Table Of ContentA
Algorithms and Combinatorics 5
Editorial Board
R. L. Graham, Murray Hill B. Korte, Bonn
L. LOV3sZ, Budapest
laroslav Nesetfil Vojtech Rodl
(Eds.)
Mathematics of
Ramsey Theory
With 15 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Jaroslav Nesetfil
Department of Applied Mathematics
Charles University
Malostranske mim. 25
11800 Praha 1
Czechoslovakia
Vojtech Rodl
Department of Mathematics
and Computer Science
Emory University
Atlanta, GA 30322
USA
Mathematics Subject Classification (1980): 00-02, OOA 10,03 F30, 05 A99,
05 B25, 05 C 15,05 C55, 05 C65, 05 C80, 11 B 13,11 B25, 28 D99, 34C35,
51E1O, 51E15, 51M99, 54H20, 68R1O
ISBN- 13: 978-3-642-72907-2 e-ISBN -13: 978-3-642-72905-8
DOl: 10.1007/978-3-642-72905-8
Library of Congress Cataloging-in-Publication Data
Mathematics of Ramsey theory / 1. Nesetfil, V. Rodl (eds.) p. cm. - (Algorithms and
combinatorics; 5)
Includes bibliographical references (p. ) and indexes.
ISBN-13: 978-3-642-72907-2
1. Ramsey theory. I. Nesetfil, J. (Jaroslav) II. Rodl, V. (Vojtech), 1949-. III. Series.
QA166.M37 1990 511'.5-dc20 90-10186 CIP
This work is subject to copyright. All rights are reserved, whether the whole or part of the
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visions of the German Copyright Law of September 9,1965, in its current version, and a
copyright fee must always be paid. Violations fall under the prosecution act of the German
Copyright Law.
© Springer-Verlag Berlin Heidelberg 1990
Softcover reprint of the hardcover 1st edition 1990
2141/3140-543210 - Printed on acid-free paper
Preface
During the last Prague Symposium on Graph Theory several people sugge
sted collecting papers which would exhibit diverse techniques of contemporary
Ramsey Theory.
The present volume is an outgrowth of this idea. Contemporary research
related to Ramsey Theory spans many and diverse areas of mathematics and
it has been our intention to demonstrate it. We decided not merely to collect
papers but also to edit the volume as a whole. In several instances we asked
for specific contributions.
Admittedly this was a bit ambitious project and as a result it took us
several years to complete it. But perhaps the time was worth it: we are pleased
that we have among the contributors many leading mathematicians.
We thank all the authors for the excellent job they have done.
We thank Jan KratochVl1 and Yvonne Hola. (Charles University, Pra
gue) for technical assistance in editing this volume. The book has been TEX
processed in the Institut fur Diskrete Mathematik (Bonn) and we thank
Mrs. H. Higgins and B. Schaschek.
Bonn, March 13, 1990 Jaroslav Nesetfil
Vojtech Rodl
Contributing Authors
J6zsef Beck Hillel Furstenberg
Mathematical Institute Institute of Mathematics
Budapest H-1364 Hebrew University
Hungary Givat Ram, Jerusalem 91904
Israel
Stefan A. Burr
Dept. of Computer Sciences
Ronald L. Graham
City College, C.U.N.Y. AT & T Bell Laboratories
New York, NY 10031 Room 2T-I02
U.S.A.
Murray Hill, NJ 07974
U.S.A.
Timothy J. Carlson
Department of Mathematics
Yitzchak Katznelson
Ohio State University
Institute of Mathematics
Columbus, OH 43210
Hebrew University
U.S.A.
Givat Ram, Jerusalem 91904
Israel
Paul Erdos and
Mathematical Instittit Department of Mathematics
Hungarian Academy of Sci. Stanford University
Budapest H-1364 Stanford, CA 92305
Hungary U.S.A.
Peter Frankl Igor Kfii
Centre National de la Dept. of Mathematics
Recherche Scientifique University of Chicago
75700 Paris Chicago, IL 60637
France U.S.A.
VIII Contributing Authors
J aroslav N eseti'il Richard Rado
Dept. of Applied Mathematics Dept. of Mathematics
Charles University University of Reading
Malostranske na.m. 25 White Knights
11800 Praha 1 Reading RG6 2AX
Czechoslovakia England
Vojtech Rodl
Alon Nilli
Department of Mathematics
c/o N. Alon Emory University
Dept. of Mathematics
Atlanta, GA 30322
Sackler Faculty of Exact
U.S.A.
Sciences
Tel Aviv University
Tel Aviv
Israel Stephen G. Simpson
Dept. of Mathematics
Pennsylvania State Univ.
University Park, PA 16802
U.S.A.
Jeff B. Paris
Dept. of Mathematics
Manchester University
Manchester MI3 9PL Robin Thomas
England
School of Mathematics
Georgia Technical University
Atlanta, GA 30332
U.S.A.
Hans J. Pro mel
Research Institute of
Discrete Mathematics
Joel Spencer
University of Bonn
Courant Institute
D-5300 Bonn 1
251 Mercer Street
West Germany
New York, NY 10012
U.S.A.
