF PAUL ERDOS

[41] On the density of some sequences of numbers, III, J. London Math. Soc. 13
(1938)119-127.
[42] (with K. Mahler) On the number of integers which can be represented in a binary
form, J. London Math. Soc. 13(1938)134-139.
[43] (with Chao Ko) Some results on definite quadratic forms, J. London Math.
Soc. 13(1938)217-224.
[44] (with G. Grunwald) Uber einen Fabeschen Satz, Ann. of Math. 39(1938)
257-261.
[45] (with P. Turan) On interpolation, II. On the distribution of the fundamental
points of Lagrange and Hermite interpolation, Ann. of Math. 39(1938)703-
724.
[46] (with G. Grunwald) Note on an elementary problem of interpolation, Bull.
Amer. Math. Soc. 44(1938)515-518.
[47] (with B. Lengyel) On fundamental functions of Lagrangean interpolation, Bull.
Amer. Math. Soc. 44(1938)828-834.
[48] (with T. Grunwald and E. Vazsonyi) Uber Euler-Linien unendlicher
Graphen, J. Math, and Phys. 17(1938)59-75.
[49] (with Chao Ko) On definite quadratic forms, which are not the sum of two
definite or semi-definite forms, Acta Arith. 1938, 101-121.
[50] On sequences of integers no one of which divides the product of two others and on
some related problems, Izv. Nauc. Isz. Inszt. Mat. Mech. Tomszk 2(1938)
74-82.
[51] On the asymptotic density of the sum of two sequences one of which forms a basis
for the integers, II, Travaux de l'lnst. Math, de Tbilissi 3(1938)217-224.
[52] On additive properties of squares of primes, I, Nederl. Akad. Wetensch. Proc.
Ser. A. 41(1938)3-7.
[53] Uber die Reihe £ -, Overdruk uit Mathematica B.7(1938)1-2.
P
1939
[54] (with K. Mahler) Some arithmetical properties of the convergents of a continued
fraction,}. London Math. Soc. 14(1939)12-18.
700 BIBLIOGRAPHY OF PAUL ERDOS

[55] Note on products of consecutive integers, I, J. London Math. Soc. 14(1939)
194-198.
[56] Note on the product of consecutive integers, II, J. London Math. Soc. 14(1939)
245-249.
[57] On the integers of the form xk + yk, J. London Math. Soc. 14(1939)250-254.
[58] On the easier Waring problem for powers of primes, II, Proc. Cambridge
Philos. Soc. 36(1939)149-165.
[59] (with H. Davenport) On sums of positive integral k-th powers, Ann. of Math.
40(1939)533-536.
[60] (with T. Griinwald) On polynomials with only real roots, Ann. of Math. 40
(1939)537-548.
[61] (with M. Kac) On the Gaussian law of errors in the theory of additive functions,
Proc. Nat. Acad. Sci. U.S.A. 25(1939)206-207.
[62] (with A. Wintner) Additive arithmetical functions and statistical independence,
Araer. J. Math. 61(1939)713-721.
[63] On the smoothness of the asymptotic distribution of additive arithmetical functions,
Araer. J. Math. 61(1939)722-725.
[64] On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61(1939)
974-976.
[65] An extremum-problem concerning trigonometric polynomials, Acta Sci. Math.
(Szeged) 9(1939)113-115.
[66] (with P. Turan) On the uniformly dense distribution of certain sequences of
points, Ann. of Math. 41(1940)163-173.
1940
[67] On extremal properties of the derivatives of polynomials, Ann. of Math. 41(1940)
310-313.
[68] (with P. Turan) On interpolation, III. Interpolator)! theory of polynomials, Ann.
of Math. 41(1940)510-553.
[69] The dimension of the rational points in Hilbert-space, Ann. of Math. 41(1940)
734-736.
701 BIBLIOGRAPHY OF PAUL ERDOS

[70] Note on some elementary properties of polynomials, Bull. Amer. Math. Soc.
46(1940)953-958.
[71] On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26
(1940)294-297.
[72] On the smoothness properties of a family of Bernoulli convolutions, Amer. J.
Math. 62(1940)180-186.
[73] (with A. Wintner) Additive functions and almost periodicity (S2), Amer. J.
Math. 62(1940)635-645.
[74] (with M. Kac) The Gaussian law of errors in the theory of additive number
theoretic functions, Amer. J. Math. 62(1940)738-742.
[75] The difference of consecutive primes, Duke Math. J. 6(1940)438-441.
[76] On a conjecture of Steinhaus, Rev. Univ. Nac. Tucuman, Ser. A. Mat. Fis.
Tedr. 1(1940)217-220.
[77] (with M. Kac, E. R. van Kempen, and A. Wintner) Ramanujan sums and
almost periodic functions, Studia Math. 9(1940)43-53.
1941
[78] (with P. Turan) On a problem of Sidon in additive number theory and on some
related problems, J. London Math. Soc. 16(1941)212-216.
[79] On some asymptotic formulas in the theory of the "factorisatio numerorum," Ann.
of Math. 42(1941)889-993.
[80] On divergence properties of the Lagrange interpolation parabolas, Ann. of Math.
42(1941)309-316.
[81] (with I. Lehner) The distribution of the number of summands in the partitions
of a positive integer, Duke Math. J. 8(1941)335-345.
1942
[82] On the uniform distribution of the roots of certain polynomials, Ann. of Math.
43(1942)59-64.
[83] On the asymptotic density of the sum of two sequences, Ann. of Math. 43(1942)
65-68.
[84] On the law of the iterated logarithm, Ann. of Math. 43(1942)419-436.
702 BIBLIOGRAPHY OF PAUL ERDOS

[85] On an elementary proof of some asymptotic formulas in the theory of partitions,
Ann. of Math. 43(1942)437-450.
[86] (withG. Szego) On a problem of I. Schur, Ann. of Math. 43(1942)451-470.
[87] Some set-theoretical properties of graphs, Rev. Univ. Nac. Tucuman, Ser. A.
Mat. Fis. Tedr. 3(1942)363-367.
1943
[88] A note on Farey series, Quart. J. Math. Oxford Ser. 14(1943)82-85.
[89] (with J. A. Clarkson) Approximation by polynomials, Duke Math. J. 10
(1943)5-11.
[90] (with A. Tarski) On families of mutually exclusive sets, Ann. of Math. 44
(1943)315-329.
[91] On some convergence properties in the interpolation polynomials, Ann. of Math.
44(1943)330-337.
[92] Some remarks on set theory, I, Ann. of Math. 44(1943)643-646.
[93] Corrections to two of my papers, Ann. of Math. 44(1943)647-649.
[94] On the convergence of trigonometric series, J. Math, and Phys. 22(1943)37-39.
[95] (with S. Kakutani) On non-denumerable graphs, Bull. Amer. Math. Soc.
49(1943)457-461.
1944
[96] On highly composite numbers, J. London Math. Soc. 19(1944)130-133.
[97] On the maximum of the fundamental functions of the idtraspherical polynomials,
Ann. of Math. 45(1944)335-339.
[98] Some remarks on connected sets, Bull. Amer. Math. Soc. 50(1944)443-446.
[99] (with L. Alaoglu) A conjecture in elementary number theory, Bull. Amer. Math.
Soc. 50(1944)881-882.
[100] (with L. Alaoglu) On highly composite and similar numbers, Trans. Amer.
Math. Soc. 56(1944)448-469.
703 BIBLIOGRAPHY OF PAUL ERDOS

1945
[101] Note on the converse of Fabry's gap theorem, Trans. Amer. Math. Soc. 57
(1945)102-104.
[102] (with* A. H. Stone) Some remarks on almost periodic transformations, Bull.
Amer. Math. Soc. 51(1945)126-130.
[103] On the least primitive root of a prime p, Bull. Amer. Math. Soc. 51(1945)
131-132.
[104] (with I. Niven) On certain variations of the harmonic series, Bull. Amer.
Math. Soc. 51(1945)433-436.
[105] Some remarks on Eider's 4> function and some related problems, Bull. Amer.
Math. Soc. 51(1945)540-544.
* [106] (with N. H. Anning) Integral distances, Bull. Amer. Math. Soc. 51(1945)
598-600.
[107] Some remarks on the measurability of certain sets, Bull. Amer. Math. Soc.
51(1945)728-731.
* [108] On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. 51(1945)
898-902.
[109] Integral distances, Bull. Amer. Math. Soc. 51(1945)966.
1946
[110] On the distribution function of additive functions, Ann. of Math. 47(1946)
1-20.
[Ill] On the Hausdorff dimension of some sets in Euclidean space, Bull. Amer. Math.
Soc. 52(1946)107-109.
[112] On the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 52
(1946)179-183.
[113] On some asymptotic formulas in the theory of partitions, Bull. Amer. Math.
Soc. 52(1946)185-188.
[114] (with I. Niven) Some properties of partial sums of the harmonic series, Bull.
Amer. Math. Soc. 52(1946)248-251.
[115] (with M. Kac) On certain limit theorems of the theory of probability, Bull.
Amer. Math. Soc. 52(1946)292-302.
704 BIBLIOGRAPHY OF PAUL ERDOS

[116] (with P. C. Rosenbloom) Toeplitz methods which sum a given sequence, Bull.
Amer. Math. Soc. 62(1946)463-464.
[117] Some remarks about additive and multiplicative functions, Bull. Amer. Math.
Soc. 52(1946)527-537.
[118] (with A. H. Copeland) Note on normal numbers, Bull. Amer. Math. Soc.
52(1946)857-860.
[119] (with A. H. Stone) On the structure of linear graphs, Bull. Amer. Math.
Soc. 52(1946)1087—1091.
* [120] (with I. Kaplansky) The asymptotic number of Latin rectangles, Amer. J.
Math. 69(1946)230-236.
* [121] On sets of distances of n points, Amer. Math. Monthly 53(1946)248-250.
[122] (with I. Niven) The a + /3 hypothesis and related problems, Amer. Math.
Monthly 53(1946)314-317.
1947
[123] (with K. L. Chung) On the lower limit of sums of independent random
variables, Ann. of Math. 48(1947)1003-1013.
* [124] Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53(1947)
292-294.
[125] Some asymptotic formulas for multiplicative functions, Bull. Amer. Math. Soc.
53(1947)537-544.
[126] Some remarks and corrections to one of my papers, Bull. Amer. Math. Soc.
53(1947)761-763.
[127] (with G. Piranian) A note on transforms of unbounded sequences, Bull. Amer.
Math. Soc. 53(1947)787-790.
[128] (with M. Kac) On the number of positive sums of independent random
variables, Bull. Amer. Math. Soc. 53(1947)1011-1020.
[129] Some remarks on polynomials, Bull. Amer. Math. Soc. 53(1947) 1169—1176.
[130] (with H. Fried) On the connection between gaps in power series and the roots
of their partial sums, Trans. Amer. Math. Soc. 62(1947)53-61.
[131] (with G. Piranian) Over-convergence on the circle of convergence, Duke Math.
J. 14(1947)647-658.
705 BIBLIOGRAPHY OF PAUL ERDOS

