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COMBINATORICS, GEOMETRY AND PROBABILITY

COMBINATORICS, GEOMETRY AND PROBABILITY A tribute to Paul Erdos Edited by

BELA BOLLOBAS ANDREW THOMASON

_ CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Cambridge University Press 1997 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1997 First paperback edition 2004 Typeset in 10/13pt Monotype Times A catalogue record for this book is available from the British Library ISBN 0 521 58472 8 hardback ISBN 0 521 60766 3 paperback

Contents Preface Farewell to Paul Erdos Toast to Paul Erdos List of Contributors

Page ix xi xiii xvii

Paul Erdos: Some Unsolved Problems

1

Aharoni, R. and R. Diestel Menger's Theorem for a Countable Source Set

11

Ahlswede, R. and N. Cai On Extremal Set Partitions in Cartesian Product Spaces

23

Aigner, M. and R. Klimmek Matchings in Lattice Graphs and Hamming Graphs

33

Aigner, M. and E. Triesch Reconstructing a Graph from its Neighborhood Lists

51

Alon, N. and R. Yuster Threshold Functions for //-factors

63

Barbour, A.D. and S. Tavare A Rate for the Erdos-Turan Law

71

Beck, J. Deterministic Graph Games and a Probabilistic Intuition

81

Bezrukov, S.L. On Oriented Embedding of the Binary Tree into the Hypercube

95

Biggs, N.L. Potential Theory on Distance-Regular Graphs

107

Bollobas, B. and S. Janson On the Length of the Longest Increasing Subsequence in a Random Permutation

121

Bollobas, B. and Y. Kohayakawa On Richardson's Model on the Hypercube

129

Cameron, P.J. and W.M. Kantor Random Permutations: Some Group-Theoretic Aspects

139

Chen, G. and R.H. Schelp Ramsey Problems with Bounded Degree Spread

145

vi

Contents

Cooper, C, A. Frieze and M. Molloy Hamilton Cycles in Random Regular Digraphs

153

de Fraysseix, H., P. Ossona de Mendez and P. Rosenstiehl On Triangle Contact Graphs

165

Deuber, W.A. and W. Thumser A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies

179

Deza, M. and V. Grishukhin Lattice Points of Cut Cones

193

Diestel, R. and I. Leader The Growth of Infinite Graphs: Boundedness and Finite Spreading

217

Dugdale, J.K. and A.J.W. Hilton Amalgamated Factorizations of Complete Graphs

223

Erdos, Paul, R.J. Faudree, C.C. Rousseau and R.H. Schelp Ramsey Size Linear Graphs

241

Erdos, Paul, A. Hajnal, M. Simonovits, V.T. S6s and E. Szemeredi Turan-Ramsey Theorems and ^-Independence Numbers

253

Erdos, Paul, E. Makai and J. Pach Nearly Equal Distances in the Plane

283

Erdos, Paul, E.T. Ordman and Y. Zalcstein Clique Partitions of Chordal Graphs

291

Erdos, Peter L., A. Seress and L.A. Szekely On Intersecting Chains in Boolean Algebras

299

Fiiredi, Z., M.X. Goemans and D.J. Kleitman On the Maximum Number of Triangles in Wheel-Free Graphs

305

Gionfriddo, M., S. Milici and Zs. Tuza Blocking Sets in SQS(2v)

319

Haggkvist, R. and A. Johansson (1,2)-Factorizations of General Eulerian Nearly Regular Graphs

329

Haggkvist, R. and A. Thomason Oriented Hamilton Cycles in Oriented Graphs

339

Halin, R. Minimization Problems for Infinite n-Connected Graphs

355

Hammer, P.L. and A.K. Kelmans On Universal Threshold Graphs

375

Hindman, N. and I. Leader Image Partition Regularity of Matrices

393

Contents

vii

Hundack, C, H.J. Promel and A. Steger Extremal Graph Problems for Graphs with a Color-Critical Vertex

421

Komjath, P. A Note on co\ -» co\ Functions

435

Komlos, J. and E. Szemeredi Topological Cliques in Graphs

439

Linial, N. Local-Global Phenomena in Graphs

449

Luczak, T. and L. Pyber On Random Generation of the Symmetric Group

463

Mader, W. On Vertex-Edge-Critically n-Connected Graphs

471

Mathias, A.R.D. On a Conjecture of Erdos and Cudakov

487

McDiarmid, C. A Random Recolouring Method for Graphs and Hypergraphs

489

Mohar, B. Obstructions for the Disk and the Cylinder Embedding Extension Problems

493

Nesetfil, J. and P. Valtr A Ramsey-Type Theorem in the Plane

525

Temperley, H.N.V. The Enumeration of Self-Avoiding Walks and Domains on a Lattice

535

Tetali, P. An Extension of Foster's Network Theorem

541

Welsh, D.J.A. Randomised Approximation in the Tutte Plane

549

Wilf, H.S. On Crossing Numbers, and some Unsolved Problems

557

Preface On Friday, 26 March 1993, Paul Erdos celebrated his 80th birthday. To honour him on this occasion, a conference was held in Trinity College, Cambridge, under the auspices of the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge. Many of the world's best combinatorialists came to pay tribute to Erdos, the universally acknowledged leader of their field. The conference was generously supported both by the London Mathematical Society and by the Heilbronn Fund of Trinity College. As at former Cambridge Conferences in honour of Paul Erdos, the day-to-day running of this conference was in the able hands of Gabriella Bollobas, with the untiring assistance of Tristan Denley, Ted Dobson, Tom Gamblin, Chris Jagger, Imre Leader, Alex Scott and Alan Stacey. The conference would not have taken place without their dedicated work. On the eve of Erdos' birthday, a sumptuous feast was held in his honour in the Hall of Trinity College. The words wherein he was toasted are reproduced in the following pages. This volume of research papers was presented to Paul Erdos by its authors as their own toast, gladly offered with their gratitude* respect and warmest wishes. Sadly, before this book reached its printed form, Paul Erdos died. Whereas it was conceived in joy it appears now tinged with sorrow. We feel his loss tremendously. But it is not appropriate that grM should overshadow this volume. Erdos lived to do mathematics and he died doing mathematics. So this work remains a tribute to the Erdos we fondly remember — the living Erdos — the mathematician. B.B. A.G.T.

IX

Farewell to Paul Erdos (26/3/1913 - 20/9/1996)

(Paul Erdos died in Warsaw on 20th September 1996. A memorial service was held for him on 18th October 1996 in the Kerepesi Cemetery in Budapest, the traditional resting place of eminent Hungarians. A great number of his friends gathered to mark his passing. Among them were colleagues and former students representing mathematics from many countries and four continents. The orations were given by Akos Csaszar, Paul Revesz, Gyula Katona, Ron Graham, Andras Hajnal, George Szekeres, and by Bela Bollobas, whose tribute is reproduced below.)

