The Mathematical Coloring Book

Alexander Soifer

The MathematicalColoring Book

Mathematics of Coloringand the Colorful Life of its Creators

Forewords by Branko Grunbaum, Peter D. Johnson Jr.,and Cecil Rousseau

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Alexander SoiferCollege of Letters, Arts and SciencesUniversity of Colorado at Colorado SpringsColorado SpringsCO 80918, [email protected]

ISBN: 978-0-387-74640-1 e-ISBN: 978-0-387-74642-5DOI 10.1007/978-0-387-74642-5

Library of Congress Control Number: 2008936132

Mathematics Subject Classification (2000): 01Axx 03-XX 05-XX

c Alexander Soifer 2009All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subject toproprietary rights.

Cover illustration: The photographs on the front cover depict, from the upper left clockwise, Paul Erdos,Frank P. Ramsey, Bartel L. van der Waerden, and Issai Schur.

Printed on acid-free paper

springer.com

This coloring book is for my late father Yuri Soifer,a great painter, who introduced colors into my life.

To Paint a Bird

First paint a cageWith wide open door,Then paint somethingBeautiful and simple,Something very pleasantAnd much neededFor the bird;Then lean the canvas on a treeIn a garden or an orchard or a forest And hide behind the tree,Do not talkDo not move. . .Sometimes the bird comes quicklyBut sometimes she needs years to decideDo not give up,Wait,Wait, if need be, for years,The length of waiting Be it short or long Does not carry any significanceFor the success of your paintingWhen the bird comes If only she ever comes Keep deep silence,Wait,So that the bird flies in the cage,And when she is in the cage,Quietly lock the door with the brush,And without touching a single featherCarefully wipe out the cage.Then paint a tree,And choose the best branch for the birdPaint green leaves

Freshness of the wind and dust of the sun,Paint the noise of animals in the grassIn the heat of summerAnd wait for the bird to singIf the bird does not sing This is a bad omenIt means that your picture is of no use,But if she sings This is a good sign,A symbol that you can beProud of and sign,So you very gentlyPull out one of the feathers of the birdAnd you write your nameIn a corner of the picture.

by Jacques Prevert1

1 [Pre]. Translation by Alexander Soifer and Maurice Stark.

Foreword

This is a unique type of book; at least, I have never encountered a book of this kind.The best description of it I can give is that it is a mystery novel, developing onthree levels, and imbued with both educational and philosophical/moral issues. Ifthis summary description does not help understanding the particular character andallure of the book, possibly a more detailed explanation will be found useful.

One of the primary goals of the author is to interest readersin particular, youngmathematicians or possibly pre-mathematiciansin the fascinating world of elegantand easily understandable problems, for which no particular mathematical knowl-edge is necessary, but which are very far from being easily solved. In fact, theprototype of such problems is the following: If each point of the plane is to begiven a color, how many colors do we need if every two points at unit distanceare to receive distinct colors? More than half a century ago it was established thatthe least number of colors needed for such a coloring is either 4, or 5, or 6 or 7.Well, which is it? Despite efforts by a legion of very bright peoplemany of whomdeveloped whole branches of mathematics and solved problems that seemed muchhardernot a single advance towards the answer has been made. This mystery, andscores of other similarly simple questions, form one level of mysteries explored. Indoing this, the author presents a whole lot of attractive results in an engaging way,and with increasing level of depth.

The quest for precision in the statement of the problems and the results and theirproofs leads the author to challenge much of the prevailing historical knowledge.Going to the original publications, and drawing in many cases on witnesses andon archival and otherwise unpublished sources, Soifer uncovers many mysteries. Inmost cases, dogged perseverance enables him to discover the truth. All this is pre-sented as following in a natural development from the mathematics to the history ofthe problem or result, and from there to the interest in the people who produced themathematics. For many of the persons involved this results in information not avail-able from any other source; in lots of the cases, the available publications present aninaccurate (or at least incomplete) data. The author is very careful in documentinghis claims by specific references, by citing correspondence between the principalsinvolved, and by accounts by witnesses.

One of these developments leads Soifer to examine in great detail the life andactions of one of the great mathematicians of the twentieth century, Bartel Leendert

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van der Waerden. Although Dutch, van der Waerden spent the years from 1931 to1945 in the Nazi Germany. This, and some of van der Waerdens activities duringthat time, became very controversial after Word War II, and led Soifer to exam-ine the moral and ethical questions relevant to the life of a scientist in a criminaldictatorship.

The diligence with which Soifer pursues his quests for information is way beyondexemplary. He reports exchanges with I am sure hundreds of people, via mail,phone, email, visits all dated and documented. The educational aspects that beginwith matters any middle-school student can understand, develop gradually into areasof most recent research, involving not only combinatorics but also algebra, topology,questions of foundations of mathematics, and more.

I found it hard to stop reading before I finished (in two days) the whole text.Soifer engages the readers attention not only mathematically, but emotionally andesthetically. May you enjoy the book as much as I did!

University of Washington Branko Grunbaum

Foreword

Alexander Soifers latest book is a fully fledged adult specimen of a new species,a work of literature in which fascinating elementary problems and developmentsconcerning colorings in arithmetic or geometric settings are fluently presented andinterwoven with a detailed and scholarly history of these problems and develop-ments.

This history, mostly from the twentieth century, is part memoir, for ProfessorSoifer was personally acquainted with some of the principals of the story (the greatPaul Erdos, for instance), became acquainted with others over the 18 year inter-val during which the book was written (Dima Raiskii, for instance, whose story isparticularly poignant), and created himself some of the mathematics of which hewrites.

Anecdotes, personal communications, and biography make for a good read, andthe readability in Mathematical Coloring Book is not confined to the accountsof events that transpired during the authors lifetime. The most important and fas-cinating parts of the book, in my humble opinion, are Parts IV, VI, and VII, inwhich is illuminated the progress along the intellectual strand that originated withthe Four-Color Conjecture and runs through Ramseys Theorem via Schur, Baudet,and Van der Waerden right to the present day, via Erdos and numbers of others,including Soifer. Not only is this account fascinating, it is indispensable: it can befound nowhere else.

The reportage is skillful and the scholarship is impressive this is what SeymourHersh might have written, had he been a very good mathematician curious to thepoint of obsession with the history of these coloring problems.

The unusual combination of abilities and interests of the author make the speciesof which this book is the sole member automatically endangered. But in the worldsof literature, mathematics and literature about mathematics, unicorns can have off-spring, even if the offspring are not exactly unicorns. I think of earlier books ofthe same family as Mathematical Coloring Book G. H. Hardys A Mathemati-cians Apology, James R. Newmans The World of Mathematics, Courant andRobbins What Is Mathematics?, Paul Halmos I Want to Be a Mathematician:an Automathography, or the books on Erdos that appeared soon after his death allof them related at least distantly to Mathematical Coloring Book by virtue of theattempt to blend (whether successfully or not is open to debate) mathematics with

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history or personal memoir, and it seems to me that, whatever the merits of thoseworks, they have all affected how mathematics is viewed and written about. And thiswill be a large part of the legacy of Mathematical Coloring Book besides pro-viding inspiration and plenty of mathematics to work on to young mathematicians,a priceless source to historians, and entertainment to those who are curious aboutthe activities of mathematicians, Mathematical Coloring Book will (we can hope)have a great and salutary influence on all writing on mathematics in the future.

Auburn University Peter D. Johnson

Foreword

What is the minimum number of colors required to color the points of the Euclideanplane in such a way that no two points that are one unit apart receive the same color?Mathematical Coloring Book describes the odyssey of Alexander Soifer and fellowmathematicians as they have attempted to answer this question and others involvingthe idea of partitioning (coloring) sets.

Among other things, the book provides an up-to-date summary of our knowledgeof the most significant of these problems. But it does much more than that. It givesa compelling and often highly personal account of discoveries that have shaped thatknowledge.

Soifers writing brings the mathematical players into full view, and he paints theirlives and achievements vividly and in detail, often against the backdrop of worldevents at the time. His treatment of the intellectual history of coloring problems iscaptivating.

Memphis State University Cecil Rousseau

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Acknowledgments

My first thank you goes to my late father Yuri Soifer, a great painter, who introducedcolors into my life, and to whom this book is gratefully dedicated. As the son of apainter and an actress, I may have inherited artistic genes. Yet, it was my parents,Yuri Soifer and Frieda Hoffman, who inspired my development as a connoisseur andstudent of the arts. I have enjoyed mathematics only because it could be viewed asan art as well. I thank Maya Soifer for restarting my creative engine when at timesit worked on low rpm (even though near the end of my work, she almost broke theengine by abandoning the car). I am deeply indebted to my kids Mark, Isabelle, andLeon, and to my cousin and great composer Leonid Hoffman for the support theirlove has always provided. I thank my old friends Konstantin Kikoin, Yuri Norsteinand Leonid Hoffman for years of stimulating conversations on all themes of highculture. I am deeply grateful to Branko Grunbaum, Peter D. Johnson Jr., and CecilRousseau, the first readers of the entire manuscript, for their kind forewords andvaluable suggestions. My 16-year old daughter Isabelle Soulay Soifer, an aspiringwriter, did a fine copy-editing job thank you, Isabelle!