1948
[132] On the integers having exactly k prime factors, Ann. of Math. 49(1948)53-66.
[133] (with I. Niven) On the roots of a polynomial and its derivative, Bull. Amer.
Math. Soc. 54(1948)184-190.
[134] (with P. Turan) On some new questions on the distribution on prime numbers,
Bull. Amer. Math. Soc. 54(1948)271-278.
[135] On the density of some sequences of integers, Bull. Amer. Math. Soc. 54(1948)
685-692.
[136] On the difference of consecutive primes, Bull. Amer. Math. Soc. 54(1948)
885-889.
[137] (with R. P. Boas and R. C Buck) The set on which an entire function is
small, Amer. J. Math. 70(1948)400-402.
[138] (with I. S. Gal) On the representation of \, 2, . . . , N by differences, Nederl.
Akad. Wetensch. Proc. 51(1948)1155-1158.
[139] (with N. G. de Bruijn) On a combinatorial problem, Nederl. Akad.
Wetensch. Proc. 51(1948)1277-1279.
[140] (with P. Turan) On a problem in the theory of uniform distribution, I—H,
Nederl. Akad. Wetensch. Proc. 51(1948)1146-1154; 1262-1269.
[141] On arithmetical properties of Lambert series, J. Indian Math. Soc. 12(1948)
63-66.
[142] Some remarks on diophantine approximations, J. Indian Math. Soc. 12(1948)
67-74.
[143] Some asymptotic formulas in number theory, J. Indian Math. Soc. 12(1948)
75-78.
1949
[144] (with W. Feller and H. Pollard) A property of power series with positive
coefficients, Bull. Amer. Math. Soc. 55(1949)201-204.
[145] On some applications of Bruit's method, Acta Sci. Math. (Szeged) 13(1949)
57-63.
[146] (with A. Renyi) Some problems and results on consecutive primes, Simon
Stevin 27(1950)115-126.
706 BIBLIOGRAPHY OF PAUL ERDOS

[147] On the number of terms of the square of a polynomial, Nieuw Arch. Wisk. (2)
23(1949)63-65.
[148] On a new method in elementary number theory which leads to an elementary proof
of the prime number theorem, Proc. Nat. Acad. Sci. U.S.A. 35(1949)374-
384.
[149] On a Tauberian theorem connected with the new proof of the prime number
theorem, J. Indian Math. Soc. 13(1949)131-147.
[150] Supplementary note, J. Indian Math. Soc. 13(1949)145-147.
[151] On the converse of Fermat's theorem, Amer. Math. Monthly 56(1949)623-
624.
[152] On a tlworem of Shu and Robbins, Ann. Math. Statist. 20(1949)286-291.
[153] On the strong law of large numbers, Trans. Amer. Math. Soc. 67(1949)
51-56.
[154] Problems and results on the differences of consecutive primes, Publ. Math.
Debrecen 1(1949)33-37.
[155] (with N. G. de Bruijn) Sequences of points on a circle, Nederl. Akad.
Wetensch. Proc. 52(1949)14-17.
[156] (with J. F. Koksma) On the uniform distribution modulo 1 of lacunary
sequences, Nederl. Akad. Wetensch. Proc. 52(1949)264-273.
[157] (with J. F. Koksma) On the uniform distribution modulo 1 of sequences
(f(n, 6)), Nederl. Akad. Wetensch. Proc. 52(1949)851-854.
[158] On the coefficients of the cyclotomic polynomial, Portugal. Math. 8(1949)
63-71.
1950
[159] (with P. Turan) On the distribution of roots of polynomials, Ann. of Math.
51(1950)105-119.
* [160] (with R. Rado) A combinatorial theorem, J. London Math. Soc. 25(1950)
249-255.
[161] On almost primes, Amer. Math. Monthly 57(1950)404-407.
[162] Some theorems and remarks on interpolation, Acta Sci. Math. (Szeged) 12
(1950)11-17.
707 BIBLIOGRAPHY OF PAUL ERDOS

[163] (with A. Dvoretzky and S. Kakutani) Double points of paths of Brownian
motion in n-space, Acta Sci. Math. (Szeged) 12(1950)75-81.
[164] (with P. T. Bateman and S. Chowla) Remarks on the size of L{\, x), Publ.
Math. Debrecen 1(1950)165-182.
[165] Some remarks on set theory, II, Proc. Araer. Math. Soc. 1(1950)127-141.
[166] (with G. Piranian) Convergence fields of row-finite and row-infinite Toeplitz
transformations, Proc. Amer. Math. Soc. 1(1950)397-401.
[167] On a problem in elementary number theory, Math. Student 17(1950)32-33.
11 1
[168] Az— -\- 1---+ — = T egyenlet egesz szdmu megolddsairol, Mat. Lapok
X\ x2 xn b
1(1950)192-210.
[169] On integers of the form 2k -\- p and some related problems, Summa Brasil.
Math. 11(1950)113-123.
[170] (with F. Herzog and G. Piranian) Schlicht gap series whose convergence on
the unit circle is uniform but not absolute, Proc. Internat. Congr. Math.
1(1950)1.
1951
[171] (with K. L. Chung) Probability limit theorems assuming only the first moment,
I, Mem. Amer. Math. Soc. 6(1951)1-19.
[172] (with A. Dvoretzky) Some remarks on random walk in space, Proceedings of
the Second Berkeley Symposium on Mathematical Statistics and
Probability, 1950, Univ. of California Press, Berkeley and Los Angeles, 1951,
353-367.
[173] On a diophantine equation, J. London Math. Soc. 26(1951)176-178.
[174] On a conjecture of Klee, Amer. Math. Monthly 58(1951)98-101.
* [175] (with P. Bateman) Geometrical extrema suggested by a lemma of Besicovitch,
Amer. Math. Monthly 58(1951)306-314.
[176] (with N. G. de Bruijn) A colour problem for infinite graphs and a problem in
the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A 54(1951)
371-373.
708 BIBLIOGRAPHY OF PAUL ERDOS

177] (with N. G. de Bruijn) Some linear and some quadratic recursion formulas, I,
Nederl. Akad. Wetensch. Proc. Ser. A 54(1951)374-382.
178] (with S. Chowla) A theorem on the distribution of the values of L-functions,
J. Indian Math. Soc. 15(1951)11-18.
179] (with H. Davenport) On sequences of positive integers, J. Indian Math. Soc.
15(1951)19-24.
180] Some problems and results in elementary number theory, Publ. Math. Debrecen
2(1951)103-109.
181] On a theorem of Radstrom, Proc. Amer. Math. Soc. 2(1951)205-206.
182] (with F. Herzog and G. Piranian) Schlicht Taylor series whose convergence
on the unit circle is uniform but not absolute, Pacific J. Math. 1(1951)75-82.
183] On some problems of Bellman and a theorem of Romanoff, J. Chinese Math.
Soc. (N.S) 1(1951)409-421.
184] (with L. Mirsky) The distribution of values of the divisor function d(n), Proc.
London Math. Soc. 3(1951)257-271.
185] (with R. Rado) Combinatorial theorems on classifications of subsets of a given
set, Proc. London Math. Soc. 3(1951)417-439.
186] (with H. N. Shapiro) On the changes of sign of a certain error function,
Canad. J. Math. 3(1951)375-384.
952
187] On the sum ~t d(j(k)), J. London Math. Soc. 27(1952)7-15.
/t = i
X
188] On the greatest prime factor of II f(k), J. London Math. Soc. 27(1952)
379-384.
189] (with N. G. de Bruijn) Some linear and some quadratic recursion formulas, 11,
Indag. Math. 14(1952)152-163.
190] A theorem on the Riemann integral, Nederl. Akad. Wetensch. Proc. Ser. A
55(1952)142-144.
191] (with K. L. Chung) On the application of the Borel-Cantelli lemma, Trans.
Amer. Math. Soc. 72(1952)179-186.
709 BIBLIOGRAPHY OF PAUL ERDOS

[192] (with H. Davenport) The distribution of the quadratic and higher residues,
Publ. Math. Debrecen 2(1952)252-265.
[193] Egy kongruenciarendszerekrbl szolo problemdrol, Mat. Lapok 3(1952)122-128.
[194] On the uniform but not absolute convergence of power series with gaps, Ann.
Soc. Polon. Math. 25(1952)162-168.
[195] (with H. Davenport) Xote on normal decimals, Canad. J. Math. 4(1952)
58-63.
[196] On a Tauberian theorem for Euler summability, Acad. Serbe Sci. Publ. Inst.
Math. 4(1952)51-56.
1953
[197] On a conjecture of Hammer si ey, J. London Math. Soc. 28(1953)232-236.
[198] (with C. A. Rogers) The covering of n-dimensional space by spheres, J.
London Math. Soc. 28(1953)287-293.
[199] Arithmetical properties of polynomials, J. London Math. Soc. 28(1953)416—
425.
[200] (with R. Rado) A problem on ordered sets, J. London Math. Soc. 28(1953)
426-438.
[201] (with G. A. Hunt) Changes of sign of sums of random variables. Pacific J.
Math. 3(1953)637-687.
[202] (with E. G. Straus) On linear independence of sequences in a Banach-space,
Pacific J. Math. 3(1953)689-698.
[203] (with N. H. Shapiro) On the least primitive root of a prime, Pacific J. Math.
7(1953)861-865.
[204] (N. G. de Bruijn) On a recursion formula and on some Tauberian theorems,
J. Res. Nat. Bur. Standards 50(1953)161-164.
[205] Some remarks on set theory, III, Michigan Math. J. 2(1953)51-57.
[206] Some remarks on set theory, IV, Michigan Math. J. 2(1953)169-173.
[207] (with F. Bagemihl and W. Seidel) Sur quelques proprietes frontihes des
fonctions holomorphes definies par certains produits dans le cercle-urat'e, Ann.
Sci. Ecole Norm. Sup. Paris (1953)135-147.
710 BIBLIOGRAPHY OF PAUL ERDOS