Paul Erdos was one of the most brilliant and probably the most remarkable of mathematicians of this century. Not only was his output prodigious, with fundamental papers in many branches of mathematics, including number theory, geometry, probability theory, approximation theory, set theory and combinatorics, and not only did he have many more coauthors than anybody else in the history of mathematics, but he was also a personal friend of more mathematicians than anybody else. The vast body of problems he has left behind will influence mathematics for many years to come. Many of us are lucky to have known him and to have benefited from his incisive mind, fertile imagination and desire to help. But hardly any one of us knew him in his prime, from the mid-thirties to the early sixties. He was hardly twenty when he took the mathematical world by storm, so that the great Issai Schur of Berlin dubbed him der Zauberer von Budapest. Throughout his life, he lived modestly, despising material possessions and coveting no honours, and was always somewhat outside the mathematical establishment. Nevertheless, he was showered with honours. Among others, he was an Honorary Member of the London Mathematical Society and an Honorary Fellow of the Royal Society. These illustrious institutions have sent wreaths to express their grief at his loss. But I am here mainly to represent Paul's many friends, colleagues and, above all, his students. Thinking of him, David's psalm springs to mind: "surely goodness and mercy shall follow me all the days of my life." For decades, he was the window to the West for the Hungarian mathematicians, and has helped more mathematicians all over the world than anybody else. He was especially kind to young people. I was just over fourteen when he called me to him and so changed the course of my life. There is no doubt that I became a combinatorialist only because of him, and I owe him a tremendous debt of gratitude for all his kindness and inspiration. Many people owe their careers to him. As David in his psalm, he could also have said: "though I walk in the valley of the shadow of death, I will fear no evil." Sadly, he was always in the shadow of death. When he was born, his two sisters died; when he was a year-and-a-half his father was taken prisoner of war and spent six years in Siberia; when his father died of a heart attack, he could not come to Hungary to comfort his mother; most of his relatives perished in the xi

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Farewell to Paul Erdos

Holocaust; in the fifties even America abandoned him and he was saved only by Israel; finally, the loss of his mother was a terrible blow to him, from which he never really recovered. But whatever happened, he always had a passionate desire to be free: he could not tolerate constraint of any kind, he was never willing to compromise. Perhaps there were only two happy periods in his adult life: from 1934 to 1939 when he was in Manchester and Princeton, and from 1964 to 1971, when he travelled around the world with his beloved mother. I was lucky enough to have known him in this second happy period. The death of Paul Erdos marks the end of an era. No conference will be the same without the p.g.o.m., the poor great old man, as he called himself, no mathematical discussion will be as much fun as it was with him. Our beloved Pali Bdcsi has left us all orphans. This exceptional man did think about what will happen after him. Endre Ady, the famous Hungarian poet, wrote: "Let him be cursed who takes my place!" Paul's wish was rather different, reflecting his character: "Let him be blessed who takes my place!" Now, when we have to say our final goodbye to Paul Erdos, we all know that there is no chance of that. His death is a tremendous loss to us all, and this sense of loss will stay with us for ever. But we should console ourselves that he has had a marvellous life, in which he has produced an exceptional amount of outstanding mathematics, and we are privileged to have known him. Kerepesi Cemetery, Budapest, 18/10/1996 Bela Bollobas

Toast to Paul Erdos

(The following is the toast of the Banquet for the 80th Birthday of Paul Erdos, held in Trinity College, Cambridge, on 25 March 1993, the eve of the birthday. The banquet was attended by many of Erdos' other friends, including Lady Jeffreys, Mrs Davenport and Peter Rado, in addition to the conference participants. Trinity College was represented by Sir Andrew Huxley, OM, former president of the Royal Society and former Master of the College, who presided at the feast. Cambridge mathematics was represented by the present and former Sadlerian Professors, John Coates, FRS, and J.W.S. Cassels, FRS.)

Professor Erdos, Sir Andrew, Ladies and Gentlemen, Mathematics is rich in unusual characters, as everyone here at this dinner will know. Nevertheless, most of us would agree that there is none whose achievement and lifestyle are more extraordinary than those of the man we are celebrating tonight, on the eve of his birthday, following a Hungarian custom. For over 60 years, his fertile mind has maintained a staggering output in many branches of mathematics: he has made notable contributions and broken fresh ground in set theory, number theory, probability theory, classical analysis, geometry, approximation theory and combinatorics. Most of us are particularly aware of his contributions to the last of these subjects: he has done more than anyone else to establish combinatorics; many branches of the subject find their origin in his ideas; the stimulus of his striking theorems and inspiring problems is one that we have all felt, and for which we owe him an incalculable debt of gratitude. It is also true that, as well as being so remarkably gifted intellectually, he has the most admirable and attractive personal qualities. He is generous to a fault, gentle, unassuming, always eager to fight for the downtrodden. Many a young student has been delighted to discover that this famous man is so easily approachable and so interested in their work. He has always made it his business to nurture young talent, possibly his greatest find being Posa. What anybody, who has ever heard of this unique man, knows is that he is unceasingly on the move. It is hardly an exaggeration that he has not slept in the same bed for more than a week in over 50 years. As a constant globe-trotter, he is the living link between mathematicians across the world, carrying with him news of theorems, conjectures and problems. Paul Erdos was a precocious child: at the age of three he was good at arithmetic to the point of discovering for himself negative numbers. Much of Paul's education was done in private; altogether he spent less than four years in schools. At the age of 17, he proceeded to university, where he soon became the focus of a wonderfully talented group of mathematicians. At the age of 21, he completed his degree, and as was the custom, he looked to spend a year abroad. In the world of 1934, the country that most attracted him was Britain. As an undergraduate, he had corresponded with Louis Mordell, the great American number