This is a singular book for me, a result of 18 years of mathematical and his-torical research, and thinking over the moral and philosophical issues surround-ing a mathematician in the society. The long years of writing have produced oneimmense benefit that a quickly baked book would never fathom to possess. I havehad the distinct pleasure to communicate on the mathematics and the history forthis book with senior sages Paul Erdos, George Szekeres, Esther (Klein) Szekeres,Martha (Wachsberger) Sved, Henry Baudet, Nicolaas G. de Bruijn, Bartel L. van derWaerden, Harold W. Kuhn, Dirk Struick, Hilde Brauer (Mrs. Alfred Brauer), HildeAbelin-Schur (Issai Schurs daughter), Walter Ledermann, Anne Davenport (Mrs.Harold Davenport), Victor Klee, and Branko Grunbaum. Many of these great peo-ple are no longer with us; others are near or in their 80s. Their knowledge and theirmemories have provided blood to the body of my book. I am infinitely indebted tothem all, as well as to the younger contributors Ronald L. Graham, Edward Nelson,John Isbell, Adriano Garsia, James W. Fernandez, and Renate Fernandez, who aremerely in their 70s.

Harold W. Kuhn wrote a triple essay on the economics of Frank P. Ramsey, Johnvon Neumann and John F. Nash Jr. especially for this book, which can be found inChapter 30. Steven Townsend wrote and illustrated a new version of his proof espe-

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cially for this book (Chapter 24). Kenneth J. Falconer wrote a new clear expositionof his proof especially for this book (Chapter 9). Your contributions are so great,and I thank you three so very much!

I am grateful to several colleagues for their self-portraits written for this book:Nicolaas de Bruijn, Hillel Furstenberg, Vadim Vizing, Kenneth Falconer, PaulODonnell, Vitaly Bergelson, Alexander Leibman, and Michael Tarsi.

I am thrilled to be able to present results and writings of my colleagues I haveknown personally: Paul Erdos, Paul ODonnell, Rob Hochberg, Kiran Chilakamarri,Ronald L. Graham, Joel H. Spencer, Paul Kainen, Peter D. Johnson, Jr., NicholasWormald, David Coulson, Paul Seymour, Neil Robertson, Robin J. Wilson, EdwardPegg, Jr., Jaroslav Nesetril, Branko Grunbaum, Saharon Shelah, Dmitry Raiskii,Dmytro Karabash, Willie Moser, George Szekeres, Esther Klein, Alfred W. Hales,Heiko Harborth, Solomon W. Golomb, Jan Mycielski, Miklos Simonovits, and col-leagues I have gotten to know through correspondence: Bruce L. Rothschild, VadimG. Vizing, Mehdi Behzad, Douglas R. Woodall, and Jan Kyncl.

I am grateful to George Szekeres and MIT Press for their permission to reproducehere the most lyrical Georges Reminiscences (part of Chapter 29). I thank Bartel L.van der Waerden and Academic Press, London, for their kind permission to repro-duce here Van der Waerdens insightful How the Proof of Baudets Conjecture WasFound (Chapter 33).

I thank all those who have provided me with the rare, early photographs of them-selves: Paul Erdos (as well as a photograph with Leo Moser); George Szekeres andEsther Klein; Vadim Vizing; Edward Nelson; Paul ODonnell. Hilde Abelin-Schurhas kindly provided photographs of her father Issai Schur. Dorith van der Waerdenand Theo van der Waerden have generously shared rare photographs of their uncleBartel L. van der Waerden and the rest of their distinguished family. Henry BaudetII has generously supplied photographs of his father P. J. H. Baudet and also ofhis family with the legendary world chess champion Emanuel Lasker. Ronald L.Graham has kindly provided a photograph of him presenting Timothy Gowers withthe check for $1000. Alice Bogdan has provided a photograph of her great brother-in-law Tibor Gallai.

I thank numerous archives for documents and photographs and permissions touse them. Dr. Mordecai Paldiel and Yad Vashem, The Holocaust Martyrs andHeroes Remembrance Authority, Jerusalem, provided documents related to thegranting Senta Govers Baudet the title of a Righteous Among the Nations. JohnWebb and Mathematics Department of the University of Cape Town provided aphotograph of Francis Guthrie. By kind permission of the Provost and Scholars ofKings College, Cambridge, I am able to reproduce here two photographs of FrankPlumpton Ramsey, from the papers of John Maynard Keynes. The Board of TrinityCollege Dublin has kindly allowed me to reproduce the correspondence betweenAugustus De Morgan and William R. Hamilton. Humboldt University of Berlinshared documents from the personnel file of Issai Schur.

The following archivists and archives provided invaluable documents andsome photographs related to Bartel L. van der Waerden: John Wigmans, Rijk-sarchief in Noord-Holland; Dr. Peter J. Knegtmans, The University Historian,

Acknowledgments xvii

Universiteit van Amsterdam; Prof. Dr. Gerald Wiemers and Martina Geigenmuller,Universitatsarchiv Leipzig; Prof. Dr. Holger P. Petersson and his personal archive;Gertjan Dikken, Het Parool; Madelon de Keizer; Dr. Wolfram Neubauer, AngelaGastl and Corina Tresch De Luca, ETH-Bibliothek, ETH, Zurich; Dr. HeinzpeterStucki, Universitatsarchiv, Universitat Zurich; Drs. A. Marian Th. Schilder, Uni-versiteitsmuseum de Agnietenkapel, Amsterdam; Maarten H. Thomp, CentraleArchiefbewaarplaats, Universiteit Utrecht; Nancy Cricco, University Archivist, andher graduate students assistants, New York University; James Stimpert, Archivist,Milton S. Eisenhower Library, Special Collections, The Johns Hopkins University;Prof. Mark Walker; Thomas Powers; Nicolaas G. de Bruijn; Henry Baudet II,Dr. Helmut Rechenberg, Director of the Werner Heisenberg Archiv, Munich;G. G. J. (Gijs) Boink, Het Nationaal Archief, Den Haag; Library of Congress,Manuscript Division, Washington D.C.; Centrum voor Wiskunde en Informatica,Amsterdam; Prof. Nicholas M. Katz, Chair, Department of Mathematics, PrincetonUniversity; and Mitchell C. Brown and Sarah M. LaFata, Fine Library, PrincetonUniversity.

My research on the life of Van der Waerden could not be based on the archivalmaterial alone. I am most grateful to individuals who provided much help inthis undertaking: Dorith van der Waerden; Theo van der Waerden; Hans van derWaerden; Helga van der Waerden Habicht; Annemarie van der Waerden; Prof.Dr. Henry Baudet (19191998); Prof. Dr. Herman J. A. Duparc (19182002);Prof. Dr. Nicolaas G. de Bruijn; Prof. Dr. Benno Eckmann; Prof. Dr. Dirk vanDalen; Prof. Dirk J. Struik (18942000); Dr. Paul Erdos (19131996); Rein-hard Siegmund-Schultze; Thomas Powers; Prof. Mark Walker, Prof. James andDr. Renate Fernandez; Dr. Maya Soifer; and Princeton University Professors HaroldW. Kuhn, Simon Kochen, John H. Conway, Edward Nelson, Robert C. Gunning,Hale Trotter, Arthur S. Wightman, Val L. Fitch, Robert Fagles, Charles Gillispieand Steven Sperber.

I thank my wonderful translators of Dutch, Dr. Stefan van der Elst and Prof.Marijke Augusteijn; and of German: Prof. Simon A. Brendle; Prof. Robertvon Dassanowsky; Prof. Robert Sackett; Prof. Dr. Heiko Harborth; and Prof.Dr. Hans-Dietrich Gronau.

I thank those who converted my pencil doodles into computer-aided illustrations:Steven Bamberger, Phil Emerick and Col. Dr. Robert Ewell.

The research quarterly Geombinatorics provided a major forum for the essaysrelated to the chromatic number of the plane. Consequently, it is cited numeroustimes in this book. I wish to thank the editors of Geombinatorics Paul Erdos, BrankoGrunbaum, Ron Graham, Heiko Harborth, Peter D. Johnson, Jr., Jaroslav Nesetril,and Janos Pach.

My University of Colorado bosses provided support on our campus and oppor-tunity to be away for long periods of time at Princeton and Rutgers UniversitiesIthank Tom Christensen, Tom Wynn, and Pam Shockley-Zalabak.

I am grateful to my Princeton-Math colleagues and friends for maintaining aunique creative atmosphere in the historic Fine Hall, and Fred Roberts for the tran-quility of his DIMACS at Rutgers University. Library services at Princeton have

xviii Acknowledgments

provided an invaluable swift service: while working there for 3 years, I must haveread thousands of papers and many books.