1954
[208
[209
[210
[211
[212
[213
[214
[215
[216
1955
[217
[218
[219;
[220
[221
[222
(with I. Niven) The number of multinomial coefficients, Amer. Math.
Monthly 61(1954)37-39.
(with A. Dvoretzky and S. Kakutani) Multiple points of paths of Brownian
motion in the plane, Bull. Res. Counc. Israel 3(1954)364-371.
(with N. C. Ankeny) The insolubility of classes of diophantine equations,
Amer. J. of Math. 76(1954)488-496.
On a problem of Sidon in additive number theory, Acta Sci. Math. (Szeged)
15(1954)255-259.
(with A. J. Maclntyre) Integral functions with gap power series, Proc.
Edinburgh Math. Soc. 10(1954)62-70.
Some results on additive number theory, Proc. Amer. Math. Soc. 5(1954)
847-853.
(with F. Bagemihl) Rearrangements of C\ summable series, Acta Math. 92
(1954)35-53.
(with F. Bagemihl) Intersections of prescribed power, type or measure, Fund.
Math. 41(1954)57-67.
(with F. Herzog and G. Piranian) Sets of divergence of Taylor series and of
trigonometric series, Math. Scand. 2(1954)262-266.
(with L. Gillman and M. Henriksen) An isomorphism theorem for real-
closed fields, Ann. of Math. 61(1955)542-554.
On the product of consecutive integers, III, Nederl. Akad. Wetensch. Proc.
Ser. A 58(1955)85-90.
(with I. S. Gal) On the law of the iterated logarithm, I, Nederl. Akad.
Wetensch. Proc. Ser. A 58(1955)65-76.
(with I. S. Gal) On the law of the iterated logarithm, II, Nederl. Akad.
Wetensch. Proc. Ser. A. 58(1955)77-84.
(with M. Golomb) Functions which are symmetric about several points, Nieuw
Arch. Wisk. 3(1955)13-19.
On consecutive integers, Nieuw Arch. Wisk. 3(1955)124-128.
711 BIBLIOGRAPHY OF PAUL ERDOS

[223] (with J. C. Oxtoby) Partitions of the plane into sets having positive measure
in every non-null measurable product set, Trans. Amer. Math. Soc. 79(1955)
91-102.
[224] On amicable numbers, Publ. Math. Debrecen 4(1955)108-111.
[225] (with P. Turan) On the role of the Lebesgue functions in the theory of the
Lagrange interpolation, Acta Math. 6(1955)47-66.
[226] Uber die Anzahl der Lbsungen von [p — 1, q — 1] f^ x, Monatsh. Math.
59(1955)318-319.
[227] (with F. Herzog and G. Piranian) Polynomials whose zeros lie on the unit
circle, Duke Math. J. 22(1955)347-362.
[228] Problems and results in additive number theory, Colloque sur la Theorie des
Nombres, Bruxelles, George Thone, Liege; Masson and Cie, Paris,
1955, 127-137.
[229] Some remarks on number theory, Riveon Lematematika 9(1955)45-48.
[230] Some theorems on graphs, Hebrew Univ. Jerusalem 10(1955)13-16.
[231] (with A. Dvoretzky) On power series diverging everywhere on the circle of
convergence, Michigan Math. J. 3(1955-56)31-35.
1956
[232] (with A. C. Offord) On the number of real roots of a random algebraic equation,
Proc. London Math. Soc. 6(1956)139-160.
[233] (with D. A. Darling) A limit theorem for the maximum of normalized sums of
independent random variables, Duke Math. J. 23(1956)143-156.
[234] (with P. T. Bateman) Monotonicity of partition functions, Mathematica
3(1956)1-14.
[235] (with W. H. J. Fuchs) On a problem of additive number theory, J. London
Math. Soc. 31(1956)67-73.
[236] On perfect and multiply perfect numbers, Ann. Mat. Pura Appl. 42(1956)
253-258.
[237] (with L. Fejes Toth) Pontok elhelyezese egy tartomdnyban, Magyar Tud.
Akad. Mat. Fiz. Oszt. Kozl. 6(1956)185-191.
[238] Megjegyzesek Kbvdry Tamds egy dolgozatdhoz, Mat. Lapok 7(1956)214-217.
712 BIBLIOGRAPHY OF PAUL ERDOS

[239] Megjegyzesek a Matematikai Lapok kit feladatdhoz, Mat. Lapok 7(1956)
10-17.
[240] Uber arithmetische Eigenschaften der Substitutionswerte eines Polynoms fur
ganzzahlige Werte des Arguments, Acad, la Roumaine 1(1956)189-194.
[241] (with P. T. Bateman) Partitions into primes, Publ. Math. Debrecen 4
(1956)198-200.
[242] On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4(1956)
201-206.
* [243] (with A. Renyi) On some combinatorial problems, Publ. Math. Debrecen
4(1956)398-405.
[244] (with A. Renyi) On the number of zeros of successive derivatives of analytic
functions, Acta Math. Acad. Sci. Hungar. 7(1956)125-144.
[245] (with R. Rado) A partition calculus in set theory, Bull. Amer. Math. Soc.
62(1956)427-489.
[246] (with G. Fodor) Some remarks on set theory, V, Acta Sci. Math. (Szeged)
17(1956)250-260.
[247] (with J. Karamata) Sur la majorabilite C des suites de nombres reels, Publ.
Inst. Math. (Beograd) (1956)37-52.
1957
[248] (with A. Dvoretzky, S. Kakutani, and S. J. Taylor) Triple points of
Brownian paths in 3-space, Proc. Cambridge Philos. Soc. 53(1957)856-
862.
[249] On the irrationality of certain series, Nederl. Akad. Wetensch. 60(1957)
212-219.
[250] Nehdny geometriai problemdrol, Mat. Lapok 8(1957)86-92.
[251] Some unsolved problems, Michigan Math. J. 4(1957)291-300.
[252] On a high-indices theorem in Borel summability, Acta Math. Acad. Sci.
Hungar. 7(1957)265-281.
[253] (with T. Kovari) On the maximum modulus of entire functions, Acta Math.
Aj-arJ S.H Hunrar 7^19^7^/^-^17
713 BIBLIOGRAPHY OF PAUL ERDOS

[254] (with A. Renyi) On the number of zeros of successive derivatives of entire
functions of finite order, Acta Math. Acad. Sci. Hungar. 8(1957)223-225.
[255] (with S. Marcus) Sur la decomposition de Pespace Euclidien en ensembles
homogener, Acta Math. Acad. Sci. Hungar. 8(1957)443-452.
[256] Uber eine Art von Lakunaritat, Colloq. Math. 4(1957)6-7.
[257] (with S. Kakutani) On a perfect set, Colloq. Math. 4(1957)195-196.
[258] Einige Bemerkungen zur Arbeit von A. Stbhr: "Gelbste und ungelbste Fragen
liber Basen der natiirlichen Zahlenreihe," J. Reine Angew. Math. Berlin
197(1957)216-219.
[259] (with A. Renyi) A probabilistic approach to problems of diophantine
approximation, Illinois J. Math. 1(1957)303-315.
[260] (with S. J. Taylor) On the set of points of convergence of a lacunary
trigonometric series and the equidistribution properties of related sequences, Proc.
London Math. Soc. 7(1957)598-615.
[261] (with G. Fodor) Some remarks on set theory, VI, Acta Sci. Math. (Szeged)
18(1957)243-260.
[262] Uber eine Fragestellung von Gaier und Meyer-Kbnig. Jber. Deutsch. Math.-
Verein. 60(1957)89-92.
[263] On the growth of the cyclotonic polynomial in the interval (0,1), Proc. Glasgow
Math. Assoc. 3(1957)102-104.
[264] Remarks on a theorem of Ramsey, Bull. Res. Council Israel 7(1957)21-24.
1958
[265] (with G. Piranian) The topologization of a sequence space by Toeplitz
matrices, Michigan Math. J. 5(1958)139-148.
[266] (with I. Vincze) Konve.x, zdrt sikgorbek megkozeliteserol, Mat. Lapok 9
(1958)19-36.
[267] Solution of two problems of Jankowska, Bull. Acad. Polon. Sci. Ser. Sci.
Math. Astr. Phys. 6(1958)545-547.
[268] (with A. Renyi and P. Sziisz) On Engel's and Sylvester's series, Ann. Univ.
Sci. Budapest. Eotvos Sect. Math. 1(1958)7-32.
714 BIBLIOGRAPHY OF PAUL ERDOS

[269] (with F. Herzog and G. Piranian) Metric properties of polynomials, J.
Analyse Math. 6(1958)125-148.
[270] On the distribution function of additive arithmetical functions and on some related
problems, Rend. Sem. Mat. Fis. Milano 27(1958)3-7.
[271] Sur certaines series h valeur irrationnelle, Enseignement Math. 4(1958)
93-100.
[272] On an elementary problem in number theory, Canad. Math. Bull. 1(1958)5—8.
[273] Some remarks on a paper of McCarthy, Canad. Math. Bull. 1(1958)71-75.
[274] Asymptotic formulas for some arithmetical functions, Canad. Math. Bull. 1
(1958)149-153.
[275] Concerning approximation with nodes, Colloq. Math. 6(1958)25-27.
[276] (with P. Sziisz and P. Turan) Remarks on the theory of diophantine
approximation, Colloq. Math. 6(1958)119-126.
[277] (with A. Hajnal) On the structure of set-mappings, Acta Math. Acad. Sci.
Hungar. 9(1958)111-131.
[278] Problems and results on the theory of interpolation, I, Acta Math. Acad. Sci.
Hungar. 9(1958)381-388.
[279] (with E. Jabotinsky) On sequences of integers generated by a sieving process,
I, Nederl. Akad. Wetensch. 61(1958)115-123.
[280] (with E. Jabotinsky) On sequences of integers generated by a sieving process,
II, Nederl. Akad. Wetensch. 61(1958)124-128.
[281] Some remarks on Euler's <p-function, Acta Arith. 4(1958)10-19.
[282] Remarks on number theory, I. On primitive <x-abundant numbers, Acta Arith.
5(1958)25-33.
[283] (with G. G. Lorentz) On the probability that n and g(n) are relatively prime,
Acta Arith. 5(1958)35-44.
[284] (with P. Scherk) On a question of additive number theory, Acta Arith. 5
(1958)45-55.
[285] (with K. Urbanik) On sets which are measured by multiples of irrational
numbers, Bull. Acad. Polon. Ser. Sci. Math. Astr. Phys. 6(1958)743-748.
715 BIBLIOGRAPHY OF PAUL ERDOS