xiii

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Toast to Paul Erdos

theorist, who by that time had left St John's College, Cambridge to work in Manchester. Mordell offered Erdos a Fellowship in his department, and the offer was gladly accepted. On 1 October 1934, Erdos arrived in London, from where he took the train to Cambridge. At the station he was met by two outstanding young mathematicians who for many years to come were to be his closest friends, Harold Davenport and Richard Rado. Sadly, Harold Davenport and Richard Rado are no longer with us, but it is indeed a pleasure to see Anne Davenport and Peter Rado at this banquet tonight. In fact, it is due to Erdos's friendship with Davenport that my own connection with Trinity came about. At that time Erdos stayed in Cambridge only for a couple of days, but long enough to meet Hardy and Littlewood, the leading English mathematicians. He then travelled on to Manchester, to Mordell, who became his mentor and friend. In the 1930s Mordell gathered a remarkable group of mathematicians to Manchester: in addition to Erdos, and later Davenport, the group included Mahler, Heilbronn, du Val and Chao Ko. It is extremely fitting that this conference has been supported by Heilbronn's generous bequest to the mathematicians of this college. On looking down on us, Heilbronn must be smiling that we are celebrating his great friend tonight. Another prominent member of the Manchester group was the eminent fluid dynamicist Miss Swirles, who befriended Paul soon after his arrival. It is a great pleasure that Miss Swirles, by now Lady Jeffreys, can share in this happy celebration tonight. Paul stayed in Manchester for four years, first as the Bishop Harvey Goodwin Fellow, and then as a Royal Society Fellow. During that time he made frequent visits to Cambridge and other centres of mathematics. In 1938 Paul left England for the States to take up a Fellowship at Princeton. It was to be ten more years before Paul returned to Hungary, and he would never again stay there for more than a few months at a time. After a year or two at the Institute, the travelling began in earnest, and the now familiar pattern was soon set. In a short space of time, he visited Philadelphia, Purdue, Stanford, Syracuse and Johns Hopkins, and many other universities for even shorter periods. Since then Paul has been travelling from university to university, from country to country, bringing news, inventing problems, writing joint papers, stimulating the minds of mathematicians everywhere, and generally being the Erdos we know and love so well. By now he has over 300 coauthors, and it has often been said that if a train journey is long enough, he will write a joint paper with the conductor. His 1300 research papers place him in a league of his own among research mathematicians. It has been said that the world wants geniuses but it wants them to behave just like other people. Paul found this out when one apocryphal, but not too far-fetched, night in Chicago he was out walking by himself. Suddenly a police car appeared and the officers began to question Paul. "So what are you doing out here, all by yourself?" "I am thinking" came the reply. "What do you mean you are thinking? What are you thinking about?" "I am proving a theorem." "You'd better come with us back to the station, Sir." Back at the station, the officer in charge said "Now, what's all this about your theorem? Tell me about it." "It doesn't matter anymore" grumbled Paul testily, "I've found a counterexample." In fact, this incident is atypical for, as we know, Paul is remarkably successful in proving theorems. A striking example is quoted by Mark Kac.

Toast to Paul Erdos

xv

"As a mathematician Erdos is what in other fields is called a 'natural'. If a problem can be stated in terms he can understand, though it may belong to afieldwith which he is not familiar, he is as likely as, or even more likely than, the experts to find a solution. An example of this is his solution of a problem in dimension theory, a part of topology of which in 1939 he knew absolutely nothing. The late Witold Hurewicz and a younger colleague, Henry Wallman, were writing a book on dimension theory which later became an acknowledged classic. They were interested in the unsolved problem of the dimension of the set of rational points in Hilbert space. What all this means is unimportant except that the problem seemed very difficult and that the 'natural' conjectures were that the answer is either zero or infinity. Erdos overheard several mathematicians discussing the problem in the common room of the old Fine Hall at Princeton. "What is the problem?" asked Erdos. Somewhat impatiently he was told what the problem was. "What is dimension?" he asked, betraying complete ignorance of the subject matter. To pacify him, he was given the definition of dimension. In a little more than an hour he came with the answer, which, to everyone's immense surprise, turned out to be T!" In addition to being successful in his own personal research, one of Paul's greatest gifts to mathematics has been his ability to stimulate the creativity of others through his fascinating and penetrating conjectures. His offer of monetary rewards for solutions is legendary. The winner of the largest reward to date is Szemeredi, for finding long arithmetic progressions in sets of positive density. It is a pleasure to see him here tonight. The biggest sum on offer is $10000, for proving that the gap between two consecutive primes is rather large infinitely often. Although Schonhage, Rankin, Maier and Tenenbaum have proved exciting results in this direction, they haven't yet managed to claim the prize. Paul is also offering $3000 for finding long arithmetic progressions in sequences of natural numbers whose reciprocals diverge, and so, in particular, among the primes. A group of Swedish computers has just discovered an arithmetic progression of 22 primes but I doubt that any payment will be forthcoming from Paul. Paul worked with most of the leading Hungarian mathematicians, especially the number theorist Paul Turan and the probabilist Alfred Renyi, who were his great friends. Turan's wife, Vera Sos, has also been a close friend and collaborator for many years, and it is fitting that she too should be celebrating tonight. My own friendship with Paul is also of many years standing. We met when I was 14, and I was tremendously impressed by his willingness to talk to me about his fascinating problems. To me he seemed to be from a different planet, a flamboyant man with an air of the exotic, with his expensive foreign suits and ready cash, brought from the unattainable free Western world. Now I know better; I think it was Paul who inspired the saying: "The man who leaves footprints on the sands of time never wears expensive shoes." In those days, I also got to know Paul's mother, Annus neni, a charming lady who adored Paul, and was, in turn, adored by her son. She kept his reprints in immaculate order, and sent copies to those who requested them. A year or two later they got to know my family, and were frequent visitors to our house whenever Paul was in Hungary. In 1964, at the age of 84, Annus neni began to travel with Paul. Their first trip was to Israel; soon Western Europe followed, including England a year later. In 1968, when she was 88, Annus neni accompanied Paul to Hawaii and Australia. When asked whether she liked to travel, she used to reply: "You know I don't travel because I like it, but to be with my son." It was moving too see their affectionate care for each other, catching up

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with those lost years, when they couldn't see each other. Annus neni greatly enjoyed her role as Queen Mother of mathematics, meeting and entertaining all the people coming to see Paul; her cocoa cake with coffee cream was especially delicious. Erdos's own tastes in food are well known to be frugal, and he doesn't care for wine, which he calls poison. It has been suggested that the College should on this occasion produce a meal of bread and water. Unfortunately when I checked with the Kitchens, they could not find the recipe, so we had to use the second choice menu. Paul Erdos has always kept up his close links with Trinity and Cambridge. Some years ago he was a Visiting Fellow Commoner of Trinity College, and in 1991 Cambridge awarded him an Honorary Doctorate - the first citizen of Hungary to receive this honour. At the ceremony it was charming to see the great actor Sir Alec Guinness taking it upon himself to shepherd Paul through the long ritual. Since his youth, Paul Erdos has had catholic interests: in particular, he has maintained an enthusiasm for history and medicine. It is always fascinating to engage him in discussion pf his favourite historical events. Nevertheless, Paul is the quintessential mathematician: he breathes, eats, drinks, and sleeps mathematics, if he sleeps at all. It could have been Erd5s, whom Littlewood had in mind when he wrote: "There is much to be said for being a mathematician. To begin with, he has to be completely honest in his work, not from any superior morality, but because he cannot get away with a fake. It has been cruelly said of arts dons, especially in Oxford, that they believe there is a polemical answer to everything; nothing is really true, and in controversy the object is to prove your opponent a fool. We escape all this. Further, the arts man is always on duty as a great mind; if he drops a brick, as we say in England, it reverberates down the years. After an honest day's work a mathematician goes off duty. Mathematics is very hard work, and dons tend to be above average in health and vigour. Below a certain threshold a man cracks up; but above it, hard mental work makes for health and vigour (also - on much historical evidence throughout the ages - for longevity)."

If hard mental work be the secret of longevity then Paul Erdos will live forever and continue to enrich us all with the brightness of his intellect and the warmth of his heart. In the meantime, we honour him on his 80th birthday. Ladies and Gentlemen, please rise and toast Paul Erdos. B.B.

