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Applied Mathematical Sciences Yusheng Li Qizhong Lin Elementary Methods of Graph Ramsey Theory Applied Mathematical Sciences Volume 211 Series Editors Anthony Bloch, Department of Mathematics, University of Michigan, Ann Arbor, MI, USA C. L. Epstein, Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA Alain Goriely, Department of Mathematics, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, NY, USA Advisory Editors J. Bell, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA, USA P. Constantin, Department of Mathematics, Princeton University, Princeton, NJ, USA R. Durrett, Department of Mathematics, Duke University, Durham, CA, USA R. Kohn, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA R. Pego, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA L. Ryzhik, Department of Mathematics, Stanford University, Stanford, CA, USA A. Singer, Department of Mathematics, Princeton University, Princeton, NJ, USA A. Stevens, Department of Applied Mathematics, University of Münster, Münster, Germany S.Wright,ComputerSciencesDepartment,UniversityofWisconsin,Madison,WI, USA Founding Editors F. John, New York University, New York, NY, USA J. P. LaSalle, Brown University, Providence, RI, USA L. Sirovich, Brown University, Providence, RI, USA The mathematization of allsciences,the fading oftraditional scientiﬁcboundaries, theimpactofcomputertechnology,thegrowingimportanceofcomputermodeling and the necessity of scientiﬁc planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purposeofthisseriesistoprovidesuchbooks,suitablefortheuserofmathematics, themathematicianinterestedinapplications,andthestudentscientist.Inparticular, this series will provide an outlet for topics of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. These books should be accessible to readers versed in mathematics or science and engineering, and will feature a lively tutorial style, a focus on topics of current interest, and present clear exposition of broad appeal. AcomplimenttotheAppliedMathematicalSciencesseriesistheTextsinApplied Mathematicsseries,whichpublishestextbookssuitableforadvancedundergraduate and beginning graduate courses. Yusheng Li • Qizhong Lin Elementary Methods of Graph Ramsey Theory 123 Yusheng Li Qizhong Lin Department of Mathematics Center for Discrete Mathematics Tongji University Fuzhou University Shanghai, China Fuzhou, China ISSN 0066-5452 ISSN 2196-968X (electronic) Applied Mathematical Sciences ISBN 978-3-031-12761-8 ISBN 978-3-031-12762-5 (eBook) https://doi.org/10.1007/978-3-031-12762-5 Mathematics Subject Classification (2020): 05D10, 05C35, 05D40 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Toourfamilies Preface RamseytheoryisnamedafterBritishmathematicianFrankP.Ramsey(February22, 1903–January19,1930)whopublishedapaper“Onaproblemofformallogic”in 1929. Ramsey theory has become a flourishing branch of extremal combinatorics. JustasTheodoreS.Motzkinpointedout,themainspiritofRamseytheoryisthat “Completedisorderisimpossible!” Ramsey theory was subsequently developed by Paul Erdős (March 26, 1913– September20,1996),aHungarianmathematician,whowasworkingonmanymath- ematicalproblems,particularlyincombinatorics,graphtheoryandnumbertheory. EarlierthanFrankP.Ramsey,IssaiSchur(January10,1875–January10,1941)and van der Waerden (February 2, 1903–January 2, 1996) obtained similar results in number theory. We refer the reader to the book Ramsey Theory by Graham, Roth- schildandSpencer(1990)forasystematicallyintroductionandthebookErdőson Graphs:HisLegacyofUnsolvedProblemsbyChungandGraham(1999)formany unsolved problems. As an important method on Ramsey theory, we would like to referthereadertothebookTheProbabilisticMethodbyAlonandSpencer(2016)for asystematicallyintroduction.Foracomprehensiveunderstandingofrandomgraphs whicharecloselyrelatedtoRamseytheory,wereferthereadertothreebooksonthis field:ThebooksRandomGraphsbyBollobás(2001,2nded.),RandomGraphsby Janson,ŁuczakandRuciński(2000),andIntroductiontoRandomGraphsbyFrieze