[286] Uti elmenyek Moszkva-Peking-Singapore, Magyar Tudomany 1961, 193-
197.
[287] (with A. Dvoretzky and S. Kakutani) Points of multiplicity c of plane
Brownian paths, Bull. Res. Counc. Israel 7(1958)175-180.
[288] Elbadokbruton Kanaddban, Magyar Tudomany 8-9(1958)335-341.
1959
[289] (with R. Rado) Partition relations connected with the chromatic number of
graphs, J. London Math. Soc. 34(1959)63-72.
[290] (with R. Rado) A theorem on partial well-ordering of sets of vectors, J. London
Math. Soc. 34(1959)222-224.
[291] Remarks on number theory, II, Acta Arith. 4(1959)171-177.
[292] Some results on diophantine approximation, Acta Arith. 5(1959)359-369.
[293] (with G. Fodor and A. Hajnal) On the structure of inner set mappings, Acta
Sci. Math. (Szeged) 1(1959)81-90.
[294] (with J. Suranyi) Megjegyzesek egy versenyfeladathoz, Mat. Lapok 10(1959)
39-48.
[295] (with Y. N. Dowker) Some examples in ergodic theory, Proc. London Math.
Soc. 9(1959)39-48.
*[296] Graph theory and probability, Canad. J. Math. 11(1959)34-38.
[297] Some remarks on prime factors of integers, Canad. J. Math. 11(1959)161-167.
[298] (with G. Piranian) Sequences of linear fractional transforms, Michigan Math.
J. 6(1959)205-209.
[299] (with A. Dvoretzky) Divergence of random power series, Michigan Math.
J. 6(1959)343-347.
[300] (with A. Renyi) Some further statistical properties of the digits in Cantor's
series, Acta Math. Acad. Sci. Hungar. 10(1959)21-29.
* [301] (with T. G-allai) On the maximal paths and circuits of graphs, Acta Math.
Acad. Sci. Hungar. 10(1959)337-357.
[302] (with Gy. Szekeres) On the product £(1- za'), Publ. Inst. Math.
(Beograd) 13(1959)29-34.
716 BIBLIOGRAPHY OF PAUL ERDOS

[303] (with J. H. H. Chalk) On the distribution of primitive lattice points in the
plane, Canad. Math. Bull. 2(1959)91-96.
[304] (with A. Renyi) On Cantor's series with convergent £ l/<7>>> Ann. Univ. Sci.
Budapest R. Eotvos Sect. Math. 2(1959)93-109.
[305] (with A. Renyi) On singular radii of power series, Magyar Tud. Akad.
Mat. Kut. Int. Kozl. 3(1959)159-169.
[306] (with A. Renyi) On the central limit theorem for samples from a finite
population, Magyar Tud. Akad. Mat. Kut. Int. Kozl. 4(1959)49-61.
[307] Uber einige Probleme der additiven Zahlentheorie, Akademie-Verlag, Berlin,
1959, 116-119.
[308] (with K. L. Chung and T. Sirao) On the Lipschitz's condition for Brownian
motion, J. Math. Soc. Japan 11(1959)263-274.
[309] (with A. Renyi) On random graphs, I, Publ. Math. Debrecen 6(1959)
290-297.
[310] (with J. Suranyi) Egy additiv szdmelmeleti problema, Mat. Lapok 10(1959)
286-290.
[311] A remark on the iteration of entire functions, Reveon Lematematika 13(1959)
13-16.
1960
[312] (with R. Rado) Intersection theorems for systems of sets, J. London Math.
Soc. 35(1960)85-90.
[313] (with S. J. Taylor) Some problems concerning the structure of random walk
paths, Acta Math. Acad. Sci. Hungar. 11(1960)137-162.
[314] (with S. Chowla and E. G. Straus) On the maximal number of pairuise
orthogonal Latin squares of a given order, Canad. J. Math. 12(1960)204-208.
[315] On an asymptotic formula in number theory, Vestnik Leningrad Univ. 15
(1960)41-49.
[316] (with A. Renyi) On the evolution of random graphs. 32. Session of the ISI,
Tokyo, 1960, 1-15.
[317] (with A. Renyi) Additive properties of random sequences of positive integers,
Acta Arith. 6(1960)83-110.
717 BIBLIOGRAPHY OF PAUL ERDOS

[318
[319;
[320
[321
[322
[323
:[324
:[325
[326
[327
[328
[329;
[330
[331
[332
1961
About an estimation of Zahorsky, Colloq. Math. 7(1960)167-170.
On random interpolation, J. Austral. Math. Soc. 1(1960).
Remarks and corrections to my paper "Some remarks on a paper of McCarthy,"
Canad. Math. Bull. 3(1960)127-130.
Remarks on number theory, III. On addition chains, Acta Arith. 6(1960)
77-81.
(with J. Suranyi) Vdlogatott fejezetek a szdmelmelel kbrebol, Tankonyvkiado,
Vallalat, Budapest, 1960. 250 pp.
(with A. Hajnal) Some remarks of set theory, VIII, Michigan Math. J.
7(1960)187-191.
(with A. Renyi) On the evolution of random graphs, Magyar Tud. Akad.
Mat. Kut. Int. Kozl. 5(1960)17-61.
On sets of distances of n points in Euclidean space, Magyar Tud. Akad. Mat.
Kut. Int. Kozl. 5(1960)165-169.
(with R. Rado) A construction of graphs without triangles having pre-assigned
order and chromatic numbers, J. London Math. Soc. 35(1960)445-448.
(with A. Hajnal) Some remarks on set theory, VII, Acta Sci. Math. (Szeged)
21(1960)154-163.
Uber die kleinste quadratfreie Zahl einer arithmetischen Reihe, Monatsh. Math.
64(1960)314-315.
(with C. A. Rogers and S. J. Taylor) Scales of functions, J. Austral. Math.
Soc. 1(1960)396-418.
(with S. J. Taylor) Some intersection properties of random walk, Acta Math.
Acad. Sci. Hungar. 11(1960)231-248.
Megjegyzesek a Matematikai Lapok ket problemdjahoz, Mat. Lapok 11(1960)
26-32.
(withG. Piranian) Restricted cluster sets, Math. Nachr. 22(1960)155-158.
[333] (with E. Specker) On a theorem in the theory of relations and a solution of a
problem of Knaster, Colloq. Math. 8(1961).
718 BIBLIOGRAPHY OF PAUL ERDOS

[334] (with A. Hajnal) On a property of families of sets, Acta Math. Acad. Sci.
Hungar. 12(1961)87-123.
[335] (with P. Turan) An extremal problem in the theory of interpolation, Acta
Math. Acad. Sci. Hungar. 12(1961)221-234.
[336] Problems and results in the theory of interpolation, II, Acta Math. Acad. Sci.
Hungar. 12(1961)235-244.
* [337] (with A. Renyi) On the strength of connectedness of a random graph, Acta
Math. Acad. Sci. Hungar. 12(1961)261-267.
[338] Uber einige Probletne der additiven Zahlentheorie, J. Reine Angew. Math.
1-2(1961)61-66.
* [339] Graph theory and probability, II, Canad. J. Math. 13(1961)346-352.
[340] On a problem of G. Golomb, J. Austral. Math. Soc. 2(1961)1-8.
[341] A problem about prime numbers and the random walks, II, Illinois J. Math.
5(1961)342-353.
[342] (with T. Gallai) Grdfok eloirt foku pontokkal, Mat. Lapok 11.4(1960)
264-274.
[343] (with A. Schinzel) Distributions of the values of some arithmetical functions,
Acta Arith. 6(1961)473-485.
* [344] (with T. Gallai) On the minimal number of vertices representing the edges of a
graph, Magyar Tud. Akad. Mat. Kut. Int. Kozl. 6(1961)181-204.
[345] (with A. Renyi) On a classical problem of probability theory, Magyar Tud.
Akad. Mat. Kut. Int. Kozl. 6(1961)215-220.
*[346] Some unsolved problems, Magyar Tud. Akad. Mat. Kut. Int. Kozl. 6(1961)
221-254.
[347] (with A. Jabotinsky) On analytic iteration, J. Analyse Math. 8(1 "61)
361-376.
[348] (with F. Herzog and G. Piranian) On Taylor series of functions regular in
Gaier regions, Arch. Math. (Basel) 5(1954)39-51.
[349] On additive arithmetical functions and applications of probability to number
theory, Proc. Internat. Congress Math. Amsterdam, 1956, 1961, 13-19.
719 BIBLIOGRAPHY OF PAUL ERDOS