List of Contributors Ron Aharoni Department of Mathematics, Technion, Haifa 32000, ISRAEL Rudolf Ahlswede Universitat Bielefeld, Fakultat fur Mathematik, Postfach 100131, 33501 Bielefeld, GERMANY Martin Aigner Freie Universitat Berlin, Fachbereich Mathematik, WE2, Arnimallee 3, 1000 Berlin 33, GERMANY Noga Alon Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, ISRAEL A. D. Barbour Institut fur Angewandte Mathematik, Universitat Zurich, Winterthurerstrasse 190, CH-8057, Zurich, SWITZERLAND Jozsef Beck Department of Mathematics, Rutgers University, Busch Campus, Hill Center, New Brunswick, NJ 08903, USA Sergej L. Bezrukov Fachbereich Mathematik, Freie Universitat Berlin, Arnimallee 2-6, D-14195 Berlin, GERMANY Norman L. Biggs London School of Economics, Houghton St, London WC2A 2AE, UK Bela Bollobas Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, UK and Louisiana State University, Baton Rouge, LA 70803 USA Ning Cai Universitat Bielefeld, Fakultat fur Mathematik, Postfach 100131, 33501 Bielefeld, GERMANY Peter J. Cameron School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London, El 4NS, UK G. Chen North Dakota State University, Fargo, ND 58105, USA Colin Cooper School of Mathematical Sciences, University of North London, London, UK xvii

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List of Contributors

Walter A. Deuber Universitat Bielefeld, Fakultat fur Mathematik, Postfach 100131, 33501 Bielefeld 1, GERMANY Michel Deza CNRS-LIENS, Ecole Normale Superieure, Paris, FRANCE Reinhard Diestel Faculty of Mathematics (SFB 343), Bielefeld University, 4-4800, Bielefeld, GERMANY J. K. Dugdale Department of Mathematics, West Virginia University, PO Box 6310, Morgantown, WV 26506-6310, USA Paul Erdos^ late, Mathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY Peter L. Erdos Centrum voor Wiskunde en Informatica, PO Box 4079, 1009 AB Amsterdam, The NETHERLANDS R. J. Faudree Department of Mathematical Science, Memphis State University, Memphis, TN 38152, USA Hubert de Fraysseix CNRS, EHESS, 54 Boulevard Raspail, 75006, Paris, FRANCE Alan Frieze Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA Zoltan Furedi Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Mario Gionfriddo Dipartimento di Matematica, Cittd Universitaria, Viale A, Doria 6, 95125 Catania, ITALY Michel X. Goemans Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Viatcheslav Grishukhin Central Economic and Mathematical Institute of Russian Academy of Sciences (CEMI RAN), Moscow, RUSSIA Roland Haggkvist Department of Mathematics, University of Umed, S-901 87 Umed, SWEDEN A. Hajnal Mathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY R. Halin Mathematisches Seminar der Universitat Hamburg, Bundesstrafie 55, D-20146, Hamburg, GERMANY P. L. Hammer RUTCOR, Rutgers University, New Brunswick, NJ 08903, USA

List of Contributors

xix

A. J. W. Hilton Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Reading RG6 2AX, UK Neil Hindman Department of Mathematics, Howard University, Washington, DC 20059, USA Christoph Hundack Institut fur Diskrete Mathematik, Universitdt Bonn, Nassestr. 2, 53113 Bonn, GERMANY Svante Janson Department of Mathematics, Uppsala University, PO Box 480, S-751 06, Uppsala, SWEDEN Anders Johannson Department of Mathematics, University of Umed, S-901 87 Umed, SWEDEN William M. Kan tor Department of Mathematics, University of Oregon, Eugene, OR 97403, USA A. K. Kelmans RUTCOR, Rutgers University, New Brunswick, NJ 08903, USA Daniel J. Kleitman Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA R. Klimmek c/o M. Aigner, Freie Universitdt Berlin, Fachbereich Mathematik, WE2, Arnimallee 3, 1000 Berlin 33, GERMANY Y. Kohayakawa Instituto de Matemdtica e Estatistica, Universidade de Sao Paulo, Caixa Postal 20570, 01452-990 Sao Paulo, SP, Brazil Peter Komjath Dept. Comp. Sci. Eotvos University, Budapest, Muzeum krt 6-8, 1088, HUNGARY Janos Komlos Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Imre Leader Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, UK Nathan Linial Institute of Computer Science, Hebrew University, Jerusalem, ISRAEL Tomasz Luczak Adam Mickiewicz University, Poznan, POLAND W. Mader Institut fur Mathematik, Universitdt Hanover, 30167 Hanover, Weifengarten 1, GERMANY Endre Makai Mathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY A. R. D. Mathias Peterhouse College, Cambridge, UK

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List of Contributors

Colin McDiarmid Department of Statistics, University of Oxford, Oxford, UK Patrice Ossona de Mendez CNRS, EHESS, 54 Boulevard Raspail, 75006, Paris, FRANCE Salvatore Milici Dipartimento di Matematica, Cittd Universitaria, Viale A, Doria 6, 95125 Catania, ITALY Bojan Mohar Department of Mathematics, University of Ljubljana, Jadranska 19, 61111 Ljubljana, SLOVENIA Michael Molloy Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA Jaroslav Nesetfil Department of Applied Mathematics, Charles University, Malostranske ndm. 25, 118 00 Praha 1, CZECH REPUBLIC Edward T. Ordman Memphis State University, Memphis, TN 38152, USA Janos Pach Department of Computer Science, City University, New York, USA and the Mathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY Hans Jurgen Promel Institut fur Diskrete Mathematik, Universitdt Bonn, Nassestr. 2, 53113 Bonn, GERMANY Laszlo Pyber Mathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY Pierre Rosenstiehl CNRS, EHESS, 54 Boulevard Raspail, 75006, Paris, FRANCE C. C. Rousseau Department of Mathematical Science, Memphis State University, Memphis, TN 38152, USA R. H. Schelp Department of Mathematical Science, Memphis State University, Memphis, TN 38152, USA Akos Seress The Ohio State University, Colombus, OH 43210, USA M. Simonovits Mathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY V. T. Sos Mathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY Angelika Steger Institut fur Diskrete Mathematik, Universitdt Bonn, Nassestr. 2, 53113 Bonn, GERMANY Laszlo A. Szekely University of New Mexico, Albuquerque, NM 87131, USA Endre Szemeredi Mathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY

List of Contributors

xxi

Simon Tavare Department of Mathematics, University of Southern California, Los Angeles, CA 90089-113, USA H. N. V. Temperley Thorney House, Thorney, Langport, Somerset, UK Prasad Tetali AT & T Bell Labs, Murray Hill, NJ 07974, USA Andrew Thomason DPMMS, 16, Mill Lane, Cambridge, CB2 1SB, UK Wolfgang Thumser Universitdt Bielefeld, Fakultdt fur Mathematik, Postfach 100131, 33501 Bielefeld 1, GERMANY Eberhard Triesch Forschungsinsitut fur Diskrete Mathematik, Nassestrafie 2, 5300 Bonn 1, GERMANY Zsolt Tuza Computer and Automation Institute, Hungarian Academy of Sciences, H-llll Budapest, Kende u. 13-17, HUNGARY Pavel Valtr Department of Applied Mathematics, Charles University, Malostranske ndm. 25, 118 00 Praha 1, CZECH REPUBLIC and Graduiertenkolleg Algorithmische Diskrete Mathematik', Fachbereich Mathematik, Freie Universitdt Berlin, Takustrasse 9, 14195 Berlin, GERMANY D. J. A. Welsh Mathematical Institute and Merton College, University of Oxford, Oxford, UK Herbert S. Wilf University of Pennsylvania, Philadelphia, PA 19104-6395, USA Raphael Yuster Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, ISRAEL Yechezkel Zalcstein Division of Computer and Computation Research, National Science Foundation, Washington, DC 20550, USA

Some Unsolved Problems

PAUL ERDOS1

During my long life I have written many papers on my favourite unsolved problems (see, for example, Baker et a\. [2]). In the collection below, all the problems are either new ones, or they are problems about which there have been recent developments.