I thank my Springer Editor Mark Spencer, who initiated our contact, showed trustin me and this project based merely on the table of contents and a single section,and in 2004 proposed to publish this book in Springer. At a critical time in my life,Springers Executive Editor Ann Kostant made me believe that what I am and whatI do really mattersthank you from the bottom of my heart, Ann!

There is no better place to celebrate the completion of the book than the land ofPythagoras, Euclid, and Archimedes. I thank Prof. Takis Vlamos for organizing myvisit and lectures on the Island of Corfu, Thessaloniki, and Athens.

Contents

Foreword by Branko Grunbaum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Foreword by Peter D. Johnson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Foreword by Cecil Rousseau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Greetings to the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi

Part I Merry-Go-Round . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 A Story of Colored Polygons and Arithmetic Progressions . . . . . . . . . . 31.1 The Story of Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Problem of Colored Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Translation into the Tongue of APs . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Prehistory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Completing the Go-Round . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Part II Colored Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Chromatic Number of the Plane: The Problem . . . . . . . . . . . . . . . . . . . . 13

3 Chromatic Number of the Plane:An Historical Essay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Polychromatic Number of the Plane and Results Near the LowerBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 De BruijnErdos Reduction to Finite Sets and Results Near theLower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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6 Polychromatic Number of the Plane and Results Near the UpperBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.1 Stechkins 6-Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2 Best 6-Coloring of the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 The Age of Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Continuum of 6-Colorings of the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8 Chromatic Number of the Plane in Special Circumstances . . . . . . . . . . 57

9 Measurable Chromatic Number of the Plane . . . . . . . . . . . . . . . . . . . . . . 609.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609.2 Lower Bound for Measurable Chromatic Number of the Plane . . . 609.3 Kenneth J. Falconer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

10 Coloring in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

11 Rational Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Part III Coloring Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

12 Chromatic Number of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7912.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7912.2 Chromatic Number and Girth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8212.3 Wormalds Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

13 Dimension of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8813.1 Dimension of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8813.2 Euclidean Dimension of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

14 Embedding 4-Chromatic Graphs in the Plane . . . . . . . . . . . . . . . . . . . . . 9914.1 A Brief Overture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9914.2 Attaching a 3-Cycle to Foundation Points in 3 Balls . . . . . . . . . . . . 10114.3 Attaching a k-Cycle to a Foundation Set of Type (a1, a2, a3, 0) . . 10214.4 Attaching a k-Cycle to a Foundation Set of Type (a1, a2, a3, 1) . . 10414.5 Attaching a k-Cycle to Foundation Sets of Types (a1, a2, 0, 0)

and (a1, 0, a3, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10414.6 Removing Coincidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10614.7 ODonnells Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10714.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

15 Embedding World Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11015.1 A 56-Vertex, Girth 4, 4-Chromatic Unit Distance Graph . . . . . . . . 11115.2 A 47-Vertex, Girth 4, 4-Chromatic, Unit Distance Graph . . . . . . . . 11615.3 A 40-Vertex, Girth 4, 4-Chromatic, Unit Distance Graph . . . . . . . . 117

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15.4 A 23-Vertex, Girth 4, 4-Chromatic, Unit Distance Graph . . . . . . . . 12115.5 A 45-Vertex, Girth 5, 4-Chromatic, Unit Distance Graph . . . . . . . . 124

16 Edge Chromatic Number of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12716.1 Vizings Edge Chromatic Number Theorem . . . . . . . . . . . . . . . . . . . 12716.2 Total Insanity around the Total Chromatic Number Conjecture . . . 135

17 Carsten Thomassens 7-Color Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Part IV Coloring Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

18 How the Four-Color Conjecture Was Born . . . . . . . . . . . . . . . . . . . . . . . . 14718.1 The Problem is Born . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14718.2 A Touch of Historiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15618.3 Creator of the 4 CC, Francis Guthrie . . . . . . . . . . . . . . . . . . . . . . . . . 15818.4 The Brother . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

19 Victorian Comedy of Errors and Colorful Progress . . . . . . . . . . . . . . . . 16319.1 Victorian Comedy of Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16319.2 2-Colorable Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16519.3 3-Colorable Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16819.4 The New Life of the Three-Color Problem . . . . . . . . . . . . . . . . . . . . 173

20 KempeHeawoods Five-Color Theorem and Taits Equivalence . . . . . 17620.1 Kempes 1879 Attempted Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17620.2 The Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18020.3 The Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18020.4 KempeHeawoods Five-Color Theorem . . . . . . . . . . . . . . . . . . . . . 18220.5 Taits Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18220.6 Frederick Guthries Three-Dimensional Generalization . . . . . . . . . 185

21 The Four-Color Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

22 The Great Debate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19522.1 Thirty Plus Years of Debate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19522.2 Twenty Years Later, or Another Time Another Proof . . . . . . . . . . 19922.3 The Future that commenced 65 Years Ago: Hugo Hadwigers

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

23 How Does One Color Infinite Maps? A Bagatelle . . . . . . . . . . . . . . . . . . . 207

24 Chromatic Number of the Plane Meets Map Coloring:TownsendWoodalls 5-Color Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 20924.1 On Stephen P. Townsends 1979 Proof . . . . . . . . . . . . . . . . . . . . . . . 20924.2 Proof of TownsendWoodalls 5-Color Theorem . . . . . . . . . . . . . . . 211

xxii Contents

Part V Colored Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

25 Paul Erdos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22725.1 The First Encounter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22825.2 Old Snapshots of the Young . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

26 De BruijnErdoss Theorem and Its History . . . . . . . . . . . . . . . . . . . . . . . 23626.1 De BruijnErdoss Compactness Theorem . . . . . . . . . . . . . . . . . . . . 23626.2 Nicolaas Govert de Bruijn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

27 Edge Colored Graphs: Ramsey and Folkman Numbers . . . . . . . . . . . . . 24227.1 Ramsey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24227.2 Folkman Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Part VI The Ramsey Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

28 From Pigeonhole Principle to Ramsey Principle . . . . . . . . . . . . . . . . . . . 26328.1 Infinite Pigeonhole and Infinite Ramsey Principles . . . . . . . . . . . . . 26328.2 Pigeonhole and Finite Ramsey Principles . . . . . . . . . . . . . . . . . . . . . 267

29 The Happy End Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26829.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26829.2 The Story Behind the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27229.3 Progress on the Happy End Problem . . . . . . . . . . . . . . . . . . . . . . . . . 27729.4 The Happy End Players Leave the Stage

as Shakespearian Heroes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

30 The Man behind the Theory: Frank Plumpton Ramsey . . . . . . . . . . . . . 28130.1 Frank Plumpton Ramsey and the Origin of the Term Ramsey

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28130.2 Reflections on Ramsey and Economics, by Harold W. Kuhn . . . . . 291

Part VII Colored Integers: Ramsey Theory Before Ramsey and ItsAfterMath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

31 Ramsey Theory Before Ramsey: Hilberts Theorem . . . . . . . . . . . . . . . . 299

32 Ramsey Theory Before Ramsey: Schurs Coloring Solution of aColored Problem and Its Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 30132.1 Schurs Masterpiece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30132.2 Generalized Schur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30432.3 Non-linear Regular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Contents xxiii

33 Ramsey Theory before Ramsey: Van der Waerden Tells the Story ofCreation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

34 Whose Conjecture Did Van der Waerden Prove? Two LivesBetween Two Wars: Issai Schur and Pierre Joseph Henry Baudet . . . . 32034.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32034.2 Issai Schur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32134.3 Argument for Schurs Authorship of the Conjecture . . . . . . . . . . . . 33034.4 Enters Henry Baudet II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33434.5 Pierre Joseph Henry Baudet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33634.6 Argument for Baudets Authorship of the Conjecture . . . . . . . . . . . 34034.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

35 Monochromatic Arithmetic Progressions: Life After Van derWaerden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34735.1 Generalized Schur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34735.2 Density and Arithmetic Progressions . . . . . . . . . . . . . . . . . . . . . . . . 34835.3 Who and When Conjectured What Szemeredi Proved? . . . . . . . . . 35035.4 Paul Erdoss Favorite Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35335.5 Hillel Furstenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35635.6 Bergelsons AG Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35835.7 Van der Waerdens Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36035.8 A Japanese Bagatelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

36 In Search of Van der Waerden: The Early Years . . . . . . . . . . . . . . . . . . . 36736.1 Prologue: Why I Had to Undertake the Search

for Van der Waerden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36736.2 The Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36936.3 Young Bartel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37336.4 Van der Waerden at Hamburg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37736.5 The Story of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38036.6 Theorem on Monochromatic Arithmetic Progressions . . . . . . . . . . 38336.7 Gottingen and Groningen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38536.8 Transformations of The Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38636.9 Algebraic Revolution That Produced Just One Book . . . . . . . . . . . 38736.10 Epilogue: On to Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