[350] Szdmelmeleti megjegyzesek, I, Mat. Lapok 12(1961)10-17.
[351] (with A. Ginsburg and A. Ziv) Theorem in the additive number theory, Bull.
Res. Counc. Israel 10(1961).
[352] (with S. J. Taylor) On the Hausdorff measure of Brownian paths in the plane,
Proc. Cambridge Philos. Soc. 57(1961)209-222.
[353] (with A. Tarski) On some problems involving inaccessible cardinals, Essays on
the Foundations of Mathematics, Magnes Press, Hebrew Univ.,
Jerusalem 1961, 50-82.
1962
[354] (with A. Dvoretsky and S. Kakutani) Nonincrease everywhere of the Brown-
ian motion process, Proceedings of the Fourth Berkeley Symp. Math.
Statist, and Prob., Vol. II, Univ. of California Press, Berkeley (1962)
103-116.
[355] Szdmelmeleti megjegyzesek, II. Az Euler-fele 4>-fiiggveny nehdny tulajdonsdgdrol,
Mat. Lapok 12(1961)161-169.
[356] (with Gy. Szekeres) On some extremum problems in elementary geometry, Ann.
Univ. Sci. Budapest. Eotvos Sect. Math. 3-4(1961)53-62.
* [357] (with Chao Ko and R. Rado) Intersection theorems for systems of finite sets,
Quart. J. Math. Oxford Ser. 12(1961)313-320.
[358] (with H. Hanani) On d summability of series, Michigan Math. J. 9(1962)
1-14.
[359] Representation of real numbers as sums and products of Liouville numbers,
Michigan Math. J. 9(1962)59-60.
* [360] On a theorem of Rademacher-Turdn, Illinois J. Math. 6(1962)122-127.
[361] Nehdny elemi geometriai problemdrol. Kozepisk, Mat. Lapok 24.5(1962)
193-201.
[362] (with C. A. Rogers) Covering space with convex bodies, Acta Arith. 7(1962)
281-285.
[363] (with A. Hajnal) Some remarks concerning our paper "On the structure of set
mappings and Non-existence of two-valued a-measure for the first uncountable
inaccessible cardinal, Acta Math. Acad. Sci. Hungar. 13(1962)223-226.
720 BIBLIOGRAPHY OF PAUL ERDOS

[364] Szdmelmeleti megjegyzesek, III. Nehdny additiu szdmelmeleti problemdrol, Mat.
Lapok 13(1962)28-38.
[365] (with B. Bollobas) Grdfelmeleti szelsoertekekre vonatkozo problemdkrol, Mat.
Lapok 13(1962)193-201.
[366] Vber ein Extretnalproblem in der Graphentheorie, Arch. Math. (Basel) 13
(1962)222-227.
[367] (with L. Posa) On the maximal number of disjoint circuits of a graph, Publ.
Math. Debrecen 9(1962)3-12.
[368] (with L. Kaplansky) Sequences of plus and minus, Scripta Math. 12(1962)
73-75.
[369] (with N. H. Shapiro) The existence of a distribution function for an error term
to the Euler function, Canad. J. Math. 7(1962)63-76.
[370] On a probletn of Sidon in additive number theory and on some related questions,
J. London Math. Soc. 19(1944)208.
[371] (with K. Prachar) Sdtze und Probleme uber pk/k, Abh. Math. Sem. Univ.
Hamburg 26(1962)51-56.
[372] Applications of probability to combinatorial problems, Colloq. on
combinatorial problems in probability theory, Aarhus, 1962, 90-92.
[373] Beantwortung einer Frage von Teufel, Elem. Math. 1962.
[374] On the topological product of discrete \-compact spaces, Proc. Symp. Topol.
Prague, 1962, 148-151.
[375] On the integers relatively prime to n and on a number theoretic function considered
by Jacobsthal, Math. Scand. 10(1962)163-170.
[376] (with C. A. Rogers) The construction of certain graphs, Canad. J. Math.
1962.
[377] On trigonometric sums with graphs, Magyar Tud. Akad. Mat. Kut. Int.
Kozl. 7(1962)37-42.
[378] Remarks on a paper of Posa, Magyar Tud. Akad. Mat. Kut. Int. Kozl.
7(1962)441-457.
[379] (with A. Hajnal) On a classification of denumerable order types and an
application to the partition calculus, Fund. Math. 5(1962)117-129.
721 BIBLIOGRAPHY OF PAUL ERDOS

[380] On the representation of large integers as sums of distinct summands taken from
a fixed set, Acta Arith. 7(1962)345-354.
[381] An inequality for the maximum of trigonometric polynomials, Ann. Polon.
Math. 12(1962)151-154.
[382] (with A. Renyi) On a problem of A. Zygmund, Studies in Mathematical
Analysis and Related Topics, Stanford Univ. Press, Stanford, Calif.,
1962, 110-116.
[383] Theone dei numeri. On a problem of Sierpinski, Atti. Accad. Naz. Lincei
Rend. CI. Sci. Fis. Mat. Nat. 33(1962)122-124.
[384] Szdmehneleti megjegyzesek, IV. Extremdlis problemdk a szdnielmeletben, Mat.
Lapok 13(1962)228-255.
1963
[385] Some remarks on the functions <p and a, Bull. Acad. Polon. Sci. 10(1962)
617-619.
* [386] On circuits and subgraphs of chromatic graphs, Mathematika 9(1962)170-
175.
[387] (with S. Stein) Sums of distinct unit fractions, Proc. Amer. Math. Soc.
14(1963)126-131.
[388] (with J. Czipszer and A. Hajnal) Some extremal problems on infinite graphs,
Magyar Tud. Akad. Mat. Kut. Int. Kozl. 7(1962)441-457.
* [389] On the number of complete subgraphs contained in certain graphs, Magyar Tud.
Akad. Mat. Kut. Int. Kozl. 7(1962)459-474.
*[390] On a combinatorial problem, I, Nordisk Mat. Tidskr. 11(1963)5-10.
[391] (with A. Renyi) Remarks on a problem of Obreanu, Canad. Math. Bull.
6(1963)267-273.
* [392] (with G. Dirac) On the maximal number of independent circuits in a graph,
Acta Math. Acad. Sci. Hungar. 14(1963)79-94.
[393] On a problem of Sidon in additive number theory and on some related problems.
[394] (with S. J. Taylor) The Hausdorff measure of the intersection of sets of positive
Lebesgue measure, Mathematika 1(1963)1-9.
722 BIBLIOGRAPHY OF PAUL ERDOS

[395] (with A. Renyi) Egy grafelmeleti pr obl'emdr 61, Magyar Tud. Akad. Mat.
Kut. Int. Kozl. 7(1963)623-641.
[396] Quelques problemes de la th'eorie des nombres, Monographies de l'Enseigne-
ment Mathematique, No. 6, L'Enseignement Mathematique,
University, Geneva, (1963)81-135.
[397] (with H. Davenport and N. J. LeVeque) On WeyPs criterion for uniform
distribution, Michigan Math. J. 10(1963)311-314.
[398] Regulare Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z.
Martin-Luther Univ. Halle-Wittenberg Math.-Natur. Reihe 12(1963)
251-257.
* [399] On a problem in graph theory, Math. Gaz. 47(1963)220-223.
[400] On some properties of Hamel basis, Colloq. Math. 10(1963)267-269.
* [401] (with A. Renyi) Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14
(1963)295-315.
[402] Ramsey es Van der Waerden tetel'evel kapcsolatos kombinatorikai kerd'esekrbl,
Mat. Lapok 14(1963)29-37.
[403] (with H. Kestelman and C A. Rogers) An intersection property of sets with
positive measure, Colloq. Math. 10(1963)75-80.
[404] (with H. Davenport) A theorem on uniform distribution, Magyar Tud.
Akad. Mat. Kut. Int. Kozl. 8(1963)3-11.
* [405] (with A. Renyi) On two problems of information theory, Magyar Tud. Akad.
Mat. Kut, Int. Kozl. 8(1963)229-249.
* [406] On the structure of linear graphs, Israel J. Math. 1(1963)156-160.
1964
* [407] (with J. W- Moon) On subgraphs of the complete bipartite graph, Canad.
Math. Bull. 7(1964)35-39.
* [408] On the number of triangles contained in certain graphs, Canad. Math. Bull.
7(1964)53-56.
[409] (with G. Piranian) Laconicity and redundancy of Toeplitz matrices, Math. Z.
83(1964)381-394.
723 BIBLIOGRAPHY OF PAUL ERDOS

* [410] (with P. Kelly) The minimal regular graph containing a given graph, Amer.
Math. Monthly 70(1963)1961-1962.
[411] An interpolation problem associated with the continuum hypothesis, Michigan
Math. J. 11(1964)9-10.
* [412] (with H. Hanani) On a limit theorem in combinatorial analysis, Publ. Math.
Debrecen 10(1963).
* [413] (with A. Ginzburg) On a combinatorial problem in Latin squares, Magyar
Tud. Acad. Mat. Kut. Int. Kozl. 8(1964)407-411.
* [414] (with A. Renyi) On random matrices, Magyar Tud. Akad. Mat. Kut. Int.
Kozl. 8(1964)455-461.
[415] (with F. Bagemihl) A problem concerning the zeros of a certain kind of holo-
morphic functions in the unit disk, J. Reine Angew. Math. 214(1964)340-
344.
[416] (with H. Heilbronn) On the addition of residue classes mod p, Acta Arith.
9(1964)149-159.
[417] (with Gordon, A. Rubel, and G. Straus) Tauberian theorems for sum sets,
Acta Arith. 9(1964)177-189.
* [418] Extremal problems in graph theory, Theory of Graphs and Its Applications:
Proceedings of the Symposium on the Theory of Graphs and Its
Applications, held in Smolenice, Czechoslovakia, June, 1963, Publ. House
Czechoslovak Acad. Sci., Prague, 1964, 29-36.
* [419] Some applications of probability to graph theory and combinatorial problems,
Theory of Graphs and Its Applications'. Proceedings of the Symposium
on the Theory of Graphs and Its Applications, held in Smolenice,
Czechoslovakia, June, 1963, Publ. House Czechoslovak Acad. Sci.,
Prague, 1964, 137-141.
[420] (with T. Gallai) Solution of a problem of Dirac, Theory of Graphs and Its
Applications: Proceedings of the Symposium on the Theory of Graphs
and Its Applications, held in Smolenice, Czechoslovakia, June, 1963,
Publ. House Czechoslovak Acad. Sci., Prague, 1964, 167-168.
[421] (with J. Neveu and A. Renyi) An elementary inequality between the
probabilities of events, Math. Scand. 13(1964)99-104.
724 BIBLIOGRAPHY OF PAUL ERDOS