Number theory 1. As usual, let us write 2 = p\ < p2 < • • • for the sequence of consecutive primes. I proved in 1934 that there is a constant c > 0 such that for infinitely many n we have c log n log log n Pn+1-Pn>

Rankin [35] proved that for some c > 0 and infinitely many n the following inequality holds: c log n log log n log log log log n Pn+l — Pn >

7j

\

\

7?

•

(1)

(log log log n)2 I offered (perhaps somewhat rashly) $10000 for a proof that (1) holds for every c. The original value of c was improved by Schonhage [38] and later by Rankin [36]. Rankin's result was recently improved by Maier and Pomerance [30]. 2. Let a\ < a2 < ' * * be an infinite sequence of integers. Denote by f(n) the number of solutions of n = at + a,-. Assume that f(n) > 0 for all n > no, i.e. (an)^=l is an asymptotic basis of order 2. Turan and I conjectured that then lim f(n) = 00

(2)

and probably lim/(n)/log n > 0. I offer $500 for a proof of (2). Perhaps (2) and lim f(n)/ log n > 0 already follow if we only assume an < en2 for all n. Let a\ < «2 < "'" and b\ < b^ < • • • be two sequences of integers such that an/bn —• 1 and let g(n) be the number of solutions of at + bj = n. Sarkozy and I conjecture that if

2

P. Erdos

g(n) > 0 for all n > no then limg(n) = oo. The condition an/bn —• 1 can not entirely be omitted but 1 — e < an/bn < 1 + 6* (e small) may suffice. 3. I proved that there is an asymptotic basis of order 2 for which c\ log n < f(n) < c2 log n

(see Halberstam and Roth [26]). I conjecture that j^--C, logn

(0 0 there is an e > 0 such that if n is sufficiently large and m ^ Cn then for every n-element set A\, ..., Am with At nAj^=0 there is a set S with \S \ cjn, c < 1, |4: n 4 / | ^ 1. Is it then true that there is a set B for which BnAt^0 but |fl H 4 I < c/ for all /? In other words, is there a set B which meets all the 4 ' s but none in many points? 8. Here is a problem of Jean Larson and myself [19]. Is it true that there is an absolute constant c so that for every n and \Sf\ = n there is a family of subsets A\9 • • •, Am of y , | 4 | > ft1^2 ~~ c? 14 ^ ^ / l ^ 1 a n d every x, j; G y is contained in some 4 ? Shrikhande and Singhi [39] have proved that every pairwise balanced design on n points in which each block is of size ^ n^ — c can be embedded in a projective plane of order n + i for some i ^ c + 2 if n is sufficiently large. This implies that if the projective plane conjecture (that the order of every projective plane is a prime power) is true then the Erdos-Larson conjecture is false. But the problem remains for which functions h(n) will the condition |4I > ^ 5 — h(n) make the conjecture true?

Graph theory 9. I offer $500 for a proof or disproof of the following conjecture of Faber, Lovasz and myself. Let G\, ..., Gn be complete graphs (each on n vertices), no two of which have an edge in common. Is it then true that x(U?=i ^ ) ^ n ? Jeff Kahn [27] recently proved that the chromatic number is n + o(ri). 10. Is it true that every triangle-free graph on 5n vertices can be made bipartite by the omission of at most 5n2 edges? Is it true that every triangle-free graph on 5n vertices can contain at most n5 pentagons? Ervin Gyori [25] proved this with l.O3n5. Gyori now proved n5 for n > no. One could ask more generally: Assume that the number of vertices is (2r + l)n and that the smallest odd cycle has size 2r + 1. Is it then true that the number of cycles of size 2r 4-1 is at most n2r+l ? 11. Let H be a graph and let Gn be a graph on n vertices which does not contain H as an induced subgraph. Hajnal and I [13] asked whether there is an absolute constant c = c(H) such that Gn contains either a complete graph or an independent set on nc vertices? If H is C4 then | ^ c < ^. 12. Let Qn be the graph of the n-dimensional cube {0,1}". I offered $100 for a proof or disproof of the conjecture that for every e > 0 there is an no such that, for n > no, every subgraph of Qn with at least (\ + e)e(Qn) edges contains C4. It is easy to find subgraphs with more than \e(Qn) edges and no C4; Guan (see Chung [9]) has constructed an example with (1 + o(l))(n + 3)2n~2 edges. Chung has given an upper bound of (a + o(l))n2" -1 , where a « 0.623. I also conjectured that every subgraph of Qn with ee(Qn) edges contains a Ce, for n sufficiently large. Chung [9] and Brouwer, Dejter and Thomassen [7] disproved this by constructing an edge-partition of Qn into four subgraphs containing no C^. 13. Suppose that G is a graph of order n with the property that every set of p vertices spans at least q edges. We let H(n;p,q) be the largest integer such that G necessarily contains a clique of that order. In the case where q = 1 this corresponds to the standard

4

P. Erdos

finite Ramsey problem: the condition is precisely that G contains no independent set of size p. Faudree, Rousseau, Schelp and I investigated the behaviour of H(n;p, q) as a function of n. We set (log H(n;p,qy c(P>1)) ^ cvS9 so c(p,(p^1)) < 1/2. 14. For e > 0, Rodl [37] constructed graphs with chromatic number Ko such that every subgraph of order n can be made bipartite by omitting en edges, for every n; another construction was given by Lovasz. Now let f(n) —> oo as slowly as we please. Is there a graph of chromatic number No such that every subgraph of n vertices can be made bipartite by omitting f(n) edges? Perhaps for every e > 0, there is a graph with chromatic number Ki for which every subgraph of order n can be made bipartite by omitting en edges, but this seems unlikely and I would guess that there is a subgraph of size n which cannot be made bipartite by omitting nh(n) edges, where h(n) —> oo. But perhaps h(n) does not have to tend to infinity fast. See also the paper with Hajnal and Szemeredi [17]. Hajnal, Shelah and I [16] proved that if G has chromatic number Ki then for some rc0 it contains a cycle of length n for every n > no. Now if F(n) tends to infinity sufficiently fast, then is it true that every graph of chromatic number Ki has a subgraph on at most F(n) vertices with chromatic number n, for all n sufficiently large? Geometry 15. Let xi, ..., xn be n distinct points in the plane, and let s\ ^ 52 ^ • • • ^ Sk be the multiplicities of the distances they determine, so

I conjectured [12] that k

J^sj 0. The lattice points show that we must have a ^ 1. In forthcoming papers Fishburn and I conjecture that if xi, ..., xn form a convex set

Some Unsolved Problems

5

then (3) can be improved to k

J2j ? < Cn3+c/lo^°^n. Is it true that the number of incongruent sets of n points with f(n) unit distances exceeds one for n > 3 and tends to infinity with w? Leo Moser and I conjectured that if xi, ..., xn is a convex n-gon then A(xu •-., xn) < en.

