AND PROBABILITY
A tribute to Paul Erdos
Edited by
BELA BOLLOBAS
ANDREW THOMASON
_ CAMBRIDGE
UNIVERSITY PRESSCAMBRIDGE 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
Preface Page ix Farewell to Paul Erdos xi Toast to Paul Erdos xiii List of Contributors 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
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
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 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.
(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 mathe-maticians 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
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
(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
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.
"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 a field with 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
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.
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
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
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
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
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
P A U L E R D O S1
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
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 , ( 0 < C < o o ) , logn
is not possible and I offer $500 for a proof or disproof of this conjecture. Sarkozy and I proved that
logn cannot hold.
4. Is it true that
is irrational? I conjectured that
2n - 3
is irrational. This assertion and its generalizations have been proved by Peter Borwein [6]. Denote by co(n) the number of distinct prime factors of n. Is it true that
E
cojn)is irrational?
5. Is it true that if n ^ 0 (mod 4) then there is a squarefree natural number 0 such that n = 2k + 61 I could only prove that almost all integers n ^ 0 (mod 4) can be written in the form 2k + 9.
Combinatorics
6. Let m = m(n) be the smallest integer for which there are n-element sets A\, ..., Am such
that AtnAj ^ 0 for all 1 ^ i < j ^ m, and such that every set S with at most n—\ elements
is disjoint from some At. (Note that the lines of finite geometry have this property.) I
conjectured with Lovasz that m(n)/n —• oo, but it is not even known whether m(n) > 3n if n sufficiently large. In the other direction, we could prove only that m(ri) < m+e, but Jeff Kahn [27] very recently proved m(n) < en.
Perhaps more is true: for every C > 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 \<n(l—e) which meets all At.
7. Is it true that in a finite geometry there is always a blocking set which meets every line in at most c points where c is an absolute constant?
More generally: Let \Sf\ = n, A\, ..., Am be a family of subsets of y , |4I > 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 nd 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
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><l) = lim :
t=^o \ log n
Standard bounds on Ramsey numbers (see, for example, Bollobas [5]) tell us that
We conjecture that with p fixed, c(p,q) is a strictly increasing function of q for 1 ^ q < (p^1) + 1 . It is easy to see that if g = (p^1) + 1 then c(p, g) = 1, which is as large as possible. For in this case, the complement of G cannot contain any connected subgraphs of size p, so all components of the complement have size less than p. Hence the complement contains at least n/(p — 1) independent vertices so G contains a clique of size at least n/(p-\). On the other hand, we have shown that H(n;p,(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 <cn
3(\ognf (3)
/=i
for some a > 0. The lattice points show that we must have a ^ 1.
then (3) can be improved to
k
J2j<cn\ (4)
and that ^ sj is maximal for the regular n-gon, for n ^ 8.
A weaker inequality than (3) would follow easily from the following conjecture. Let A(x\, ..., xn) be the number of pairs x,-,x;- whose distance is 1, and let f(n) be the
maximum A(x\, ..., xn) over all sets of n distinct points in the plane. I conjecture that
f(n) < n{+c/lozl°zn. (5) The best bound found to date is due to Spencer, Szemeredi and Trotter [40], who proved f(n) < en5. It would follow from (5) that $ > ? < 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)
Furedi [22] proved that A(x\, • • •, xn) < en log n; this gives an upper bound of en3 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 ) and F3(n) = \ - O(n).
o 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.
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 <€' of the number of distinct distances || xt — yt ||. Is F identically 1 in four
dimensions (in this case f(2n) > n€)l How does f/F behave at dimension 2 or 3? Do we have f/F —• oo or is the ratio bounded? Possibly it is unbounded in ]R3 but bounded in 1R2. 19. Let ^ be a set of points in the plane such that distinct distances between the points always differ by at least 1. I conjecture that the diameter of ^ is at least n — \ provided n is large enough. Note that if # = ((J,0))"=1 we obtain equality. However for n < 10 some
configurations have diameter less than n — 1.
The best result in this direction so far is due to Kanold [29], who proved that
20. Let ^ be a set of n points in Euclidean space among which all distances differ by at least 1. A conjecture independent of dimension is that diam^ ^ (1 + o(\))n2. Clearly diam^ is always at least g).
The conjecture is settled only for ^ cz ]R (not even for ]R2). To prove it for ]R, let ^ = {x\)\ with 0 = x\ < X2 < ... < xn. Further, let ykj = x,-+fc — x, and let
Yk = YMII yk,i = xn + xn-\ + . . . + xn-t+i < kxn. Because the y^t are all distinct (even over
/c), we have ^Y s y ± y ^ , ' 2 n - 2 5Xn > I\ -h 12 and for k ^ n/2, 4- 1 \ n> 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,
n — 1 satisfying
/(»>(*(»)) = 1 a n d l ln\ x{f ) = O i f 7 ^ 1 s o
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 ifn(/)(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 ||ifw(/)||= oo.