[422] (with A. Hajnal) Some remarks on set theory, IX. Combinatorial problems in
measure theory and set theory, Michigan Math. J. 11(1964)107-127.
[423] (with C. A. Rogers) The star of coverings of space with convex bodies, Acta
Anth. 9(1964)41-45.
[424] (with L. Moser) A problem on tournaments, Canad. Math. Bull. 7(1964)
351-356.
[425] On some applications of probability to analysis and number theory, J. London
Math. Soc. 39(1964)692-696.
[426] (with L. Moser) On the representation of directed graphs as unions of orderings,
Magyar Tud. Akad. Mat. Kut. Int. Kozl. 9(1964)125-132.
[427] On a problem in elementary number theory and combinatorial problems, Math.
Comp. 18(1964)644-646.
[428] Some problems on the distribution of prime numbers, Math. Congr. Varenna,
1964, 1-8.
[429] Problems and results of diophantine approximations, Compositio Math. 16
(1964)52-65.
[430] On a combinatorial problem, II, Acta Math. Acad. Sci. Hungar. 15(1964)
445-447.
[431] (with L. Few and C. A. Rogers) The amount of overlapping in partial
covering of space by equal spheres, Mathematika 11(1964)171-184.
[432] On some divisibility properties of ( ), Canad. Math. Bull. 7(1964)513-
518. W
[433] Some remarks on Ramsey's theorem, Canad. Math. Bull. 7(1964)619-622.
: [434] (with A. Hajnal and J. W- Moon) A problem in graph theory, Amer. Math.
Monthly 71(1964)1107-1110.
[435] On two problems of S. Marcus concerning functions with the Darboux property,
Rev. Roumaine Math. Pures Appl. 9(1964)803-804.
[436] (with E. Ingham) Arithmetical Tauberian theorem, Acta Arith. 9(1964)
341-356.
725 BIBLIOGRAPHY OF PAUL ERDOS

* [437] (with A. Hajnal) On complete topological subgraphs of certain graphs, Ann.
Univ. Sci. Budapest. Eotvos Sect. Math. 7(1964)143-149.
[438] (with E. G. Straus) On the irrationality of certain Ahmes series, J. Indian
Math. Soc. 28(1964)130-133.
1965
[439] (with A. Renyi) Probabilistic methods in group theory, J. Analyse Math.
4(1965)127-138.
* [440] (with L. Posa) On independent circuits contained in a graph, Canad. J. Math.
17(1965)347-352.
[441] Some recent advances and current problems in number theory, Lectures on
Modern Mathematics, Vol. Ill, Wiley, New York, 1965, 196-244.
[442] Extremal problems in number theory, Proc. Symposia in Pure Math., Vol.
VIII, Theory of Numbers, Amer. Math. Soc, Providence, R.I. (1965)
181-189.
[443] Remarks on a theorem of Zygmund, Proc London Math. Soc. 14(1965)
81-85.
[444] (with A. Hajnal and R. Rado) Partition relations for cardinal numbers, Acta
Math. Acad. Sci. Hungar. 16(1965)93-196.
* [445] On extremal problems of graphs and generalized graphs, Israel J. Math. 2
(1965)183-190.
[446] (with H. S. Shapiro and A. L. Shield) Large and small subspaces of Hilbert
space, Michigan Math. J. 12(1965)169-178.
[447] On the multiplicative representation of integers, Israel J. Math. 2(1965)251-
261.
[448] On the distribution of divisors of integers in the residue classes (mod d), Bull.
Soc. Math. Grece, 1(1965)27-36.
[449] (with A. Sharma) On Tchebycheff quadrature, Canad. J. Math. 17(1965)
652-658.
* [450] (with J. W- Moon) On sets of consistent arcs in a tournament, Canad. Math.
Bull. 8(1965)269-271.
726 BIBLIOGRAPHY OF PAUL ERDOS

[451] (with P. Turan) On some problems of a statistical group theory, I, Z. Wahrsch-
einlichkeitstheorie und Verw. Gebiete 4(1965)175-186.
* [452] On an extremal problem in graph theory, Colloq. Math. 13(1965)251-254.
[453] (with A. Renyi) On the mean value of nonnegative multiplicative number
theoretical functions, Michigan Math. J. 12(1965)321-338.
[454] Some remarks on number theory, Israel J. Math. 3(1965)6—12.
[455] On a problem of Sierpinski, Acta Arith. 14(1965)189-192.
[456] (with I. Csiszar) On the function git) = lim sup (f(x -\- t) — f(x)),
Magyar Tud. Akad. Mat. Kut. Int. Kozl. 9(1965)603-606.
[457] On some extremal problems in graph theory, Israel J. Math. 3(1965)113-116.
[458] (with J. Aczel) The non-existence of a Hamel-basis and the general solution of
Cauchy's functional equation for nonnegative numbers, Publ. Math. Debrecen
12(1965)259-263.
[459] (with F. Harary and W. T. Tutte) On the dimension of a graph, Mathe-
matika 12(1965)118-122.
[460] A problem of independent r-tuples, Ann. Univ. Sci. Budapest. Eotvos Sect.
Math. 8(1965)93-96.
1966
[461] (with A. Hajnal) On chromatic number of graphs and set-systems, Acta Math.
Acad. Sci. Hungar. 17(1966)61-99.
[462] (with A. Hajnal and E. C. Milner) On the complete subgraphs of graphs
defined by systems of sets, Acta Math. Acad. Sci. Hungar. 17(1965)159-
229.
[463] (with A. Hajnal) On a problem of B. Jonsson, Bull. Acad. Polon. 14(1966)
19-23.
* [464] (with A. W. Goodman and L. Posa) The representation of a graph by set
intersections, Canad. J. Math. 18(1966)106-112.
[465] (with E. Harter) Konstruktion von nichtperiodischen Minimalbasen mil der
Dichte x/2 fur die Menge der nichtnegativen ganzen Zahlen, J. Reine Angew.
Math. 221(1966)44-47.
727 BIBLIOGRAPHY OF PAUL ERDOS

[466] (with B. Volkmann) Additive Gruppen mit vorangegebener Haussdorffschen
Dimension, J. Reine Angew. Math. 221(1966)203-208.
* [467] On the construction of certain graphs, J. Combinatorial Theory 1(1966)
149-153.
[468] (with A. Sarkozi and E. Szemeredi) On the solvability of the equations
[ai,ai\ = ar and {a i,a',) = a'r in sequences of positive density, J. Math.
Ana. Appl. 15(1966)60-64.
[469] An example concerning open everywhere discontinuous functions, Rev. Roumaine
Math. Pures Appl. 11(1966)621-622.
[470] (with A. Sarkozi and E. Szemeredi) On the divisibility properties of
sequences of integers, Acta Arith. 11(1966)411-418.
* [471] (with M. Siminovits) A limit theorem in graph theory, Studia Sci. Math.
Hungar. 1(1966)51-57.
[472] (with M. Makkai) Some remarks on set theory, Studia Sci. Math. Hungar.
1(1966)157-159.
* [473] (with A. Renyi and V. T- Sds) On a problem of graph theory, Studia Sci.
Math. Hungar. 1(1966)215-235.
[474] (with A. Renyi) On the existence of a factor of degree one of a connected random
graph, Acta Math. Acad. Sci. Hungar. 17(1966)359-379.
[475] (with A. Sarkozi and E. Szemeredi) On the divisibility properties of
sequences of integers, II, Studia Sci. Math. Hungar. 1(1966)431-435.
[476] On some properties of prime factors of integers, Nagoya Math. J. 27(1966).
[477] On the difference of consecutive terms of sequences defined by divisibility
properties, Acta Arith. 12(1966)175-182.
[478] On some applications of probability methods to function theory and on some
extremal properties of polynomials (a jerevani kongresszus anyagabol), Izdat.
"Nauka," Moscow, 1966, 359-362.
[479] Szdmelmeleti megjegyzesek V. Problemdk a szdmelmeletben, II, Mat. Lapok
17(1966)135-155.
* [480] On cliques in graphs, Israel J. Math. 4(1966)233-234.
728 BIBLIOGRAPHY OF PAUL ERDOS

1967
[481] Asymptotische Untersuchungen uber die Anzahl der Teiler von n, Math. Ann.
169(1967)230-238.
[482] (with P. Turan) A statisztikus csoportelmelet egyes problemdirol, Magyar
Tud. Akad. Mat. Fiz. Oszt. Kozl. 17(1967)51-57.
[483] (with A. Sarkozi and E. Szemeredi) On a theorem of Behrend, J. Austral.
Math. Soc. 7(1967)9-16.
[484] Some remarks on number theory, II, Israel J. Math. 5(1967)57-64.
[485] (with P. Turan) On some problems of a statistical group theory, II, Acta
Math. Acad. Sci. Hungar. 18(1967)151-164.
[486] (with J. H. van Lint) On the number of positive integers < x and free of
prime factors > y, Simon Stevin 40(1967)73-76.
[487] Some remarks on chromatic graphs, Colloq. Math. 16(1967)253-256.
[488] (with G. Piranian) Essential Haussdorff cores of sequences, J. Indian Math.
Soc. 30(1967)93-115.
[489] (with A. Sarkozi and E. Szemeredi) On an extremal problem concerning
primitive sequences, J. London Math. Soc. 49(1967)484-488.
[490] Some recent results on extremal problems in graph theory, Theory of Graphs
(Internat. Sympos., Rome, 1966), Gordon and Breach, New York, 1967,
117-130.
[491] (with A. Hajnal) Kromatikus grdfokrol, Mat. Lapok 18(1967)1-2.
[492] (with J. L. Selfridge) Some problems on the prime factors of consecutive
integers, Illinois J. Math. 11(1967)428-430.
[493] (with R. Rado) Partition relations and transitivity domains of binary
relations, J. London Math. Soc. 42(1967)624-633.
[494] (with J. Clunie) On the partial sums of power series, Proc. Roy. Irish Acad.
Sect. A, 5. No. 11(1967)113-123.
[495] On some applications of graph theory to geometry, Canad. J. Math. 1967,
968-971.
729 BIBLIOGRAPHY OF PAUL ERDOS