andKaroński(2016). ThenumberofresearchpapersonRamseytheorybefore1970swasnotsubstan- tial.TheCombinatorialConferenceatBalatonfüred,Hungary1973,inhonorofPaul Erdősforhis60thbirthday,wasamilestoneinRamseytheoryhistory.Therewere morethantwodozentalksdevotedtowhatisnowcalledRamseytheory.Manypa- pershavebeenpublishedafterthisconference.Onestrikingfeatureistheinvention ofmanymodernmethodsthatinvolveideasfromvariousbranchesofmathematics suchasprobability,algebra,geometry,andanalysis. Graph Ramsey theory is an important area that serves not only as an abundant sourcebutalsoasatestinggroundofthesemethodsandmanyothernewmethods. vii viii Preface Despite substantial advances in graph Ramsey theory, most outstanding problems arefarfrombeingsolved.Thenewinsightsgeneratedbytacklingtheseproblemswill most likely lead to new tools and techniques. Due to these reasons, graph Ramsey theoryisfullofvitalityandhencedeservesmuchmoreresearchefforts. Thebookemphasizesmakingthetexteasierforthestudentstolearn.Toovercome difficultiestoaccesssporadicresultsinanextensiveliterature,wesetouttodescribe thematerialinthiselementarybook,whichaimstoprovideanintroductiontograph Ramsey theory. The prerequisites for this book are minimal: we only require that the reader be familiar with elementary level of graph theory, calculus, probability and linear algebra. To make this book as self-contained as possible, we attempt to introducethetheoryfromscratch,forinstance,someresultsrelyonthepropertiesof finitefields,sowelaiddownthebackgroundbeforehand.Webelievethatthisbook, intendedforbeginninggraduatestudents,canserveasanentrancetothisbeautiful theory. To facilitate better understanding of the material, this book contains some standard exercises in which a large part of the exercises are not difficult since our bookservesasaprimeronthistopic. Wehaveusedthemanuscriptofthebookaslecturenotesmorethan20yearsin many universities including these in mainland of China, Hong Kong and Taiwan, etc.WealsouseditmanytimesforsummerschoolssupportedbyNaturalScience FoundationofChina.Theselectedtopicsarealmostindependentsothatbeginners may skip some chapters, sections, and proofs, particularly that are marked with asterisks.Wearesorryfornotbeingabletoincorporatemanydeepresultsintothis book. As most listeners in the short terms are preferably interested in the specific topics, they can obtain a clearer picture on the topics from the selected chapters insteadofthewholebook. There are thirteen chapters in this book, divided mainly according to both the content of the book and methods used for the problems. In Chapter 1, we will introducesomebasicdefinitionsanddiscusstheexistencesofRamseynumbersby givingupperbounds.InChapter2,wewillconsiderseveralsmallRamseynumbers and a Ramsey number on integers, i.e., Schur number on integers. For algebraic constructions in this chapter, we shall recall some basics of finite fields briefly. In Chapter3,wewillfocusonthebasicmethodsuchasverticesarelabeledorpicked randomlyorsemi-randomly,inwhichwealwayscomputetheexpectationsofrandom variables.Thefrequently-usedmethodstoestimatetheprobabilityofavariablefrom expectation including Markov’s inequality and Chernoff bound will be introduced inthischapter.InChapter4,wewillgiveanoverviewonrandomgraphswhichnow has become a flourishing branch. Applications to classic Ramsey numbers due to Erdős (1947) will be given in this chapter, which is always considered as the first consciousapplicationoftheprobabilisticmethod,andthegraphRamseytheoryis always refereed to as the birthplace of random graphs. This chapter also contains thresholdfunctionsforrandomgraphswithcertainproperties.InChapter5,wewill introduceLovászLocalLemmathatrelaxestheindependenceofpairwiseeventsto partialindependence.WewillalsogiveanoverviewoftheMartingalesandtriangle- free process. In Chapter 6, we shall consider some constructive lower bounds of Ramseynumbers,whichtellsusthattheprobabilisticmethodismorepowerfulthan Preface ix constructivemethodforlowerboundsofmostnon-linearRamseyfunctions.Also,we