(6) 3

Furedi [22] proved that A(x\, • • •, xn) < en log n; this gives an upper bound of en log n in (4). The inequality (4) would follow from (6). 16. Let xi, ..., xn be n distinct points in the plane. Denote by Fk(n) the maximum number of distinct lines passing through at least k of our points and by fk(n) the maximum number of lines passing through exactly k of our points. Clearly fk(n) ^ Fk(n). Determine or estimate fk(n) and Fk(n) as well as possible. Trivially fi{ri) = F2(n) = (JJ). The problem with k = 3 is the Orchard problem, and really goes back to Sylvester. Burr, Grimbaum and Sloane [8] proved that 2

2

/3(w) = ^ - O ( n ) o

and

F3(n) = \ - O(n). o

Determine lim,i_>00 Fk(n)/n2 and limn_KX)//c(rc)/n2, if they exist. The upper bound Fk(n) ^ (2)7(2) follows from an obvious counting argument; a lower bound can be obtained by considering a rectangle of k by n/k lattice points. Are the limits attained by the lattice points? 17. Let f(n) denote the minimum number of distinct distances among a set # = (x,-)" of points in the plane. In 1946 [12] I proved that 4 ~~ 2 and conjectured that the upper bound gave the true order of f(n). So far, the best lower bound is n^(\ogn)~c, due to Chung, Szemeredi and Trotter [10]. The question also arises whether, in general, a particular point of the configuration is associated with a large number of distances. I conjecture that in any configuration there is some point with at least cn/^Jlogn distinct distances to other points. In fact this may be true for all but a few of the points. Altman [1] showed that if # is convex then there are at least n/2 distinct distances

6

P. Erdos

between the points; I conjecture that there is some point associated with at least n/2 distinct distances. Szemeredi conjectured there are at least n/2 distinct distances among the points of # provided only that # has no three points collinear, but could only prove this with a bound of w/3. 18. Consider two configurations ^ = (x,)", *€' = (j^)", and define F(2n) to be the minimum over all # and

Yi+...

+ YK

Now let k = \yjri]. Roughly, we get

so xn ^ n2(l + o(l)) as desired. Analysis 21. We let / = [—1,1] and supose / : / —• 1R is a continuous function which we wish to approximate by a polynomial. Suppose we are given, for each 1 ^ n < oo,

so we have a triangular matrix X = (xjn|). Let /,'"' be the unique polynomial of degree

Some Unsolved Problems n — 1 satisfying /(»>(*(»)) = 1 a n d l l n \ x { f ) = O

if 7 ^ 1

so

Then we denote by «£?„(/, X), or simply j£?n(/), the polynomial given by

1=1

so this is the unique polynomial of degree n — 1 agreeing with / on x^\.. x%\ It is known that for certain choices of X, if/ is of bounded variation then if n (/)(x) —• /(x) uniformly. However, for more general continuous / the behaviour is not so good and, as we now describe, a number of authors have examined how bad this behaviour can be. With a fixed choice of X, we can regard £?n as a linear map from C(I) to itself. Let us write down its norm. Let

Then we easily see that rr so if we let Xn = max Xn(x), then \\&n\\=kn. _ Faber [21] proved that for any choice of X, lim^oo^ = oo. It therefore follows from the Principle of Uniform Boundedness that there exists an / with lim^oc ||if w (/)||= oo. This result was strengthened by Bernstein [4] who showed that for any X, there exist / G C [ - l , 1] and x 0 G [-1,1] such that lim ||J2\,(/)(x 0 )||=oo, n—>oo

i-e.

\imXn(x0) = 00. rc—•oo

In several papers (Bernstein [3], Grlinwald ([23], [24]), Marcinkiewicz [31] and Privalov ([32], [33])) it was shown that for particular choices of X, this kind of bad behaviour can occur almost everywhere and, in certain cases, everywhere. In 1980 Vertesi and I [20] showed that given any X, there exists an / with hm ||J£\,(/)(x)||= 00 for almost all x. n—+oo

Certainly this result cannot be extended from almost all x to all x. For example, if xo appears in all but finitely many rows of X - i.e. is equal to some x) for all n ^ no then we have j£?n(/)(xo) = /(xo) for n ^ no. Does there, however, exist an X, such that for every / , there is some point xo where divergence would be possible, i.e. where oo

yet

n—>oo

22. Let f(z) = zn + ... be a monic polynomial of degree n.

8

P. Erdos

Is it true that the length of {z e C : | /(z) |= 1} is maximal in the case when /(z) = zn—1 ? This problem was posed, along with many others, in my paper with Herzog and Piranian [18]. 23. Let \zn\ = 1(1 ^ n < oo). Put

k=\

and Mn = max|/ n (z)|. Is it true that limMw = oo? This conjecture was settled by Wagner: he proved that there is a c > 0 such that Mn > (log nf holds for infinitely many values of n. I further conjectured that Mn > nc for some c > 0 and infinitely many n and, in fact, for every n we have n

Y,Mk>nl+c.

(7)

/c=l

Inequality (7), if true, may very well be difficult, so I offer $100 for a solution. 24. Let xi,X2,... be a sequence of real numbers tending to 0. We call (yn)%L\ similar to (x«)S=i ^ y« = a x n + b f° r some a, b G IR and all n. Is it true that there is a set £ c R of positive measure which contains no subsequence (yn)^Li similar to (xn)™=ll Komjath proved that if xn —>• 0 slowly (xn > c/n) then there is a set of positive measure which contains no subsequence similar to (xn)™=1.

Set theory 25.1 have not included our many problems on set theory with Hajnal since undecidability raises its ugly head everywhere and many of our problems have been proved or disproved or shown to be undecidable (this happened most often). However, I think that the following simple problem is still open. Let a be a cardinal or ordinal number or an order type. Assume a —• (a, 3)2. Is it then true that, for every finite n, a —> (a, n)2 also holds? Here a —> (a, n)2 is the well-known arrow symbol of Rado and myself: if G is a graph whose vertices form a set of type a then either G contains a complete graph Kn or an independent set of type a. See Erdos, Hajnal and Milner [15] and Erdos, Hajnal, Mate and Rado [14]. Group theory 26. Let G be a group. Assume that it has at most n elements which do not commute pairwise. Denote by h(n) the smallest integer for which any such G can be covered by h(n) Abelian subgroups. Determine or estimate h(n) as well as possible. Pyber [34] proved that

for some positive constants c\ and c^. The lower bound was already known to Isaacs.