Pavel Pudlak
Mathematical Institute Bernd Voigt
Czechoslovak Academy Research Institute of
of Sciences Discrete Mathematics
Zitna. 25 University of Bonn
11567 Praha D-5300 Bonn 1
Czechoslovakia West Germany
Contributing Authors IX
Benjamin Weiss William Weiss
Institute of Mathematics Mathematics Department
Hebrew University University of Toronto
Givat Ram, Jerusalem 91904 Toronto, Ontario M55 IAI
Israel Canada
Table of Contents
Introduction
Ramsey Theory Old and New ......................................... 1
J. Nesetril and V. Rocil
1. Ramsey Numbers .................................................. 2
2. Transfinite Ramsey Theory ......................................... 3
3. Chromatic Number ................................................. 3
4. Classical Theorems ................................................. 3
5. Other Classical Theorems ........................................... 4
6. Structural Generalizations .......................................... 5
7. Infinite Ramsey Theorem ........................................... 6
8. Unprovability Results ............................................... 6
9. Non-Standard Applications ......................................... 7
Part I. Classics
Problems and Results on Graphs and Hypergraphs:
Similarities and Differences ........................................... 12
P. Erdos
1. Extremal Problems of Turan Type .................................. 13
2. Density Problems ................................................. 15
3. Ramsey's Theorem ................................................ 17
4. Ramsey-Turan Type Problems ..................................... 22
5. Chromatic Numbers ............................................... 25
Note on Canonical Partitions ......................................... 29
R. Rado
XII Table of Contents
Part II. Numbers
On Size Ramsey Number of Paths, Trees and Circuits. II ............... 34
J. Beck
1. Introduction ....................................................... 34
2. Proof of Theorem 1 - Part One ..................................... 36
3. Proof of Theorem 1 - Part Two .................................... 40
4. Proof of Theorem 2 ............................................... 42
5. Proof of Theorem 3 ............................................... 44
On the Computational Complexity of Ramsey-Type Problems ........... 46
S.A. Burr
1. Introduction ....................................................... 46
2. NP-Complete Ramsey Problems ................................... 47
3. Polynomial-Bounded Ramsey Problems ............................. 49
4. Discussion ........................................................ 51
Constructive Ramsey Bounds and Intersection Theorems for Sets ........ 53
P. Frankl
1. Introduction ....................................................... 53
2. Families of Sets with Prescribed Intersections ........................ 54
3. The Proof of Theorem 2.3 .......................................... 55
4. The Proof of Theorem 1.1 .......................................... 55
Ordinal Types in Ramsey Theory and Well-Partial-Ordering Theory ..... 57
I. KHz and R. Thomas
1. Introduction ....................................................... 58
2. Sheaves .......................................................... 61
3. Ramsey Systems .................................................. 64
4. Well-Partial-Ordering .............................................. 71
5. Erdos-Szekeres Theorem ........................................... 77
6. Ramsey Systems .................................................. 85
7. Canonical Ramsey Theorem ........................................ 92
Part III. Structural Theory
Partite Construction and Ramsey Space Systems ....................... 98
J. Nesetfil and V. Rodl
1. Introduction ....................................................... 98
2. Statement of Results ............................................. 100
3. Partite Lemma ................................................... 101
4. Partite Construction .............................................. 106
5. Applications ..................................................... 108
Table of Contents XIII
Graham-Rothschild Parameter Sets ................................... 113
H.J. Promel and B. Voigt
1. Introduction ...................................................... 113
2. Parameter Sets and Parameter Words (Definition and Basic Examples) 114
3. Hales-Jewett's Theorem ........................................... 117
4. Graham-Rothschild's Theorem .................................... 128
5. Infinite Versions .................................................. 136
6. Other Structures ................................................. 143
Shelah's Proof of the Hales-Jewett Theorem .......................... 150
A. Nilli
Part IV. Noncombinatorial Methods
Partitioning Topological Spaces ...................................... 154
W. Weiss
1. Introduction ..................................................... 154
2. Partitioning Singletons ........................................... 155
3. Partitioning Pairs ................................................ 167
4. Open Problems .................................................. 169
Topological Ramsey Theory ......................................... 172
T.J. Carlson and S.G. Simpson
1. Introduction ...................................................... 172
2. Ramsey Spaces and Ellentuck's Theorem ........................... 173
3. Finitary Consequences of Ellentuck Type Theorems ................. 176
4. The Axiom of Choice and the Construction of Non-Ramsey Sets ..... 177
5. Finite Dimensional Analogues of Ellentuck Type Theorems .......... 179
6. Canonical Partitions .............................................. 180
Ergodic Theory and Configurations in Sets of Positive Density ......... 184
H. Furstenberg, Y. Katznelson and B. Weiss
1. Introduction ..................................................... 184
2. Correspondence Between Subsets of lR? and ru?-Actions ............. 185
3. Ergodic Averages for Subsets of R2 ................................ 187
4. First Application to Subsets of Positive Density in R2 ............... 188
5. Proof of Theorem A .............................................. 189
6. A Recurrence Property of R2-Actions .............................. 191
7. Proof of Theorem B .............................................. 197