introduceadisproofoftheconjectureofBorsukingeometrythatisasurprisingby- productofgraphRamseytheory.Additionally,thischaptercontainsbasicproperties ofintersectinghypergraphs.InChapter7,Turánnumberswillbeintroduced,inwhich theTuránnumbersofbipartitegraphsaretightlyrelatedtothecorrespondingRamsey numbersinmanycolors.InChapter8,wewillintroducecommunicationchannel,and theconnectionbetweenRamseytheoryandcommunicationchannelwillberevealed. InChapter9,wewillintroducethemethodofthedependentrandomchoice,which can be applied to embed a small or sparse graph into a dense graph. Chapter 10 focusesonquasi-randomgraphsandregulargraphswithsmallsecondeigenvalues, for which some deep applications especially some graph Ramsey numbers will be included. In Chapter 11, we will introduce an important Ramsey number on integers, i.e. van der Waerden number on arithmetic progression. We will also introduceSzemerédi’sregularitylemmawhichassertsthateverylargegraphcanbe decomposedintoafinitenumberofpartssothattheedgesbetweenalmosteverypair ofpartsformsa“random-looking”graph.Wewillgivesomeapplicationsincluding aclassicapplicationongraphswithboundedmaximumdegreeandaRamsey-Turán numberbyusingtheregularitylemma.Severalextensionsontheregularitylemma will be given. In Chapter 12, we shall discuss some more examples on Ramsey linearfunctions.Thefirstsectionofthechapterdiscussesthelinearityofsubdivided graphs,andthesecondisonaspeciallinearity:socalledRamseygoodness,proposed byBurrandErdős(1983).TherearealotofvariantsongraphRamseytheory,some ofwhichwillbeintroducedinChapter13,includingsizeRamseynumbers,induced Ramseytheorem,bipartiteRamseynumbers,andFolkmannumbers,etc. WearedeeplyindebtedtotheseprofessorswhohelpedustolearnRamseytheory, andcolleagueswhoorganizedseminarsandsummerschools,aswellasstudentswho attendedtheclasses.Inparticular,wearedeeplyindebtedtoProfessorWenanZang whoshouldbeacoauthorifheisnotsobusysincealargepartofthebookischosen fromthelecturenotesIntroductiontoGraphRamseyTheorybyY.LiandW.Zang. Finally, we would like to thank the National Science Function of China and the ResearchGrantsCouncilofHongKongfortheirfinancialsupport. YushengLi, TongjiUniversity Jan.2022 QizhongLin, FuzhouUniversity Contents 1 Existence...................................................... 1 1.1 Terminology............................................... 1 1.2 GeneralUpperBounds ...................................... 3 1.3 UpperBoundsfor𝑟𝑘(3) ..................................... 7 1.4 SomeEarlyRamseyNumbers ................................ 10 1.5 HypergraphRamseyNumber................................. 14 1.6 Exercises ................................................. 17 2 SmallRamseyNumbers......................................... 19 2.1 RamseyFolklore ........................................... 19 2.2 FiniteFieldand𝑟 (3) ....................................... 21 3 2.3 SchurNumbers ............................................ 25 2.4 PaleyGraphs .............................................. 32 2.5 CombinationofPaleyGraphs ................................ 37 2.6 SpectrumandIndependenceNumber .......................... 40 2.7 Exercises ................................................. 44 3 BasicProbabilisticMethod...................................... 47 3.1 SomeBasicInequalities ..................................... 47 3.2 ALowerBoundof𝑟(𝑛,𝑛) ................................... 50 3.3 PickVerticesSemi-Randomly ................................ 51 3.4 IndependenceNumberofSparseGraphs ....................... 54 3.5 UpperBoundsfor𝑟(𝑚,𝑛).................................... 57 3.6 OddCycleversusLarge𝐾𝑛 .................................. 59 3.7 TheFirstTwoMoments ..................................... 65 3.8 ChernoffBounds ........................................... 67 3.9 Exercises ................................................. 72 4 RandomGraph ................................................ 75 4.1 Preliminary ............................................... 75 4.2 LowerBoundsfor𝑟(𝑚,𝑛) ................................... 78 xi