37 In Search of Van der Waerden: The Nazi Leipzig, 19331945 . . . . . . . . 39337.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39337.2 Before the German Occupation of Holland: 19311940 . . . . . . . . . 39437.3 Years of the German Occupation of the Netherlands: 19401945 . 40637.4 Epilogue: The War Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

38 In Search of Van der Waerden: The Postwar Amsterdam, 1945 . . . . . . 41838.1 Breidablik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

xxiv Contents

38.2 New World or Old? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42138.3 Defense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42738.4 Van der Waerden and Van der Corput: Dialog in Letters . . . . . . . . . 43438.5 A Rebellion in Brouwers Amsterdam . . . . . . . . . . . . . . . . . . . . . . . 446

39 In Search of Van der Waerden: The Unsettling Years, 19461951 . . . . 44939.1 The Het Parool Affair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44939.2 Job History 19451947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45839.3 America! America! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46239.4 Van der Waerden, Goudsmit and Heisenberg:

A Letteral Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46539.5 Professorship at Amsterdam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47239.6 Escape to Neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47439.7 Epilogue: The Drama of Van der Waerden . . . . . . . . . . . . . . . . . . . . 480

Part VIII Colored Polygons: Euclidean Ramsey Theory . . . . . . . . . . . . . . . 485

40 Monochromatic Polygons in a 2-Colored Plane . . . . . . . . . . . . . . . . . . . . 487

41 3-Colored Plane, 2-Colored Space, and Ramsey Sets . . . . . . . . . . . . . . . 500

42 Gallais Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50542.1 Tibor Gallai and His Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50542.2 Double Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50942.3 Proof of Gallais Theorem by Witt . . . . . . . . . . . . . . . . . . . . . . . . . . 50942.4 Adriano Garsia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51442.5 An Application of Gallai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51642.6 Hales-Jewetts Tic-Tac-Toe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Part IX Colored Integers in Service of Chromatic Numberof the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

43 Application of BaudetSchurVan der Waerden . . . . . . . . . . . . . . . . . . . 521

44 Application of BergelsonLeibmans and MordellFaltingsTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

45 Solution of an Erdos Problem: ODonnells Theorem . . . . . . . . . . . . . . . 52945.1 ODonnells Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52945.2 Paul ODonnell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

Part X Predicting the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

46 What If We Had No Choice? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

Contents xxv

46.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53546.2 The Axiom of Choice and its Relatives . . . . . . . . . . . . . . . . . . . . . . . 53746.3 The First Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54046.4 Examples in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54346.5 Examples in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54446.6 AfterMath & ShelahSoifer Class of Graphs . . . . . . . . . . . . . . . . . . 54646.7 An Unit Distance ShelahSoifer Graph . . . . . . . . . . . . . . . . . . . . . . 549

47 A Glimpse into the Future: Chromatic Number of the Plane,Theorems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55347.1 Conditional Chromatic Number of the Plane Theorem . . . . . . . . . . 55347.2 Unconditional Chromatic Number of the Plane Theorem . . . . . . . . 55447.3 The Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

48 Imagining the Real, Realizing the Imaginary . . . . . . . . . . . . . . . . . . . . . . 55748.1 What Do the Founding Set Theorists Think about

the Foundations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55748.2 So, What Does It All Mean? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56048.3 Imagining the Real vs. Realizing the Imaginary . . . . . . . . . . . . . . . 562

Part XI Farewell to the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

49 Two Celebrated Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Greetings to the Reader

I bring here all: what have I lived thru,And that what keeps my soul alive,My rectitude and aspirations,And what have seen my own eyes.

Boris Pasternak, The Waves, 19312

When the form is realized, it is here to live its ownlife.

Pablo Picasso

Pasternaks epigraph describes precisely my work on this bookI gave it all ofmyself, without reservation. August Renoir believed that just as many people readone book all their lives (the Bible, the Koran, etc.), so can he paint all his life onepainting. Likewise I could write one book all my lifein fact, I almost have, for Ihave been working on this book for 18 years.

It is unfair, however, to keep the book all to myselfmany colleagues have beenwaiting for the birth of this book. In fact, it has been cited and even reviewed manyyears ago. The first mention of it appears already in 1991 on page 336 of the bookby Victor Klee and Stan Wagon [KW], where the authors recommend the book forsurvey of later developments of the chromatic number of the plane problem. Onpage 150 of their 1995 book [JT], Tommy R. Jensen and Bjarne Toft announced thata comprehensive survey [of the chromatic number of the plane problem]. . .will begiven by Soifer [to appear]. Once in the 1990s my son Mark told me that he sawmy Mathematical Coloring Book available for $30 for special order at the Bordersbookstore. I offered to buy a copy!

I started writing this book when copies of my How Does One Cut a Triangle?[Soi1] arrived from the printer, in early 1990. I told my father Yuri Soifer then

2 [Pas], Translated for this book by Ilya Hoffman. The original Russian text is: : , , , ,

.

xxvi

Greetings to the Reader xxvii

that this book would be dedicated to him, and so it is. This coloring book is formy late father, a great painter and man. Yuri lived with his sketchpad and drawingutensils in his pocket, constantly and intensely looking at people and making sharpmomentary sketches. He was a great artist and my lifelong example of searching forand discovering life around him, and creating art that challenged real life herself.Yuri never taught me his trade, but during our numerous joint tours of art in muse-ums and exhibitions, he pointed out beauties that only true artists could notice: adream of harvest in Van Goghs Sower, Rodins distortions in a search of greaterexpressiveness. These timeless lessons allowed me to become a student of beauty,and discover subtleties in paintings, sculptures, and movies throughout my life.

This book includes not just mathematics, but also the process of investigation,trains of mathematical thought, and where possible, psychology of mathematicalinvention. The book does not just include history and prehistory of Ramsey The-ory and related fields, but also conveys the process of historical investigationthekitchen of historical research if you will. It has captivated me, and made me feellike a Sherlock HolmesI hope my reader will enjoy this sense of suspense anddiscovery as much as I have.

The epigraph for my book is an English translation of Jacques Preverts geniusand concise portrayal of creative processI know of no better. I translated it withthe help of my friend Maurice Starck from Nouvelle Caledonie, the island in thePacific Ocean to which no planes fly from America, but to paraphrase RudyardKipling, Id like to roll to Nouvelle Caledonie some day before Im old!

This book is dedicated to problems involving colored objects, and results aboutthe existence of certain exciting and unexpected properties that occur regardlessof how these objects (points in the plane, space, integers, real numbers, subsets,etc.) are colored. In mathematics, these results comprise Ramsey Theory, a flour-ishing area of mathematics, with a motto that can be formulated as follows: anycoloring of a large enough system contains a monochromatic subsystem of given inadvance structure, or simply put, absolute chaos is absolutely impossible. RamseyTheory thus touches on many fields of mathematics, such as combinatorics, geom-etry, number theory, and addresses new problems, often on the frontier of two ormore traditional mathematical fields. The book will also include some problems thatcan be solved by inventing coloring, and results that prove the existence of certaincolorings, most famous of the latter being, of course, The Four-Color Theorem.

Most books in the field present mathematics as a flower, dried out between pagesof an old dusty volume, so dry that the colors are faded and only theoremproofnarrative survives. Along with my previous books, Mathematical Coloring Bookwill strive to become an account of a live mathematics. I hope the book will presentmathematics as a human endeavor: the reader should expect to find in it not onlyresults, but also portraits of their creators; not only mathematical facts, but alsoopen problems; not only new mathematical research, but also new historical inves-tigations; not only mathematical aspirations, but also moral dilemmas of the timesbetween and during the two horrific World Wars of the twentieth century. In myview, mathematics is done by human beings, and knowing their lives and culturesenriches our understanding of mathematics as a product of human activity, rather

xxviii Greetings to the Reader

than an abstraction which exists separately from us and comes to us exclusively as acatalog of theorems and formulas. Indeed, new facts and artifacts will be presentedthat are related to the history of the Chromatic Number of the Plane problem,the early history of Ramsey Theory, the lives of Issai Schur, Pierre Joseph HenryBaudet, and Bartel Leendert van der Waerden.

I hope you will join me on a journey you will never forget, a journey full ofpassion, where mathematics and history are researched in the process of solvingmysteries more exciting than fiction, precisely because those are mysteries of realaffairs of human history. Can mathematics be received by all senses, like a vibrantflower, indeed, like life itself? One way to find out is to experience this book.