[496] (with P. Turan) On some problems of a statistical group theory, III, Acta
Math. Acad. Sci. Hungar. 18(1967)309-326.
[497] (with A. Hajnal) On decomposition of graphs, Acta Math. Acad. Sci.
Hungar. 18(1967)359-376.
[498] (with S- Hartman) On sequences of distances of a sequence, Colloq. Math.
17(1967)191-193.
[499] Some remarks on the iterates of the (f and a functions, Colloq. Math. 17(1967)
115-202.
[500] Grdfok pdros kbriiljdrdsu reszgrdfjairol, Mat. Lapok 18(1967)283—288.
1968
[501] (with E. Szemeredi and A. Sarkozi) On the divisibility properties of
sequences of integers, II, Acta Math. 14(1968)1-12.
[502] (with E. G. Straus) Uber eine geometrische Frage von Fejes Toth, Elem.
Math. 23(1968)11-14.
[503] (with E. Szemeredi) On a problem of P. Erdos and S. Stein, Acta Arith.
15(1968)85-90.
* [504] On some new inequalities concerning extremal properties of graphs, Theory of
Graphs: Proceedings of the Colloquium Held at Tihany, Hungary,
September 1966, edited by P. Erdos and G. Katona, Akademiai Kiadd,
Budapest-Academic Press, New York, 1968, 77—81.
[505] (with A. Hajnal) On chromatic number of infinite graphs, Theory of Graphs:
Proceedings of the Colloquium Held at Tihany, Hungary, September
1966, edited by P. Erdos and G. Katona, Akademiai Kiado, Budapest-
Academic Press, New York, 1968, 83-98.
[506] (with A. Sarkozi and E. Szemeredi) On the solvability of some equations in
dense sequences of integers, Soviet Math. Dokl. 8(1967)1160—1163-
[507] On the boundedness and unboundedness of polynomials, J. Analyse Math. 19
(1967)135-148.
[508] (with A. Hajnal and E. C Milner) On sets of almost disjoint subsets of a set,
Acta Math. Acad. Sci. Hungar. 19(1968)209-218.
730 BIBLIOGRAPHY OF PAUL ERDOS

[509] (with A. Sarkozi and E. Szemeredi) On the solvability of certain equations in
sequences of positive upper logarithmic density, J. London Math. Soc. 43
(1968)71-78.
[510] (with D. A. Darling) On the recurrence of a certain chain, Proc. Amer. Math.
Soc. 19(1968)336-338.
[511] Problems and results on the convergence and divergence properties of the Lagrange
interpolation polynomials and some extremal problems. Mathematica (Cluj)
10(1968)65-73.
[512] (with A. Renyi) Some remarks on the large sieve of Ju. V. Linnik, Ann. Univ.
Sci. Budapest Eotvos Sect. Math. 11(1968)3-13.
[513] (with A. Hajnal) Egy kombinatorikus problemdrol, Mat. Lapok 19(1968)
345-348.
[514] (with A. Hajnal) Hilbert-terben lev'6ponthalmazok nehdny geometriai is halma-
zelmeleti tulajdonsdgdrol, Mat. Lapok 19(1968)255-258.
[515] (with A. Sarkozi and E. Szemeredi) Uber Folgen ganzer Zahlen, Abhan-
dlungen aus Zahlentheorie und Analysis, zur Erinnerung an E. Landau
(1877-1938), VEB Deutscher Verlag d. Wissenschaften, Berlin, 1968,
79-86.
[516] (with P. Turan) On some problems of a statistical group-theory, IV, Acta
Math. Acad. Sci. Hungar. 19(1968)413-435.
* [517] (with A. Renyi) On random matrices, II, Studia Sci. Math. Hungar.
3(1958)459-464.
* [518] (with D. J. Kleitman) On coloring graphs to maximize the proportion of
multicolored k-edges, J. Combinatorial Theory 5(1968)164-169.
[519] (with A. Sarkozi and E. Szemeredi) On divisibility properties of sequences of
integers, Number Theory, Colloq. Janos Bolyai Math. Soc, North-
Holland, Amsterdam, 2(1968)36-49.
1969
* [520] Uber die in Graphen enthaltenen saturierten planaren Graphen, Math. Nachr.
40(1969)13-17.
[521] (with R. Rado) Intersection theorems for systems of sets, II, J. London Math.
Soc. 44(1969)467-479.
731 BIBLIOGRAPHY OF PAUL ERDOS

[522] On the distribution of prime divisors of n, Aequationes Math. 2(1969)177—
183.
[523] On the number of complete subgraphs and circuits contained in graphs, Casopis
Pest. Mat. 94(1969)290-296.
[524] (with A. Renyi) On random entire functions, Zastosowania Matematyki.
Applicationes Mathematicae, Hugo Steinhaus Jubilee Volume 10
(1969)47-55.
* [525] Problems and results in chromatic graph theory, Proof Techniques in Graph
Theory, edited by F. Harary, Academic Press, New York-London,
1969, 27-35.
[526] (with I. Katai) On the growth of dk (n), Fibonacci Quart. 7(1969)267-274.
*[527] On a combinatorial problem, III, Canad. Math. Bull. 12(1969)413-416.
[528] (with A. Sarkozi and E. Szemeredi) On some extremal properties on
sequences of integers, Ann. Univ. Sci. Budapest Eotvos Sect. Math. 12(1969)
131-135.
[529] (with A. Hajnal and E. C. Milner) Polarized partition relations for ordinal
numbers, Canadian Math. Congress 1969. 63-87.
[530] On the irrationality of certain series, Math. Student 36(1968)222-226
(1969).
[531] On the sum C=i d(d(n)), Math. Student 36(1968)227-229.
[532] (with J. Denes and P. Turan) On some statistical properties of the alternating
group of degree, Enseignement Math. 15(1969)89-99.
[533] (with I. Katai) On the sum £ d4(n), Acta Sci. Math. (Szeged) 30(1969)
314-325.
[534] Some applications of graph theory to number theory, The Many Facets of
Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich.,
1968), Springer-Verlag, Berlin (1969)77-82.
[535] (with A. Hajnal and E. C. Milner) A problem on well ordered sets, Acta
Math. Acad. Sci. Hungar. 20(1969)323-329.
[536] On some applications of graph theory to number theoretic problems, Publ. Rama-
nujan Inst. 1(1969)131-136.
732 BIBLIOGRAPHY OF PAUL ERDOS

[537] (with A. Sarkozi and E. Szemer^di) Uber Folgen ganzer Zahlen, Number
Theory and Analysis: A Collection of Papers in Honor of Edmund
Landau, edited by P. Turan, Plenum, New York 1969, 77-86. (Reprint
of paper 515.)
1970
* [538] (with L. Moser) An extremal problem in graph theory, J. Austral. Math.
Soc. 11(1970)42-47.
* [539] On a lemma of Hajnal-Folkrnan, Colloq. Math. Soc. J. Bolyai 4.
Combinatorial theory and its applications I, 1970, 311-317.
[540] (with L. Gerencser and A. Mat6) Problems of graph theory concerning
optimal design, Colloq. Math. Soc. J. Bolyai 4. Combinatorial theory
and its applications I, 1970, 317-327.
[541] (with A. Hajnal and E. C. Milner) Set mappings and polarized partition
relations. Colloq. Math. Soc. J. Bolyai 4. Combinatorial theory and its
applications I, 1970, 327-365.
* [542] (with J. Komlds) On a problem of Moser, Colloq. Math. Soc. J. Bolyai 4.
Combinatorial theory and its applications I, 1970, 365-369.
[543] (with J. Schonheim) On the set of non pairwise coprime divisors of a number,
Colloq. Math. Soc. J. Bolyai 4. Combinatorial theory and its
applications I, 1970, 369-377.
* [544] (with M. Simonovits) Some extremal problems in graph theory, Colloq. Math.
Soc. J. Bolyai 4. Combinatorial theory and its applications I, 1970,
378-392.
* [545] (with V. T. Sds) Some remarks on Ramsey's and Turan's theorem, Colloq.
Math. Soc. J. Bolyai 4. Combinatorial theory and its applications II,
1970, 393-405.
[546] (with E. G. Straus) Xonaveraging sets II, Colloq. Math. Soc. J. Bolyai
4. Combinatorial theory and its applications II, 1970, 405-413.
[547] Problems in combinatorial set theory. Combinatorial Structures and Their
Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969),
edited by H. Guy, H. Hanani, N. Sauer, and J. Schonheim, Gordon
and Breach, New York, 1970, 97-100.
733 BIBLIOGRAPHY OF PAUL ERDOS

* [548] Some unsolved problems in graph theory and combinatorial analysis,
Combinatorial Mathematics and Its Applications (Proc. Conf., Oxford, 1969),
Academic Press, London and New York, 1971, 97-109,
[549] On the distribution of the convergents of almost all real numbers, J. Number
Theory 2(1970)425-441.
[550] (with A. Sarkozi) On the divisibility properties of sequences of integers, Proc.
London Math. Soc. 21(1970)97-101.
[551] (with M. Herzog and J. Schonheim) An extremal problem on the set of non-
coprime divisors of a number, Israel J. Math. 8(1970)408-412.
[552] (with P. Hapial and D. Neuman) Some unsolved problems.
[553] On the distribution of the roots of orthogonal polynomials, Proc. of the
Conference on Constructive Theory of Functions (1970)145-150.
[554] (with A. H. Stone) On the sum of two Sorel-sets, Proc. Arner. Math. Soc.
25(1970)304-306.
[555] On the application of combinatorial analysis to number theory, geometry and
analysis, Acta Congr. Intern. Math. (1970)201-210.
[556] (with A. Hajnal) Some results and problems on certain polarized partitions,
Acta Math. Acad. Sci. Hungar. 21(1970)369-392.
[557] (with A. R6nyi) On a new law of large numbers, J. Analyse Math. 23(1970)
103-111.
[558] (with A. Hajnal) Problems and results in finite and infinite combinatorial
analysis, Ann. New York Acad. Sci. 175(1970)115-124.
[559] (with E. G. Straus) Some number theoretic results, Pacific J. Math. 36(1971)
635-646.
[560] On sets of distances of n points, Araer. Math. Monthly 77(1970)738-740.
[561] Tux an Pal graf-tetelerol, Mat. Lapok 21(1970)249-251.
[562] (with A. R6nyi) On some applications of probability methods to additive number
theoretic problems, Contributions to Ergodic Theory and Prob. Proc.
Conf., Ohio State Univ., Columbus, Ohio 1970, Springer, Berlin, 1970,
37-44.
734 BIBLIOGRAPHY OF PAUL ERDOS

