STUDENT MATHEMATICAL LIBRARY Volume 73 Ramsey Theory on the Integers Second Edition Bruce M. Landman Aaron Robertson http://dx.doi.org/10.1090/stml/073 Ramsey Theory on the Integers Second Edition STUDENT MATHEMATICAL LIBRARY Volume 73 Ramsey Theory on the Integers Second Edition Bruce M. Landman Aaron Robertson American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10. For additional information and updates on this book, visit www.ams.org/bookpages/stml-73 Library of Congress Cataloging-in-Publication Data Landman, Bruce M., 1951– Ramsey theory on the integers / Bruce M. Landman, Aaron Robertson. – Second edition. pages cm. – (Student mathematical library ; volume 73) Includes bibliographical references and index. ISBN 978-0-8218-9867-3 (alk. paper) 1. Ramsey theory. 2. Combinatorial analysis. I. Robertson, Aaron, 1971– II. Title. 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In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 To To Eleanor Elisa, Quinn, Ava Emma and Sarah Pearl, Doug, Jason – Bruce – aaron Contents List of Tables xi Preface to the Second Edition xiii Acknowledgements xv Preface to the First Edition xvii Chapter 1. Preliminaries 1 §1.1. The Pigeonhole Principle 3 §1.2. Ramsey’s Theorem 6 §1.3. Some Notation 9 §1.4. Three Classical Theorems 11 §1.5. A Little More Notation 14 §1.6. Exercises 17 §1.7. Research Problems 20 §1.8. References 21 Chapter 2. Van der Waerden’s Theorem 23 §2.1. The Compactness Principle 27 §2.2. Alternate Forms of van der Waerden’s Theorem 29 §2.3. Computing van der Waerden Numbers 31 vii viii Contents §2.4. Bounds on van der Waerden Numbers 38 §2.5. The Erd˝os and Tur´an Function 47 §2.6. On the Number of Monochromatic Arithmetic Progressions 51 §2.7. Proof of van der Waerden’s Theorem 53 §2.8. Exercises 57 §2.9. Research Problems 59 §2.10. References 62 Chapter 3. Supersets of AP 67 §3.1. Quasi-Progressions 68 §3.2. Generalized Quasi-Progressions 77 §3.3. Descending Waves 81 §3.4. Semi-Progressions 83 §3.5. Iterated Polynomials 92 §3.6. Arithmetic Progressions as Recurrence Solutions 102 §3.7. Exercises 104 §3.8. Research Problems 107 §3.9. References 111 Chapter 4. Subsets of AP 113 §4.1. Finite Gap Sets 115 §4.2. Infinite Gap Sets 121 §4.3. Exercises 140 §4.4. Research Problems 142 §4.5. References 145 Chapter 5. Other Generalizations of w(k; r) 147 §5.1. Sequences of Type x, ax + d, bx +2d 147 §5.2. Homothetic Copies of Sequences 163 §5.3. Sequences of Type x, x + d, x +2d + b 168 §5.4. Polynomial Progressions 175 §5.5. Exercises 177 Contents ix §5.6. Research Problems 179 §5.7. References 181 Chapter 6. Arithmetic Progressions (mod m) 183 §6.1. The Family of Arithmetic Progressions (mod m) 184 §6.2. A Seemingly Smaller Family is More Regular 187 §6.3. The Degree of Regularity 193 §6.4. Exercises 196 §6.5. Research Problems 198 §6.6. References 200 Chapter 7. Other Variations on van der Waerden’s Theorem 203 §7.1. The Function Γm(k) 203 §7.2. Monochromatic Sets a(S + b) 207 §7.3. Having Most Elements Monochromatic 209 §7.4. Permutations Avoiding Arithmetic Progressions 213 §7.5. Exercises 217 §7.6. Research Problems 217 §7.7. References 219 Chapter 8. Schur’s Theorem 221 §8.1. The Basic Theorem 222 §8.2. A Generalization of Schur’s Theorem 234 §8.3. Refinements of Schur’s Theorem 239 §8.4. Schur Inequality 243 §8.5. Exercises 245 §8.6. Research Problems 247 §8.7. References 248 Chapter 9. Rado’s Theorem 251 §9.1. Rado’s Single Equation Theorem 251 §9.2. Some Rado Numbers 268 §9.3. Generalizations of the Single Equation Theorem 279 x Contents §9.4. Solutions to Linear Recurrences 285 §9.5. Mixing Addition and Multiplication 287 §9.6. Exercises 290 §9.7. Research Problems 292 §9.8. References 294 Chapter 10. Other Topics 297 §10.1. Monochromatic Sums 297 §10.2. Doublefree Sets 302 §10.3. Diffsequences 303 §10.4. Brown’s Lemma 318 §10.5. Monochromatic Sets Free of Prescribed Differences 321 §10.6. Patterns in Colorings 326 §10.7. Rainbow Ramsey Theory on the Integers 328 §10.8. Zero-Sums and m-Sets 334 §10.9. Exercises 345 §10.10. Research Problems 346 §10.11. References 349 Notation 353 Bibliography 357 Index 381 List of Tables Table 2.1: Mixed van der Waerden values 37 Table 2.2: Values and lower bounds for w(k; r)43 Table 3.1: Values and lower bounds for Qk−i(k)77 Table 3.2: Values and lower bounds for GQf(x)(k)80 Table 3.3: Values and lower bounds for SPm(k)92 Table 3.4: Values of R(Sn,k; r)andR(AP ∪ Pn,k) 102 Table 4.1: Values of w(f(x), 3; 2) 132 Table 5.1: Values and lower bounds for T (a, b; 2) 162 Table 5.2: Degree of regularity of (a, b)-triples 163 Table 5.3: Values of R(c, AUGb, 3) 174 Table 6.1: Values and lower bounds for R(AP(m),k; 2) 186 Table 6.2: Degree of regularity of families of type AP(m) 196 Table 10.1: Values of Δ(D, k) 318 Table 10.2: Number of squarefree and cubefree colorings 328 xi Preface to the Second Edition As noted in the preface of the first edition, one of the book’s pri- mary purposes is to serve as a source of unsolved problems in the flourishing and relatively “wide-open” area of Ramsey theory on the integers. We especially hoped the book would provide beginning re- searchers, including graduate students and undergraduate students, with a range of accessible topics in which to delve. It seems that we have already been at least somewhat successful because, since the publication of the first edition, a good number of the research prob- lems have been solved or partially solved, and several of the theorems have been improved upon. As a consequence, the new edition includes many substantial revisions and additions. Various new sections have been added and others have been sig- nificantly updated. Among the newly introduced topics are: rainbow Ramsey theory, an “inequality” version of Schur’s theorem, monochro- matic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erd˝os-Ginzberg-Ziv theorem, and the number of arithmetic progres- sions under arbitrary colorings. We also offer many new results and proofs among the topics that are not new to this edition, most of xiii xiv Preface to the Second Edition which were not known when the first edition was published. Further- more, the book’s tables, exercises, lists of open research problems, and bibliography have all been significantly updated. Finally, we have repaired numerous misprints. Acknowledgements We wish to thank a number of individuals for finding errors in the first edition and for helping to improve the exposition of the text, including: Charles Baker, Arie Bialostocki, Srashti Dwivedi, Sohail Farhangi, Paul Frigge, Andraes J¨onsson, Jonathan Leprince, Ryan Matzke, James Henry Sanders, and Hunter Snevily. Special thanks go to Sarah Landman for her careful proofreading. We are grateful to all the readers of the first edition who contacted us with insightful comments and suggestions. We also express our gratitute to our fellow Ramsey theorists for enlightening us on some recent advances in the field and for pointing us to a number of intriguing problems. We appreciate the kind and professional support given to us throughout the process by Edward Dunne, Christine Thivierge, and the rest of the AMS production staff. Bruce Landman adds a special debt of gratitude to his wife, Eleanor, for her support, patience, and sense of humor throughout this endeavor, and to his two wonderful daughters, Emma and Sarah, whoaddsomuchmeaningtoeverythinghedoes. Aaron would like to thank: his parents, Pearl and Doug, for providing him with a happy and supportive upbringing; his brother Jason (sorry for waking you with the snare drum, hiding in the closet to scare you, etc.); his advisor, Doron Zeilberger, without whom he xv xvi Acknowledgements probably would have dropped out of graduate school; and the love- of-his-life, Elisa, for just being fantastic and for giving him two ab- solutely wonderful children, Quinn and Ava (and one other child ..