Some Unsolved Problems

9

References [I] Altman, (1963), On a problem of P. Erdos, Amer. Math. Monthly 70 148-157. [2] Baker, A., Bollobas, B. and Hajnal, A. eds. (1990), A Tribute to Paul Erdos, Cambridge University Press, xi;-f-478pp. [3] Bernstein, S. (1918), Quelques remarques sur Interpolation, Math. Ann. 79 1-12. [4] Bernstein, S. (1931), Sur la limitation des valeurs d'un polynome, Bull. Acad. Sci. de I'URSS 8 1025-1050. [5] Bollobas, B. (1985), Random Graphs, Academic Press, xjt;+447pp. [6] Borwein, P. B. (1991), On the irrationality of £ ( l / f a n + r)), J. Number Theory 37 253-259. [7] Brouwer, A. E., Dejter, I. J. and Thomassen C. (1993), Highly symmetric subgraphs of hypercubes (preprint). [8] Burr, S.A., Griinbaum, B. and Sloane, N.J.A. (1974), The orchard problem, Geom. Dedicata 2 397-424. [9] Chung, F. R. K. (1992), Subgraphs of a hypercube containing no small even cycles, J. Graph Theory 16 273-286. [10] Chung, F.R.K., Szemeredi, E. and Trotter, W. (1992), The number of different distances determined by a set of points in the Euclidean plane, Discrete and Computational Geometry 1 1-11. [II] Erdos, P. (1935), On the difference of consecutive primes, Quart. J. Math. Oxford 6 124-128. [12] Erdos, P. (1946), On sets of distances of n points, Amer. Math. Monthly 53 248-250. [13] Erdos, P. and Hajnal, A. (1989), Ramsey-type theorems, Discrete Applied Math. 25 37-52. [14] Erdos, P., Hajnal, A., Mate, A. and Rado, R. (1984), Combinatorial Set Theory: Partition Relations for Cardinals, North-Holland Publishing Company, Studies in Logic and the Foundations of Mathematics, Vol. 106. [15] Erdos, P., Hajnal, A. and Milner, E. C. (1966), On the complete subgraphs of graphs defined by systems of sets, Acta Math. Acad. Sci. Hungaricae 17 159-229. [16] Erdos, P., Hajnal, A. and Shelah, S. (1974), On some general properties of chromatic numbers, in Topics in Topology (Proc. Colloq. Keszthely, 1977) Colloq. Math. Soc. J. Bolyai 8 243-255. [17] Erdos, P., Hajnal, A. and Szemeredi, E. (1982), On almost bipartite large chromatic graphs, Annals of Discrete Math. 12, Theory and Practice of Combinatorics, Articles in Honor of A. Kotzig (A. Rosa, G. Sabidussi and J. Turgeon, eds.), North-Holland, 117-123. [18] Erdos, P., Herzog, F. and Piranian, G. (1958), Metric properties of polynomials, Journal d Analyse Mathematique 6 125-148. [19] Erdos, P. and Larson, J. (1982), On pairwise balanced block designs with the sizes of blocks as uniform as possible, Annals of Discrete Mathematics 15 129-134. [20] Erdos, P. and Vertesi, P. (1980), On the almost everywhere divergence of Lagrange Interpolatory Polynomials for large arbitrary systems of nodes, Acta Math. Acad. Sci. Hungaricae 36 71-89. [21] Faber, G. (1914), Uber die interpolatorische Darstellung stetiger Funktionen, Jahresber. der Deutschen Math. Ver 23 190-210. [22] Fliredi, Z. (1990), The maximum number of unit distances in a convex n-gon, J. Comb. Theory (Ser. A) 55 316-320. [23] Griinwald, G. (1935), Uber die Divergenzersheinungen der Lagrangeschen Interpolationpolynome, Acta. Sci. Math. Szeged 1 207-221. [24] Griinwald, G. (1936), Uber die Divergenzersheinungen der Lagrangeschen Interpolationpolynome stetiger Funktionen, Annals of Math. 37 908-918. [25] Gyori, E. (1989), On the number of C5's in a triangle-free graph, Combinatorica 9 101-102. [26] Halberstam, H. and Roth, K.F. (1983), Sequences, Springer-Verlag, xiii+290pp. [27] Kahn, J. (1992), Coloring nearly-disjoint hypergraphs with n + o(n) colors, J. Combinatorial Theory (Ser. A) 59 31-39. [28] Kahn, J. (1993), On a problem of Erdos and Lovasz. II n(r) = o(r), J. Amer. Math. Soc. 14.

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P. Erdos

[29] Kanold, (1981), Uber Punktmengen im /c-dimensionalen euklidischen Raum, Abh. Braunschweig. wiss. Ges. 32 55-65. [30] Maier, H. and Pomerance, C. (1990), Unusually large gaps between consecutive primes, Trans. Amer. Math. Soc. 322 201-238. [31] Marcinkiewicz, J. (1937), Sur la divergence des polynomes d'interpolation, Ada Sci. Math. Szeged 8 131-135. [32] Privalov, A. A. (1976), Divergence of Lagrange interpolation based on the Jacobi abscissas on sets of positive measure, Sibirsk. Mat. Z. 18 837-859 (in Russian). [33] Privalov, A. A. (1978), Approximation of functions by interpolation polynomials, in Fourier analysis and approximation theory I—II, North-Holland, Amsterdam 659-671. [34] Pyber, L. (1987), The number of pairwise non-commuting elements and the index of the centre in a finite group, J. London Math. Soc. 35 287-295. [35] Rankin, R. A. (1938), The difference between consecutive prime numbers, J. London Math. Soc. 13, 242-247. [36] Rankin, R. A. (1962), The difference between consecutive prime numbers. V, Proc. Edinburgh Math. Soc. 13 331-332. [37] Rodl, V. (1982), Nearly bipartite graphs with large chromatic number, Combinatorica 2 377-387. [38] Schonhage, A. (1963), Eine Bemerkung zur Konstruktion grosser Primzahllucken, Arch. Math. 14 29-30. [39] Shrikhande, S. S. and Singhi, N. M. (1985), On a problem of Erdos and Larson, Combinatorica 5 351-358. [40] Spencer, J., Szemeredi, E. and Trotter, W. (1984), Unit distances in the Euclidean plane, Graph Theory and Combinatorics, Academic Press, London 293-303.

Menger's Theorem for a Countable Source Set

RON AHARONI + and REINHARD DIESTEL* +

Department of Mathematics, Technion, Haifa 32000, Israel * Mathematical Institute, Oxford University, Oxford OX1 3LB, England

Paul Erdos has conjectured that Menger's theorem extends to infinite graphs in the following way: whenever A, B are two sets of vertices in an infinite graph, there exist a set of disjoint A-B paths and an A-B separator in this graph such that the separator consists of a choice of precisely one vertex from each of the paths. We prove this conjecture for graphs that contain a set of disjoint paths to B from all but countably many vertices of A. In particular, the conjecture is true when A is countable.