[563] Some extremal problems in combinatorial number theory, Mathematical Essays
Dedicated to A. J. Macintyre, edited by H. Shankar, Ohio Univ. Press,
Athens, Ohio, 1970, 123-133.
*[564] (with D. J. Kleitman) Extremal problems among subsets of a set, Proc.
Second Chapel Hill Conf. on Combinatorial Mathematics and Its
Applications (Univ. North Carolina, Chapel Hill, N.C., 1970), 146-170.
[565] Some applications of graph theory to number theory, Proc. Second Chapel Hill
Conf. on Combinatorial Mathematics and Its Applications (Univ.
North Carolina, Chapel Hill, N.C., 1970), 136-145.
[566] Some problems in additive number theory, Amer. Math. Monthly 77(1970)
619-621.
1971
[567] (with A. Hajnal) Ordinary partition relations for ordinal numbers, Periodica
Math. Hungar. 1(1971)369-392.
[568] On the sum £d|2„_, d~\ Israel J- Math. 9(1971)43-48.
*[569] (with D. J. Kleitman) On collections of subsets containing no 4-member
Boolean algebra, Proc. Amer. Math. Soc. 28(1971)87-90.
[570] (with I. Katai) Xon complete sums of multiplicative functions, Periodica
Math. Hungar. 1(1971)209-212.
[571] (with E. Milner and R. Rado) Partition relations for r\2-sets, J. London
Math. Soc. 3(1971)193-204.
[572] On some extremal problems on r-graphs, Discrete Math. 1(1971)1-6.
[573] (with A. Hajnal) Unsolved problems in set theory, Proceedings of Symposia
in Pure Math. XIII. Part. 1. Amer. Math. Soc, Providence, R.I.
(1971)17-48.
[574] (with J. Bosak and A. Rosa) Decompositions of complete graphs into factors
with diameter two, Mat. Casopis Sloven Akad. Vied. 21(1971)14-28.
*[575] Problems and results in combinatorial analysis, Combinatorics (Proc. Symp.
Pure Math., Vol. XIX) Amer. Math. Soc, Providence, R.I., 1971,
77-89.
*[576] (with G. Purdy) Some Extremal Problems in Geometry, J. Combinatorial
Theory 10(1971)246-252.
735 BIBLIOGRAPHY OF PAUL ERDOS

[577] (with J. L. Selfridge) Complete prime subsets of consecutive integers, Conf.
Numerical Methods, Winnipeg, 1971.
1972
[578] On the fundamental problem of mathematics, Amer. Math. Monthly 79(1972)
149.
[579] (with A. Hajnal) Unsolved and solved problems in set theory, 1972.
[580] (with E. Netanyahu) A remark on polynomials and the transfinite diameter,
Technion Israel Inst, of Technology, Haifa, 1972. Preprint.
[581] On a problem of Grunbaum, Canad. Math. Bull. 15(1972)23-25.
[582] Problems and Results on combinatorial number theory, North-Holland,
Amsterdam, 1972, 1-22.
[583] (with R. L. Graham) On Sums of Fibonacci numbers, Fibonacci Quart. 10
(1972)249-254.
[584] (with A. Hajnal) On some general properties of chromatic numbers, Israel
J. Math. 1972 (to be published).
[585] (with B. Grunbaum) Osculation vertices in arrangements of curves (to be
published).
[586] (with M. Simonovits) An extremal graph problem, Acta Math. Acad. Sci.
Hungar. 22(1971)275-282.
[587] (with J. A. Bondy) Ramsey number for cycles in graphs (to be published).
[588] (with S. Shelah) On problems of Moser and Hanson (to be published).
k
[589] (with E. Szemeredi) On the number of solutions of m = Y* x\ (to be
published).
[590] On the number of unique subgraphs of a graph, J. Combinatorial Theory
B 13(1972)112-115.
Auxiliary List
* [i] (with J. Spencer) Imbalances in k-colorations, Networks 1(1972)379-383.
* [ii] Extremal problems in graph theory, A Seminar on Graph Theory, edited by
F. Harary, Holt, Rinehart and Winston, New York, 1971, 54-59.
736 BIBLIOGRAPHY OF PAUL ERDOS

* [iii] Applications of probabilistic methods to graph theory, A Seminar on Graph
Theory, edited by F. Harary, Holt, Rinehart and Winston, New York,
1971, 60-69.
* [iv] (with R. J. McEliece and H. Taylor) Ramsey bounds for graph products,
Pacific J. Math 37 (1971) 45-46.
* [v] (with A. Renyi) On the evolution of random graphs, Bull. Inst. Internat.
Statist. 38(4)(1961), Tokyo, 343-347.
* [vi] (with Gy. Szekeres) On some extremum problems in elementary geometry, Ann.
Univ. Sci. Budapest Eotvos Sect. Math. 3(1960/61)53-62.
737 BIBLIOGRAPHY OF PAUL ERDOS

Alphabetical Listing of
Papers in this Volume
The number in square brackets preceding the title indicates the number of the
paper. The words a, an, on, some, and the are ignored in alphabetization.
[iii] Applications of probabilistic methods to graph theory 429
[419] Some applications of probability to graph theory and
combinatorial problems 23
[401] Asymmetric graphs (with A. Renyi) 333
[120] The asymptotic number of Latin rectangles (with I. Kaplan-
sky) 641
[386] On circuits and subgraphs of chromatic graphs 97
[480] On cliques in graphs 361
[569] On collections of subsets containing no 4-member Boolean
algebra (with D. J. Kleitman) 507
[518] On coloring graphs to maximize the proportion of
multicolored hedges (with D. J. Kleitman) 107
[390] On a combinatorial problem. I 439
[430] On a combinatorial problem. II 445
[527] On a combinatorial problem. Ill 448
[18] A combinatorial problem in geometry (with Gy. Szekeres) 5
[413] On a combinatorial problem in Latin squares (with A. Ginz-
burg) 653
[243] On some combinatorial problems (with A. Renyi) 513
[160] A combinatorial theorem (with R. Rado) 376
[185] Combinatorial theorems on classifications of subsets of a given
set (with R. Rado) 383
[437] On complete topological subgraphs of certain graphs (with A.
Hajnal) 167
739 ALPHABETICAL LIST OF PAPERS

The construction of certain graphs (with C. A. Rogers) 418
On the construction of certain graphs 189
On the dimension of a graph (with F. Harary and W. T. Tutte) 356
On the evolution of random graphs (with A. Renyi) 574
On the evolution of random graphs (with A. Renyi) 569
On an extremal problem in graph theory 182
An extremal problem in graph theory (with L. Moser) 240
Extremal problems among subsets of a set (with D. J. Kleit-
man) 482
Some extremal problems in geometry (with G. Purdy) 690
Extremal problems in graph theory 155
Extremal problems in graph theory 229
Some extremal problems in graph theory (with M. Simono-
vits) 246
On extremal problems of graphs and generalized graphs 174
On some extremum problems in elementary geometry (with
Gy. Szekeres) 680
Geometrical extrema suggested by a lemma of Besicovitch (with
P. Bateman) 667
Graph theory and probability. I 406
Graph theory and probability. II 411
Imbalances in /t-colorations (with J. Spencer) 122
On independent circuits contained in a graph (with L. Posa) 324
Integral distances (with N. H. Anning) 661
Intersection theorems for systems of finite sets (with Chao Ko
and R. Rado) 466
Intersection theorems for systems of sets (with R. Rado) 460
On a lemma of Hajnal-Folkman 474
740 ALPHABETICAL LIST OF PAPERS

On a lemma of Littlewood and Offord 455
On a limit theorem in combinatorial analysis (with H. Hanani) 521
A limit theorem in graph theory (with M. Simonovits) 194
On the maximal number of disjoint circuits of a graph (with
L. Pdsa) 293
On the maximal number of independent circuits in a graph
(with G. Dirac) 303
On the maximal number of pairwise orthogonal Latin squares
of a given order (with S. Chowla and E. G. Straus) 648
On the maximal paths and circuits of graphs (with T. Gallai) 273
On the minimal number of vertices representing the edges of a
graph (with T. Gallai) 57
The minimal regular graph containing a given graph (with P.
Kelly) 354
On some new inequalities concerning extremal properties of
graphs 235
On the number of complete subgraphs contained in certain
graphs 145
On the number of triangles contained in certain graphs 151
On a problem in graph theory 527
A problem in graph theory (with A. Hajnal and J. W. Moon) 163
On a problem of graph theory (with A. Renyi and V. T. Sds) 201
On a problem of Moser (with J. Kornlos) 479
A problem on independent r-tuples 186
A problem on tournaments (with L. Moser) 531
Problems and results in chromatic graph theory 113
Problems and results in combinatorial analysis 40
[iv] Ramsey bounds for graph products (with R. J. McEliece and
H. Taylor) 434
741 ALPHABETICAL LIST OF PAPERS

On random graphs. I (with A. Renyi) 561
On random matrices. I (with A. Renyi) 625
On random matrices. II (with A. Renyi) 632
Some recent results on extremal problems in graph theory 222
Some remarks on chromatic graphs 103
Remarks on a paper of Posa 143
Some remarks on Ramsey's and Turan's theorem (with V. T.
Sds) 260
Some remarks on the theory of graphs 373
On the representation of directed graphs as unions of orderings
(with L. Moser) 79
The representation of a graph by set intersections (with A. W.
Goodman and L. Pdsa) 87
On sets of consistent arcs in a tournament (with J. W. Moon) 537
On sets of distances of n points 664
On sets of distances of n points in Euclidean space 676
On the strength of connectedness of a random graph (with A.
Renyi) 618
On the structure of linear graphs 319
On subgraphs of the complete bipartite graph (with J. W-
Moon) 424
On a theorem of Rademacher-Turan 131
On two problems of information theory (with A. Renyi) 543
Uber die in Graphen enthaltenen saturierten planaren Graphen 363
Uber ein Extremalproblem in der Graphentheorie 137
Some unsolved problems (partial paper) 15
Some unsolved problems in graph theory and combinatorial
analysis 27
742 ALPHABETICAL LIST OF PAPERS