While much of the book is dedicated to results of Ramsey Theory, I did not wishto call my book Introduction to Ramsey Theory, for such a title would immediatelylose young talented readers interest. Somehow, the playfulness of MathematicalColoring Book appealed to me from the start, even though I was asked on occasionwhether 5-year olds would be able to color in my book between its lines. To be abit more serious, and on advice of Vickie Kern of the Princeton University Press,I created a subtitle Mathematics of Coloring and the Colorful Life of Its Creators.This book is not a dullster of traditional theoremprooftheoremproof kind. Itexplores the birth of ideas and searches for its creators. I discovered very quicklythat in conveying colorful lives of creators, I could not always rely on encyclo-pedias and biographical articles, but had to conduct historical investigations on myown. It was a hard work to research some of the lives, especially that of B. L. vander Waerden, which alone took 12 years of archival research and thinking over theassembled evidence. Fortunately this produced a satisfying result: we have in thisbook some definitive biographies, of Bartel L. van der Waerden, Pierre Joseph HenryBaudet, Issai Schur, autobiography of Hillel Furstenberg, and others.

I always attempt to understand who made a discovery and how it was made.Accordingly, this book tries to explore biographies of the discoverers and the psy-chology of their creative processes. Every stone has been turned: my informationcomes from numerous archives in Germany, the Netherlands, Switzerland, Ireland,England, South Africa, the United States; invaluable and irreplaceable now inter-views conducted with eyewitnesses; discussions held with creators. Cited bibliog-raphy alone includes over 800 itemsI have read thousands of publications in theprocess of writing this book. I was inspired by people I have known personally, suchas Paul Erdos, James W. Fernandez, Harold W. Kuhn, and many others, as well aspeople I have not personally met, such as Boris Pasternak, Pablo Picasso, HerbertReadto name a few of the many influencesor D. A. Smith, who in the discussionafter Alfred Brauers talk [Bra2, p. 36], wrote:

Mathematical history is a sadly neglected subject. Most of this history belongs to thetwentieth century, and a good deal of it in the memories of mathematicians still living.The younger generation of mathematicians has been trained to consider the product,mathematics, as the most important thing, and to think of the people who producedit only as names attached to theorems. This frequently makes for a rather dry subjectmatter.

Greetings to the Reader xxix

Milan Kundera, in his The Curtain: An Essay in Seven Parts [Kun], said about anovel what is true about mathematics as well:

A novelist talking about the art of the novel is not a professor giving a discourse fromhis podium. Imagine him rather as a painter welcoming you into his studio, where youare surrounded by his canvases staring at you from where they lean against the walls.He will talk about himself, but even more about other people, about novels of theirsthat he loves and that have a secret presence in his own work. According to his criteriaof values, he will again trace out for you the whole past of the novels history, and inso doing will give you some sense of his own poetics of the novel.

I was also inspired by the early readers of the book, and their feedback. StanisawP. Radziszowski, after reviewing Chapter 27, e-mailed me on May 2, 2007:

I am very anxious to read the whole book! You are doing great service to the commu-nity by taking care of the past, so the things are better understood in the future.

In his unpublished letter, Ernest Hemingway in a sense defended my writing ofthis book for a very long time:3

When I make country, or a city, or a river in a novel it is slow work because you haveto always make it, then it is alive. But nobody makes anything quickly nor easily if itis any good.

Branko Grunbaum, upon reading the entire manuscript, wrote in the February 28,2008 e-mail:

Somehow it seems that 18 years would be too short a time to dig up all this information!

This book will not strike the reader by completeness or most general results.Instead, it would give young active high school and college mathematicians anaccessible introduction to the beautiful ideas of mathematics of coloring. Mathe-matics professionals, who may believe they know everything, would be pleasantlysurprised by the unpublished or unnoticed mathematical gems. I hope young and notso young mathematicians alike will welcome an opportunity to try their handormindon numerous open problems, all easily understood and not at all easy tosolve.

If the interest of my colleagues and friends at Princeton-Math is any indica-tion, every intelligent reader would welcome an engagement in solving histori-cal mysteries, especially those from the times of the Third Reich, World War II,and de-Nazification of Europe. Historians of mathematics would find a lot of newinformation and old errors corrected for the first time. And everyone will experienceseeing, for the first time, faces they have not seen before in print: rare photographsof the creators of mathematics presented herein, from Francis Guthrie to Issai Schuras a young man, from young Edward Nelson to Paul ODonnell, from Pierre JosephHenry Baudet to Bartel L. van der Waerden and his family, and documents, such as

3 From the unpublished 1937 letter. Quoted from New York Times, February 10, 2008, p. AR 8.

xxx Greetings to the Reader

the one where Adolph Hitler commits a micromanagement of firing the Jew, IssaiSchur, from his job of professor at the University of Berlin.

This is a freely flowing book, free from a straight jacket of a typical textbook, yetuseable as a text for a host of various courses, two of which I have given to collegeseniors and graduate students at the University of Colorado: What is Mathematics?,and Mathematical Coloring Course, both presenting a laboratory of a mathemati-cian, a place where students learn mathematics and its history by researching them,and in the process realizing what mathematics is and what mathematicians do.

In writing this book, I tried to live up to the high standard, set by one of myheroes, the great Danish film director Carl Theodore Dreyer [Dre]:

There is a certain resemblance between a work of art and a person. Just as one can talkabout a persons soul, one can also talk about the work or arts soul, its personality.The soul is shown through the style, which is the artists way of giving expression ofhis perception of the material. The style is important in attaching inspiration to artisticform. Through the style, the artist molds the many details that make it whole. Throughstyle, he gets others to see the material through his eyes. . . Through the style he infusesthe work with a soul and that is what makes it art.

Mathematics is an art. It is a poor mans art: Nothing is needed to conceive it,and only paper and pencil to convey.

This long work has given me so very much, in Aleksandr Pushkins words, theheavenly, and inspiration, and life, and tears, and love.4 I have been raising thisbook for 18 years, and over the past couple of years, I felt as if the book herself wasdictating her composition and content to me, while I merely served as an obedientscribe. At 18, my book is now an adult, and deserves to separate from me to liveher own life. As Picasso put it, When the form is realized, it is here to live its ownlife. Farewell, my child, let the world love you as I have and always will!

4 In the original Russian it sounds much better: , , , , .

IMerry-Go-Round

1A Story of Colored Polygons and ArithmeticProgressions

Have you guessed the riddle yet? the Hatter said,turning to Alice again.No, I give it up, Alice replied. Whats the answer?I havent the slightest idea, said the Hatter.Nor I, said the March Hare.

Lewis Carroll, A Mad Tea-PartyAlices Adventures in Wonderland

1.1 The Story of Creation

I recall April of 1970. The thirty judges of the Fourth Soviet Union National Math-ematical Olympiad, of whom I was one, stayed at a fabulous white castle, half waybetween the cities of Simferopol and Alushta, nestled in the sunny hills of Crimea,surrounded by the Black Sea. This castle should be familiar to movie buffs: in 1934the Russian classic film Vesyolye Rebyata (Jolly Fellows) was photographed here bySergei Eisensteins long-term assistant, director Grigori Aleksandrov. The problemshad been selected and sent to printers. The Olympiad was to take place a day later,when something shocking occurred.

A mistake was found in the only solution the judges had of the problem createdby Nikolai (Kolya) B. Vasiliev, the Vice-Chair of this Olympiad and a fine problemcreator, head of the problems section of the journal Kvant from its inception in 1970to the day of his untimely passing. What should we do? This question virtuallymonopolized our lives.

We could just cross this problem out on each of the six hundred printed problemsheets. In addition, we could select a replacement problem, but we would have towrite it in chalk by hand in every examination room, since there would be no time toprint it. Both options were rather embarrassing, desperate resolutions of the incidentfor the Jury of the National Olympiad, chaired by the great mathematician AndrejN. Kolmogorov, who was to arrive the following day. The best resolution, surely,would have been to solve the problem, especially because its statement was quitebeautiful, and we had no counter example to it either.

A. Soifer, The Mathematical Coloring Book, 3DOI 10.1007/978-0-387-74642-5 1, C Alexander Soifer 2009

4 I Merry-Go-Round

Even today, 38 years later, I can close my eyes and see how each of us, thirtyjudges, all fine problem solvers, worked on the problem. A few sat at the tableas if posing for Rodins Thinker. Some walked around as if measuring the roomsdimensions. Andrei Suslin, who would later prove the famous Serres conjecture,1went out for a hike. Someone was lying on a sofa with his eyes closed. Silence wasso absolute that you could hear a fly. The intense thinking seemed to stop the timeinside of the room. However, we were unable, on to stop the time outside. Night fell,and with it our hopes for solving the problem in time.

Suddenly, the silence was interrupted by a victorious outcry: I got it! echoedthrough the halls and the watch tower of the castle. It came from Alexander Livshits,an undergraduate student at Leningrad (St. Petersburg) University, and former win-ner of the Soviet and the International Mathematical Olympiads (a perfect 42 scoreat the 1967 IMO in Yugoslavia).2 His number-theoretic solution used the method oftrigonometric sums. However, this, was the least of our troubles: we immediatelytranslated the solution into the language of colored polygons.

Now we had options. A decision was reached to leave the problem in because theproblem and its solution were too beautiful to be thrown away. We knew, though,that the chances of receiving a single solution from six hundred bright Olympianswere very slim. Indeed nobody solved it.