1. Introduction

If there is any conjecture in infinite graph theory whose fame has clearly transcended the boundaries of the field, it is the following infinite version of Menger's theorem, conjectured by Erdos: Conjecture 1.1. (Erdos) Whenever A,B are two sets of vertices in a graph G, there exist a set of disjoint A-B paths and an A-B separator in G such that the separator consists of a choice of precisely one vertex from each of the paths. Here, G may be either directed or undirected and either finite or infinite, and 'disjoint' means 'vertex disjoint'. If G is finite, the statement is clearly a reformulation of Menger's theorem. A set of A-B paths together with an A-B separator as above will be called an orthogonal paths/separator pair. We remark that the naive infinite analogue to Menger's theorem, which merely compares cardinalities, is considerably weaker and easy to prove. Indeed, consider any inclusionmaximal set & of disjoint A-B paths. If ^ can be chosen infinite, (J ^ , which is trivially an A-B separator, still has size only \g?\. If not, choose & of maximal (finite) cardinality, and there is a simple reduction to the finite Menger theorem [5]. This was in fact first observed by Erdos, and seems to have inspired his above conjecture as the 'true' generalization of Menger's theorem.

12

R. Aharoni and R. Diestel

Although Erdos's conjecture has been proved for countable graphs [2], a full proof still appears to be out of reach. However, no other conjecture in infinite graph theory has inspired as interesting a variety of partial or related results as this has; see [4] for a survey and list of references. The main aim of this paper is to prove a lemma, which, in addition to implying (with [2]) the results stated in the abstract, might play a role in an overall proof of the conjecture by induction on the size of G. Briefly, the lemma implies that if the conjecture is true for all graphs of size K, where K is any infinite cardinal, then it is true also for arbitrary graphs, provided the source set A is no larger than K. (In particular, we see that the conjecture holds for any graph if A is countable.) Now if \A\ = \G\= X and the conjecture holds for all graphs of size < 2, the lemma enables us to apply the induction hypothesis to G with A replaced by its smaller subsets A'; we may then try to combine the orthogonal paths/separator pairs obtained between these A' and B to one between A and B. We must point out, however, that such a proof of Erdos's conjecture will be by no means straightforward, and it is not the only possible approach. 2. Definitions and statement of the main result All the graphs we consider will be directed; undirected versions of our results can be recovered in the usual way by replacing each undirected edge with two directed edges pointing in opposite directions. An edge from a vertex x to a vertex y will be denoted by xy. When G is a graph, G denotes the graph obtained from G by reversing all its edges. Paths, likewise, will be directed, and we usually refer to them by their vertex sequence. If P = x... y is a path and v9 w are vertices on P in this order, vPw denotes the subpath of P from v to w. Similarly, we write Pv and vP for initial and final segments of P, Pv for Pv — v, vP for vP — v, and so on. If Q = y... z is another path, and P n Q = {y}, then PyQ denotes the path obtained by concatenating P and Q. Let X, Y be sets of vertices in a graph. An X-Y path is a path from X to Y whose inner vertices are neither in X nor in Y. If x is a vertex, a set of {x}-Y paths that are disjoint except in x is an x-Y fan; the fan is onto if every vertex in Y is hit. Similarly, a set of X-y paths that are disjoint except in y is an X-y fan. A warp is a set of disjoint paths. When W is a warp, we write V[iV] for the set of vertices of the paths in 1V, and E \iV\ for the set of their edges. Similarly, we write in \W~\ for the set of initial vertices of the paths in #", and ter [W] for the set of their terminal vertices. For a vertex x G V\iV\ we denote the path in W containing x by r(x), or briefly Q(x). For x £ V[i^]9 we put Qir(x) := {x}. A warp consisting of A-B paths is an A-B warp. By "IV we denote the warp in *G consisting of the reversed paths from W*

' Clearly, ?F = iV. We shall use this fact as an excuse to denote warps in *G, if they are introduced afresh rather than being obtained from a warp in G, by iV etc. straight away; their reversals in G will then be denoted by iV. The idea here is to avoid the counter-intuitive practice of having a warp iV in *G and a resulting warp iV in G. This convention, if not its explanation, should help the reader avoid any warps in his or her intuition when such things are discussed briefly in Section 5.

Menger's Theorem for a Countable Source Set

13

Let G = ( F , £ ) be a graph and A,B c 7. Any such triple T = (G,A,B) will be called a web. The web (G, B,A) is denoted by 1 . An A-B warp ^ with in [W] = A is a linkage in F, and F is linkable if it contains a linkage. A set S ^ F separates A from 5 in G if every path in G from A to B meets S. Note that A and 5 may intersect, in which case clearly Af)B ^ S. A warp W in G is called a wai;e in F if F[#"] n A = in [W] and ter[W] separates A from J? in G. The wave {(a) | a e A} is called the trivial wave. If ^ is a wave in F, then F / # ^ denotes the web (G-(A\in[W])-(V[W]\ter[W]),

ter[W],

*)•

In every web F = (G,,4,£) there is a wave T^* such that T/W has no non-trivial wave. (This is not difficult to see. If Wo is a wave in F and W\ is a wave in F/Wo, then # 1 defines a wave in F in a natural way: just extend its paths back to A along the paths of Wo. This wave in F is 'bigger' than Wo, and chains of waves in F with respect to this order tend to an obvious limit wave W, which consists of the paths that are eventually in every wave of the chain. If the chain was maximal, then Y/W has no non-trivial wave. See [2] for details.) A wave W in F is a hindrance if A\in [W] ^=0; if F contains a hindrance, it is called hindered. Note that every hindrance is a non-trivial wave. The following was observed in [2]: Erdos's conjecture is equivalent to the assertion that every unhindered web is linkable. We are now in a position to state the main result proved in this paper. (For the reasons explained earlier, and because it is of a technical nature, we call it a lemma, not a theorem.) Lemma 2.1. Let F = (G,A,B) be a web and / > \B\ter[f]\, then F is hindered.

an A-B warp in G (possibly empty). If

Lemma 2.1 will be proved in Sections 3 and 4. Our aim will be to turn the given warp f, step by step, into a hindrance. This will require some alternating path techniques; the definitions and lemmas needed are given in Section 3. Section 4 is devoted to the main body of the proof of Lemma 2.1. In Section 5 we look at the implications of the lemma for Erdos's conjecture.

3. Aternating paths Let F = (G,A,B) be a web, and let f be an A-B warp in G. A finite sequence P = xoeoxi^i ...en-\xn of not necessarily distinct vertices xt and distinct (directed) edges et of G will be called an alternating path (with respect to f) if the following three conditions are satisfied: (i) for every i < n, either e, = x,-x,-+i G E(G)\E[f] or et = x;+ix; G E[f]; (ii) if x,- = Xj for i ± j9 then xt e V[f]; (iii) for every i, 0 < i < n, if x,