This result was strengthened by Bernstein [4] who showed that for any X, there exist / G C [ - l , 1] and x0 G [-1,1] such that
lim ||J2\,(/)(x0)||=oo, i-e. \imXn(x0) = 00. n—>oo 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
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>n
l+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 xn + 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
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 an + r)), J. Number Theory 37 253-259. [7] Brouwer, A. E., Dejter, I. J. and Thomassen C. (1993), Highly symmetric subgraphs of
hyper-cubes (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
Rela-tions for Cardinals, North-Holland Publishing Company, Studies in Logic and the FoundaRela-tions 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 Interpolationpoly-nome, Acta. Sci. Math. Szeged 1 207-221.
[24] Griinwald, G. (1936), Uber die Divergenzersheinungen der Lagrangeschen Interpolationpoly-nome 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.
[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.
Source Set
RON AHARONI+ and R E I N H A R D 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 inclusion-maximal 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.
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 <2>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.
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 / an A-B warp in G (possibly empty). If
> \B\ter[f]\, then F is hindered.
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];
Figure 1 Two alternating paths with respect to £
All the alternating paths we consider in this section will be alternating paths in G with respect to / . Note that, by (iii) above, an alternating path starting at a vertex of / has its first edge in / . As the edges of an alternating path are pairwise distinct, it can visit any given vertex at most twice, and this happens in essentially only two ways: if x, = Xj for some i < j < n, then x,- £ V[f] by (ii), so by (iii)
either ei-\,ej e E[f] and e,-,e/_i ^ E[f] (Figure 1 left) or eueHX e £ [ / ] and <?/_!,£/ £ £ [ / ] (Figure 1 right).
Note that initial segments of alternating paths are again alternating paths, but final segments need not be. Finally, an ordinary path which avoids J* or meets it only in its last vertex is trivially an alternating path.
There are analogous alternating versions of the notions of X-Y path, X-y fan and so on.
Lemma 3.1. If a e A\in[f] and b e B\ter[/], and if P = a...b is an alternating path with respect to /, then G contains an A-B warp /' such that in [f] = in [/] U {a} and
ter[f] = ter[f] U {b} and {Q e / | P nQ = 0} c / ' .
Proof. Consider the graph on V\J\ U V(P) whose edge set is the symmetric difference of E[f] and E(P). The (undirected) components of this graph are all finite. Considering their vertex degrees, we see that they are either A-B paths or cycles avoiding Au B (possibly trivial). The assertion follows. •
Lemma 3.2. Let P\ = xo^o • • • Cn-i^n and Pi = yo/o • • -fm-iym be alternating paths. If xn =
yo, then there exists an alternating path P^ from xo to ym such that V(P^) <= V(P\)U
Proof. Let i < n be minimal such that there exists a j < m with the following two properties:
(ii) if Xi G V[/l then either e^x G E[f] or / , G £ [ / ] .
(Note that such an i exists, because xn = yo and P2 is an alternating path. Moreover, j is
easily seen to be unique.) Then xo^o• • • ei-ixtfj• --fm-iym is an alternating path as desired.
•
4. Proof of the main lemma
We now prove Lemma 2.1. As in the lemma, let F = (G,A,B) be a web, and let / be an A-B warp in F. Let us write
A\ := in [/] and A2 :=
and
#! := ter[/] and B2 :=
and put /c := |#2|. We assume that \A2\ > K, and construct a hindrance #^ in F. Again, all
the alternating paths considered in this section will be alternating paths in G with respect to f, unless otherwise stated.
Let us quickly dispose of the case when K is finite. Assume that K is minimal such that the lemma fails. By Lemma 3.1 and the minimality of K, there is no alternating path from A2 to B2. For each path Q e </, let x(Q) denote the last vertex of Q that lies on some
alternating path starting in A2\ if no such vertex exists, let x(Q) be the initial vertex of Q.
We claim that
^ := {Q* I Q e / and x = x(Q)}
is a wave in G; since \A2\ > K > 0 and hence in ]iV\ = in \J\ g A, this wave #^ will be a
hindrance and the lemma will be proved.
To show that iV is a wave, we have to prove that ter[if] separates A from B. So let P be any A-B path. Since P is not an alternating path from A2 to B2, it meets V[f]
and hence V[i^]; let y be its last vertex in K[#"], and write Q := Q/(y) and x := x(Q). Suppose P avoids ter[i^]. Then x ^ y, so there exists an alternating path R from A2 to y
that ends with an edge of iV. (Indeed, by definition of #^, there is an alternating A2-x
path B!\ if x! is the first vertex of B! on yQx, then K V followed by yQx! in reverse order is an alternating path from A2 to y.) By definition of #", # avoids V[/]\V[^] and is
thus an alternating path with respect to if. Let z be the first vertex of R on yP. Then either z = j ; or z ^ V[iT], so KzP is again an alternating path with respect to #". By definition of TT, i^zP avoids V[/]\V[iT]. Therefore zP avoids F [ / ] \ { y } 3 Bi. (Recall that y £ B, because y ^ x and hence y G Qx.) Thus KzP is an alternating path from A2
to B2, a contradiction.
We shall now assume that K is infinite. To motivate our proof, let us consider the (much easier) case of / = 0. (This is an important special case, and we recommend that the reader remain aware of it throughout the proof of Lemma 2.1.) Assume Erdos's conjecture