1.2 The Problem of Colored Polygons

Here is the problem.

Problem 1.1 (N. B. Vasiliev; IV Soviet Union National Olympiad, 1970). Verticesof a regular n-gon are colored in finitely many colors (each vertex in one color)in such a way that for each color all vertices of that color form themselves a reg-ular polygon, which we will call a monochromatic polygon. Prove that among themonochromatic polygons there are two polygons that are congruent. Moreover, thetwo congruent monochromatic polygons can always be found among the monochro-matic polygons with the least number of vertices.

I first told the above story and the problem in my 1994 Olympiad book [Soi9].It appeared in the section Further Explorations, and as such I left the pleasure ofdiscovering the proof to the readers. It is time for me to share the solution.

Solution of Problem 1.1 by Alexander Livshits (in polygonal translation): Let medivide the problem into three parts: Preliminaries, Tool, and Proof.

Preliminaries: Given a system S of vectors v1, v2, . . . , vn in the plane with aCartesian coordinate system, all emanating from the origin O . We would call the

1 Daniel Quillen proved it independently, and got Fields Medal primarily for that.2 Andrei Suslin informs me that as of 1991 Alexander worked as a computer programmer in Leningrad;I was unable to determine his later whereabouts.

1 A Story of Colored Polygons and Arithmetic Progressions 5

system S symmetric if there is an integer k, 1 k < n, such that rotation of everyvector of S about O through the angle 2k

ntransforms S into itself.

Of course, the sum vi of all vectors of a symmetric system is 0, because

vidoes not change under rotation through the angle 0 < 2k

n< 2.

Place a regular n-gon Pn in the plane so that its center coincides with the originO . Then the n vectors drawn from O to all the vertices of Pn form a symmetricsystem (Fig. 1.1).

Fig. 1.1

Let v be a vector emanating from the origin O and making the angle with theray OX (Fig. 1.1). Symbol T m will denote a transformation that maps v into thevector T mv of the same length as v, but making the angle m with OX (Fig. 1.2).

O

Fig. 1.2

To check your understanding of these concepts, please prove the following toolon your own.

Tool 1.2 Let v1, v2, . . . , vn be a symmetric system S of vectors that transformsinto itself under the rotation through the angle 0 < 2k

n< 2, 1 k < n, (you can

think of 2kn

as the angle between two neighboring vectors of S). A transformationT m applied to S produces a system T m S of vectors T mv1, T mv2, . . . , T mvn thatis symmetric if n does not divide km. If n divides km, then T mv1 = T mv2 = . . . =T mvn .Solution of Problem 1.1: We will argue by contradiction. Assume that the vertices ofa regular n-gon Pn are colored in r colors and we got subsequently r monochromaticpolygons: n1-gon Pn1 , n2-gon Pn2 , . . . , nr -gon Pnr , such that no pair of congruentmonochromatic polygons is created, i.e.,

n1 < n2 < . . . < nr .

6 I Merry-Go-Round

We create a symmetric system S of n vectors going from the origin to all ver-tices of the given n-gon Pn . In view of Tool 1.2, a transformation T n1 applied toS produces a symmetric system T n1 S. The sum of vectors in a symmetric systemT n1 S is zero, of course.

On the other hand, we can first partition S in accordance with its coloring intor symmetric subsystems S1, S2, . . . , Sr , then obtain T n1 S by applying the trans-formation T n1 to each system Si separately, and combining all T n1 Si . By Tool 1.2,T n1 Si is a symmetric system for i = 2, . . . , r , but T n1 S consists of n1 identicalnon-zero vectors. Therefore, the sum of all vectors of T n1 S is not zero. This contra-diction proves that the monochromatic polygons cannot be all non-congruent.

Prove the last sentence of Problem 1.1 on your own:

Problem 1.3 Prove that in the setting of Problem 1.1, the two congruent monochro-matic polynomials must exist among the monochromatic polynomials with the leastnumber of vertices.

Readers familiar with complex numbers may have noticed that in the proof ofProblem 1.1 we can choose the given n-gon Pn to be inscribed in a unit circle, andposition Pn with respect to the axes so that the symmetric system S of vectors could berepresented by complex numbers, which are precisely all n-th degree roots of 1. Thenthe transformation T m would simply constitute raising these roots into the m-th power.

1.3 Translation into the Tongue of APs

You might be wondering what this striking problem of colored polygons has incommon with arithmetic progressions (AP), which are part of the chapters title.Actually, everything! Problem 1.1 can be nicely translated into the language of infi-nite arithmetic progressions, or APs for short.3

Problem 1.4 In any coloring (partition) of the set of integers into finitely many infi-nite monochromatic APs, there are two APs with the same difference. Moreover,the largest difference necessarily repeats.

Equivalently:

Problem 1.5 Any partition of the set of integers into finitely many APs can beobtained only in the following way: N is partitioned into k APs, each of the samedifference k (where k is a positive integer greater than 1); then one of these APs ispartitioned into finitely many APs of the same difference, then one of these APs (atthis stage we have APs of two different differences) is partitioned into finitely manyAPs of the same difference, etc.

3 An infinite sequence a1, a2, . . . , an, . . . is called an arithmetic progression or AP, if there for anyinteger m > 1, we have the equality am = am1 + k for a fixed k, where k is a real number called thedifference of the arithmetic progression.


It was as delightful that our striking problem allowed two beautiful distinct for-mulations, as it was valuable: only because of that I was able to discover the prehis-tory of our problem.

1.4 Prehistory

Indeed, a year after I first published the story of this problem, in 1994 [Soi9],I discovered that this exquisite bagatelle of a problem actually had a prehistory! Ibecame aware of it while watching a video recording of Ronald L. Grahams mostelegant lecture Arithmetic Progressions: From Hilbert to Shelah. To my surprise,Ron mentioned our bagatelle in the language of integer partitions. Let me presentthe prehistory through the original e-mails, so that you would discover the story thesame way as I have.

April 5, 1995; Soifer to Graham:

In the beginning of your video Arithmetic Progressions, you present a problem ofpartitioning integers into APs. You refer to MirskyNewman. Can you give me a morespecific reference to their paper? You also mention that their paper may not contain theresult, but that it is credited to them. How come? When did they allegedly prove it?

April 5, 1995; Graham to Soifer:

Regarding the MirskyNewman theorem, you should probably check with Erdos. Idont know that there ever was a paper by them on this result. Paul is in Israel at TelAviv University.

April 6, 1995; Soifer to Erdos:

In the beginning of his video Arithmetic Progressions, Ron Graham presents a prob-lem of partitioning integers into arithmetic progressions (with the conclusion that twoprogressions have the same difference). Ron refers to MirskyNewman. He gives nospecific reference to their paper. He also mentions that their paper may not contain theresult, but that it is credited to them. . . Ron suggested that I ask you, which is what Iam doing.

I have good reasons to find this out, as in my previous book and in the one I writingnow, I credit Vasiliev (from Russia) with creating this problem before early 1970. Hecertainly did, which surely does not exclude others from discovering it independently,before or after Vasiliev.

April 8, 1995; Erdos to Soifer:

In 1950 I conjectured that there is no exact covering system in which all differences aredistinct, and this was proved by Donald J. Newman and [Leon] Mirsky a few monthslater. They never published anything, but this is mentioned in some papers of mine inthe 1950s (maybe in the Summa Brasil. Math. 11(1950), 113123 [E50.07], but I amnot sure).

8 I Merry-Go-Round

April 8, 1995; Erdos to Soifer:Regarding that Newmans proof, look at P. Erdos, on a problem concerning coveringsystems, Mat. Lapok 3(1952), 122128 [E52.03].I am looking at these early Erdoss articles. In the 1950 paper he introduces cov-

ering systems of (linear) congruences. Since each linear congruence x a (mod n)defines an AP, we can talk about covering system of APs and define it as a setof finitely many infinite APs, all with distinct differences, such that every integerbelongs to at least one of the APs of the system. In the 1952 paper [E52.03] Paulintroduces the problem for the first time in print (in Hungarian!):4

I conjectured that if system [of k APs with differences ni respectively] is covering thenk

i=1

1ni

> 1, (1.1)

that is, the system does not uniquely cover every integer. However, I could not provethis. For (1.1) Mirsky and Newmann [Newman] gave the following witty proof (thesame proof was found later by Davenport and Rado as well).Wow: Leon Mirsky, Donald Newman, Harold Davenport, and Richard Rado

quite a company of distinguished mathematicians worked on this bagatelle! Erdosthen proceeds [E52.03] with presenting this companys proof of his conjecture,which uses infinite series and limits.

In viewing old video recordings of Paul Erdoss lectures at the University ofColorado at Colorado Springs, I found a curious historical detail Paul mentioned inhis March 16, 1989 lecture: he created this conjecture in 1950 while traveling by carfrom Los Angeles to New York!

1.5 Completing the Go-Round

In 1959, Paul Erdos and Janos Suranyi published a book on the Theory of Numbers.In 2003 English translation [ESu2] of its 1996 2nd Hungarian edition, Erdos andSuranyi present the result from the Erdoss 1952 paper:

In a covering system of congruences [APs], the sum of the reciprocals of the moduli islarger than 1.

Erdos and Suranyi then repeat MirskyNewmanDavenportRado proof fromErdoss 1952 paper [E52.03]. Then there comes a surprise:

A. Lifsic [sic] gave an elementary solution to a contest problem that turned out to beequivalent to Theorem 3.

4 In English this result was briefly mentioned, without proof, much later, in 1973 [E73.21] and 1980[EG].


Based again on exercises 9 and 10, it is sufficient to prove that it is not possible tocover the integers by finitely many arithmetic progressions having distinct differencesin such a way that no two of them share a common element.

Erdos and Suranyi then repeat the trick that was first discovered by us, the judgesof the Soviet National Mathematical Olympiad in May 1970, the trick of convertingthe calculus solution into the Olympiads original problem about colored polygons!Here is how it goes:

Wind the number line around a circle of circumference d . On this circle, the integersrepresent the vertices of a regular d-sided polygon. . . The arithmetic progressions formthe vertices of disjoint regular polygons that together cover all vertices of the d-sidedpolygon.

Erdos and Suranyi continue by repeating, with credit, Sasha Livshitss solutionof Kolya Vasilievs Problem of Colored Polygons that we have seen at the start ofthis chapter.5 We have thus come to a full circle, a Merry-Go-Round from the SovietUnion Mathematical Olympiad to Erdos and back to the same Olympiad. I hope youhave enjoyed the ride!

5 Erdos and Suranyi obtained the translation of the problem into the language of polygons and the polyg-onal proof from the 1988 Russian book [VE] by Vasiliev and Andrei Egorov, which they credit for it. Inthis book, Vasiliev gives credit for the solution to Sasha Livshitsand in a sign of extreme modesty doesnot credit himself with creating this remarkable colored polygon problem independently from Erdos andin a different form.

In looking now at the original 1996 Hungarian 2nd edition [ESu1] of ErdosSuranyi book, I realizewith sadness that Paul Erdos did not see the beauties of Sasha Livshitss proofit did not appear inthe Hungarian edition of 1996, the year when Paul passed away. Clearly, Suranyi alone added Livshitssproof to the 2003 English translation [ESu2] of the book.

IIColored Plane

2Chromatic Number of the Plane: The Problem

A great advantage of geometry lies in the fact thatin it the senses can come to the aid of thought,and help find the path to follow.

Henry Poincare [Poi]

[I] cant offer money for nice problems of otherpeople because then I will really go broke. . .It is a very nice problem. If it were mine,I would offer $250 for it.

Paul Erdos Boca Raton, February, 1992

If Problem 8 [chromatic number of the plane] takesthat long to settle, we should know the answer by theyear 2084.

Victor Klee & Stan Wagon [KW]

Our good ole Euclidean plane, dont we know all about it? What else can there beafter Pythagoras and Steiner, Euclid and Hilbert? In this chapter we will look at anopen problem that exemplifies what is best in mathematics: anyone can understandthis problem; yet no one has been able to conquer it for over 58 years.

In August 1987, I attended an inspiring talk by Paul Halmos at Chapman Col-lege in Orange, California, entitled Some problems you can solve, and some youcannot. This problem was an example of a problem you cannot solve.

A fascinating problem . . . that combines ideas from set theory, combinatorics,measure theory, and distance geometry, write Hallard T. Croft, Kenneth J. Falconer,and Richard K. Guy in their book Unsolved Problems in Geometry [CFG].

If Problem 8 takes that long to settle [as the celebrated Four-Color Conjecture],we should know the answer by the year 2084, write Victor Klee and Stan Wagonin their book New and Old Unsolved Problems in Plane Geometry [KW].

Are you ready? Here it is:

What is the smallest number of colors sufficient for coloring the plane in sucha way that no two points of the same color are unit distance apart?

A. Soifer, The Mathematical Coloring Book, 13DOI 10.1007/978-0-387-74642-5 2, C Alexander Soifer 2009

14 II Colored Plane

This number is called the chromatic number of the plane and is denoted by . Tocolor the plane means to assign one color to every point of the plane. Please notethat here we color without any restrictions, and are not limited to nice, tiling-likeor map-like coloring. Given a positive integer n, we say that the plane is n-colored,if every point of the plane is assigned one of the given n colors.

A segment here will stand for just a 2-point set. Similarly, a polygon will standfor a finite set of point. Monochromatic set is a set whose all elements are assignedthe same color. In this terminology, we can formulate the Chromatic Number of thePlane Problem (CNP) as follows: What is the smallest number of colors sufficientfor coloring the plane in a way that forbids monochromatic segments of length 1?

I do not know who first noticed the following result. Perhaps, Adam? Or Eve?To be a bit more serious, I do not think that ancient Greek geometers, for example,knew this nice fact, for they simply did not ask this kind of questions!

Problem 2.1 (Adam & Eve?) No matter how the plane is 2-colored, it contains amonochromatic unit distance segment, i.e.,

3.

Solution: Toss on the given 2-colored plane an equilateral triangle T of side 1(Fig. 2.1). We have only 2 colors while T has 3 vertices (I trust you have not forgot-ten the Pigeonhole Principle). Two of the vertices must lie on the same color. Theyare distance 1 apart.

1

11

Fig. 2.1

We can do better than Adam:

Problem 2.2 No matter how the plane is 3-colored, it contains a monochromaticunit distance segment, i.e.,

4.

Solution by the Canadian geometers, brothers Leo and William Moser, (1961,[MM]) Toss on the given 3-colored plane what we now call The Mosers Spindle(Fig. 2.2). Every edge in the spindle has the length 1.

2 Chromatic Number of the Plane: The Problem 15

Fig. 2.2 The Mosers Spindle

Assume that the seven vertices of the spindle do not contain a monochromaticunit distance segment. Call the colors used to color the plane red, white, and blue.The solution now will faithfully follow the childrens song: A B C D E F G. . ..

Let the point A be red, then B and C must be one white and one blue, there-fore D is red. Similarly E and F must be one white and one blue, therefore G isred. We found a monochromatic segment DG of length 1 in contradiction to ourassumption.

Observe: The Mosers Spindle has worked for us in solving Problem 2.2 preciselybecause any 3 points of the spindle contain two points distance 1 apart. This impliesthat in a Mosers spindle that forbids monochromatic distance 1, at most 2 pointscan be of the same color. Remember this observation, for we will need it later inChapters 4 and 40.

When I presented the Mosers solution to high school mathematicians, everyoneagreed that it was beautiful and simple. But how do you come up with a thing likethe spindle? I was asked. As a reply, I presented a less elegant but a more natu-rally found solution. In fact, I would call it a second version of the same solution.Here we touch on a curious aspect of mathematics. In mathematical texts we oftensee second solution, third solution, but which two solutions ought to be calleddistinct? We do not know: it is not defined, and thus is a judgment call. A distinctsolution for one person could be the same solution for another. It is interesting tonotice that both versions were published in the same year, of 1961, one in Canadaand the other in Switzerland.

Second Version of the Solution (Hugo Hadwiger, 1961, [Had4]). Assume thata 3-colored redwhiteblue plane does not contain a monochromatic unit distancesegment. Then an equilateral triangle ABC of side 1 will have one vertex of eachcolor (Fig. 2.3). Let A be red, then B and C must be one white and one blue. Thepoint A symmetric to A with respect to the side BC must be red as well. If we rotateour rhombus ABAC through any angle about A, the vertex A will have to remainred due to the same argument as above. Thus, we get a whole red circle of radius

16 II Colored Plane

Fig. 2.3

AA (Fig. 2.3). Surely, it contains a cord d of length 1, both endpoints of which arered, in contradiction to our assumption.

Does an upper bound exist for ? It is not immediately obvious. Can you findone? Think of tiling the plane with square tiles!

Problem 2.3 There is a 9-coloring of the plane that contains no monochromatic unitdistance segments, i.e.,

9.

Proof We tile the plane with unit squares. Now we color one square in color 1, andits eight neighbors in colors 2, 3, . . . , 9 (Fig. 2.4). The union of these 9 squares isa 3 3 square S. Translates of S (i.e., images of S under translations) tile the planeand determine how we color it in 9 colors.

You can easily verify (do) that no distance d in the range 2 < d < 2 is realizedmonochromatically in the plane. Thus by shrinking all linear sizes by the factor of,say, 1.5, we get a coloring that contains no monochromatic unit distance segments.(Observe: due to the above inequality, we have enough cushion, so that it does notmatter in which of the two adjacent colors we color the boundaries of squares).

Fig. 2.4

Now that a tiling has helped us to solve the above problem, it is natural to askwhether another tiling can help us improve the upper bound. One can indeed!

2 Chromatic Number of the Plane: The Problem 17

Problem 2.4 There is a 7-coloring of the plane that contains no monochromatic unitdistance segments, i.e.,

7.

Solution ([Had3]): We can tile the plane by regular hexagons of side 1. Now wecolor one hexagon in color 1, and its six neighbors in colors 2, 3, . . . , 7 (Fig. 2.5).The union of these seven hexagons forms a flower P , a highly symmetric polygonP of 18 sides. Translates of P tile the plane and determine how we color the plane in7 colors. It is easy to compute (please do) that each color does not have monochro-matic segments of any length d, where 2 < d 4. I believe such an n exists but its valuemay be very large.10

A certainty comes in 1980 [E80.38] and [E80.41]:I am sure that [the chromatic number of the plane] 2 > 4 but cannot prove it.In 1981 [E81.23] and [E81.26] we read, respectively:It has been conjectured [by E. Nelson] that 2 = 4, but now it is generally believedthat 2 > 4.

It seems likely that (E2

)> 4.

In 1985 [E85.01] Paul Erdos writes:I am almost sure that h(2) > 4.Oncejust onceErdos expresses mid-value expectations, just as Ron Graham

has in his Conjectures 3.3 and 3.4. It happened on Thursday, March 10, 1994 atthe 25th South Eastern International Conference on Combinatorics, Computing andGraph Theory in Boca Raton. Following Erdoss plenary talk (9:3010:30 A.M.),I was giving my talk at 10:50 A.M., when suddenly Paul Erdos said (and I jottedit down):

9 Graham cites Paul ODonnells Theorem 45.4 (see it later in this book) as perhaps, the evidence that is at least 5.10 If the chromatic number of the plane is 7, then for G(x1, . . . , xn) = 7 such an n must be greater than6197 [Pri].

3 Chromatic Number of the Plane: An Historical Essay 31

Excuse me for interrupting, I am almost sure that the chromatic number of the planeis greater than 4. It is not a proof, but any measurable set without distance 1 in a verylarge circle has measure less than 1/4. I also do not think that it is 7.

It is time for me to speak on the record and predict the chromatic number ofthe plane. I am leaning toward predicting 7 or else 4somewhat disjointly fromGraham and Erdoss apparent expectation. Limiting myself to just one value, I con-jecture:Chromatic Number of the Plane Conjecture 3.5 11

= 7.

If you, in fact, prove the chromatic number is 7 or 4, I do not think you wouldlose Grahams prizes. I am sure Ron will pay his prizes for disproofs as well asfor proofs. On January 26, 2007 in a personal e-mail, Graham clarified the terms ofawarding his prizes:

I always assume that we are working in ZFC (for the chromatic number of the plane!).My monetary awards can vary depending on which audience I am talking to. I alwaysgive the maximum of whatever I have announced (and not the sum!).

11 See more predictions in Chapter 47.

4Polychromatic Number of the Plane and ResultsNear the Lower Bound

When a great problem withstands all assaults, mathematicians create related prob-lems. It gives them something to solve, plus sometimes there is an extra gain in thisprocess, when an insight into a related problem brings new ways to see and conquerthe original one. Numerous problems have been posed around the chromatic numberof the plane. I would like to share with you my favorite among them.

It is convenient to say that a monochromatic set S realizes distance d if S containsa monochromatic segment of length d; otherwise we say that S forbids distance d.

Our knowledge about this problem starts with the celebrated 1959 book by HugoHadwiger and Hans Debrunner ( [HD2], and subsequently its enhanced translationsinto Russian by Isaak M. Yaglom [HD3] and into English by Victor Klee [HDK]).Hadwiger reported in the book the contents of the September 9, 1958 letter hereceived from the Hungarian mathematician A. Heppes:

Following an initiative by P. Erdos he [i.e., Heppes] considers decompositions of thespace into disjoint sets rather than closed sets. For example, we can ask whether propo-sition 59 remains true in the case where the plane is decomposed into three disjointsubsets. As we know, this is still unresolved.

In other words, Paul Erdos asked whether it was true that if the plane were parti-tioned (colored) into three disjoint subsets, one of the subsets would have to realizeall distances. Soon the problem took on its current appearance. Here it is.Erdoss Open Problem 4.1 What is the smallest number of colors needed for col-oring the plane in such a way that no color realizes all distances?12

This number had to have a name, and so in 1992 [Soi5] I named it the polychro-matic number of the plane and denoted it by p. The name and the notation seemedso natural that by now it has become standard, and has (without credit) appeared insuch encyclopedic books as [JT] and [GO].

Since I viewed this to be a very important open problem, I asked Paul Erdos toverify his authorship, suggested in passing by Hadwiger. As always, Paul was verymodest in his July 16, 1991 letter to me [E91/7/16ltr]:

12 The authors of the fine problem book [BMP] incorrectly credit Hadwiger as first to study this prob-lem (p. 235). Hadwiger, quite typically for him, limited his study to closed sets.

32 A. Soifer, The Mathematical Coloring Book,DOI 10.1007/978-0-387-74642-5 4, C Alexander Soifer 2009

4 Polychromatic Number of the Plane and Results Near the Lower Bound 33

I am not even quite sure that I created the problem: Find the smallest number of col-ors for the plane, so that no color realizes all distances, but if there is no evidencecontradicting it we can assume it for the moment.

My notes show that during his unusually long 2-week visit in December 1991January 1992 (we were working together on the book of Pauls open problems, soonto be completed and published by Springer under the title Problems of pgom Erdos),Paul confirmed his authorship of this problem. In the chromatic number of the planeproblem, we were looking for colorings of the plane such that each color forbidsdistance 1. In the polychromatic number problem, we are coloring the plane in such away that eachcolor i forbidsadistancedi . Fordistinct colors i and j , thecorrespondingforbidden distances di and d j may (but do not have to) be distinct. Of course,

p .

Therefore,

p 7.

Nothing else had been discovered during the first 12 years of this problems life.Then, in 1970, Dmitry E. Raiskii, a student of the Moscow High School for WorkingYouth13 105, published ( [Rai]) the lower and upper bounds for p. We will lookhere at the lower bound, leaving the upper bound to Chapter 6.

Raiskiis Theorem 4.2 (D. E. Raiskii [Rai]) 4 p.Three years after Raiskiis publication, in 1973 the British mathematician

Douglas R. Woodall from the University of Robin Hood (I mean Nottingham),published a paper [Woo1] on problems related to the chromatic number of theplane. Among other things, he gave his own proof of the lower bound. As I showedin [Soi17], Woodalls proof stemmed from a triple application of two simple ideasof Hugo Hadwiger ( [HDK], Problems 54 and 59).

In 2003, the Russian turned Israeli mathematician Alexei Kanel-Belov commu-nicated to me an incredibly beautiful short proof of this lower bound by the newgeneration of young Russian mathematicians, all his students. The proof was foundby Alexei Merkov, a 10th grader from the Moscow High School 91, and commu-nicated by Alexei Roginsky and Daniil Dimenstein in 1997 at a Moscow PioneerPalace [Poisk]. The following is the authors proof with my gentle modifications.Proof of the Lower Bound (A. Merkov): Assume the plane is colored in three col-ors, red, white and blue, but each color forbids a distance: r, w, and b respectively.Equip the 3-colored plane with the Cartesian coordinates with the origin O , andconstruct in the plane three seven-point sets Sr , Sw and Sb each being the MosersSpindle (Fig. 2.2), such that all spindles share O as one of their seven vertices,

13 Students in such high schools hold regular jobs during the day, and attend classes at night.

34 II Colored Plane

and have edges all equal to r, w, and b respectively. This construction defines 6red vectors v1, . . . , v6 from the origin O to each remaining point of Sr ; 6 whitevectors v7, . . . , v12 from O to the points of Sw; and 6 blue vectors v13, . . . , v18from O to the points of Sb 18 vectors in all.

Introduce now the 18-dimensional Euclidean space E18 and a function M fromR18 to the plane R2 naturally defined as follows: (a1, . . . , a18) a1v1+. . .+a18v18.This function induces a 3-coloring of R18 by assigning a point of R18 the color ofthe corresponding point of the plane. The first six axes of E18 we will call red, thenext six axes white, and the last six axes blue.

Define by W the subset in E18 of all points whose coordinates include at mostone coordinate equal to 1 for each of the three colors of the axes, and the rest (15or more) coordinates 0. It is easy to verify (do) that W consists of 73 points. Forany fixed array of allowable in W coordinates on white and blue axes, we get the7-element set A of points in W having these fixed coordinates on white and blueaxes. The image M(A) of the set A under the map M forms in the plane a translate ofthe original seven-point set Sr . If we fix another array of white and blue coordinates,we get another 7-element set in E18, whose image under M would form in the planeanother translate of Sr . Thus, the set W gets partitioned into 72 subsets, each ofwhich maps into a translate of Sr .

Now recall the observation we m

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