Co mbinatorics Probability and Computing - PDF document

Co mbinatorics Probability and Computing
Co mbinatorics Probability and Computing

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2003 Cambridge University Press DOI 101017S0963548303005741 Printed in the United Kingdom Tur an Numbers of Bipartite Graphs and Related RamseyType Questions NOGA ALON MICHAEL KRIVELEVICH an BENNY SUDAKOV Institute for Advanced Study Princeton NJ 08 ID: 61561 Download Pdf

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2003 Cambridge University Press

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Alon,M.KrivelevichandB.SudakovTheseresultsandsomerelatedonesarederivedfromasimpleandyetsurprisinglypowerfullemma,proved,usingprobabilistictechniques,atthebeginningofthepaper.Thislemmaisare“nedversionofearlierresultsprovedandappliedbyvariousresearchersincludingRodl,Kostochka,GowersandSudakov.1.IntroductionAllgraphsconsideredhereare“nite,undirectedandsimple.Foragraphandaninteger,theTurannumberex(n,Hhemaximumpossiblenumberofedgesinasimplegraphverticesthatcontainsnocopyofheasymptoticbehaviourofthesenumbersforgraphsofchromaticnumberatleast3iswellknown:see,ee,4].Forbipartitegraphsever,thesituationisconsiderablymorecomplicated,andtherearerelativelyfewnontrivialbipartitegraphsforwhichtheorderofmagnitudeofex(n,H)isknown.Our“rstresulthereassertsthat,forevery“xedbipartitegraphinwhichthedegreesofallverticesinonecolourclassareatmost,ex(n,HThisresult,whichcanalsobederivedfromanearlierresultofFuredi[14],istightforevery“xed,asshownbytheconstructionsin[17]and[2].Ourproofisdierentfromthatin[14],andprovidessomewhatstrongerestimates.Agraphis-degenerateifeveryoneofitssubgraphscontainsavertexofdegreeat.AnoldconjectureofErdos([9],seealso[7],[13])assertsthat,forevery“xeddegeneratebipartitegraph,ex(n,HHereweprovethatthereisanabsoluteconstant0,suchthat,foreverysuch,ex(n,HOurtechniquehereprovidesseveralRamsey-typeresultsaswell.FortwographsheRamseynumberG,Hheminimumnumbersuchthat,inanycolouringoftheedgesofthecompletegraphonverticesbyredandblue,thereisaredcopyofbluecopyof.IfwesometimesdenoteG,G)byOur“rstRamsey-typeresultisthat,foreverygraphvertices,maximumdegreeandchromaticnumber2,andforeveryintegerH,K log(logThisisnearlytightfor=2,butisprobablyfarfrombeingtightforlargevaluesofOneofthebasicresultsinRamseyTheoryisthefactthat,forthecompletegraph .AconjectureofErdos(see[7])assertsthatthereisanabsoluteconstantsuchthat,foranygraphedges, .Hereweprovethisconjectureforbipartitegraphs,andprovethat,forgeneralgraphsedges, logomeabsolutepositiveconstantThebasictoolintheproofofmostoftheresultshereisasimpleandyetsurprisinglypowerfullemma,whoseproofisprobabilistic.Anearlyvariantofthislemmawas“rstprovedin[8]and[19],andversionsthatareclosertotheoneweproveandapplyherehavebeenprovedandappliedin[15],[23],[20]and[3].Thereisnodoubtthatvariantsofthelemmawill“ndadditionalapplicationsaswell.Ournotationismostlystandard.Hereissomelessconventionalnotation.GivenagraphV,E),forv,U)bethesetofallneighbours anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestionsv,Uv,U;letalsov,V);forasubsetv,u)foreveryhecommonneighbourhoodofTherestofthepaperisorganizedasfollows.InthenextsectionweproveourbasiclemmaandapplyittoboundingtheTurannumbersofbipartitegraphswithboundeddegreesononeside.InSection3weboundtheTurannumbersofdegeneratebipartitegraphs.InSections4and5weprovetheRamsey-typeresultsmentionedabove,andinSection6weimprovetheestimateofFuredifortheTurannumbersofcertaingenericbipartitegraphs.The“nalsectioncontainssomeconcludingremarksandopenproblems.Throughoutthepaperwemakenoattemptstooptimizevariousabsoluteconstants.Tosimplifythepresentation,weoftenomit”oorandceilingsignswheneverthesearenotcrucial.Alllogarithmsareinthenaturalbaseunlessotherwisespeci“ed.2.Turannumbersofbipartitegraphsofgivenmaximumdegreestartwiththefollowingbasiclemma,whoseproofisprobabilistic.Lemma2.1.a,b,n,rbepositiveintegers.LetV,Ebeagraphonverticeswithaveragedegree.If nrŠ1ŠnrŠ1 ontainsasubsetofatleastverticessuchthateveryverticesofhaveatmmonneighbours.Proof.beasubsetofrandomverticesofhosenuniformlywithrepetitions.denotethecardinalityoflinearityofexpectation, nr=1 nrvV|N(v)|r1 nrnvV|N(v)| nr=1 nrŠ12|E(G)| nr=dr wheretheinequalityfollowsfromtheconvexityofdenotetherandomvariablecountingthenumberof-tuplesinwithfewercommonneighbours.Foragiven-tupleheprobabilitythatwillbeasubsetofisprecisely( .Asthereareatmost(ofcardinalityforwhich1,itfollowsthat nr. Alon,M.KrivelevichandB.SudakovApplyinglinearityofexpectationonceagainandrecallingcondition(2.1)ofthelemma,weconcludethat nrŠ1ŠnrŠ1 HencethereexistsachoiceforsuchthatforthecorrespondingsetwegetPicksuchaset,andforevery-tuplefromwithfewerthancommonneighbours,deleteonevertexfrom.Denotetheobtainedsetby.The,andevery-tupleofverticesofhasatleastcommonneighbours.Thiscompletestheproof. Theorem2.2.B,Fbeabipartitegraphwithsidesofsizesespectively.Supposethatthedegreesofallverticesdonotexceed.LetV,Ebeagraphonverticeswithaveragedegree.If nrŠ1Šnr+bŠ1 ontainsacopyofProof.betheverticesof.ByLemma2.1(withplayingtheroleof)thereisasubsetardinalitysuchthatevery-subsetofatleastcommonneighboursinextwe“ndanembeddingofdescribedbyaninjectivefunction).Startbyde“ningtobeanarbitrarybijection.Nowembedtheverticesofne.Supposethatthecurrentvertextobeembeddedisheassumptiononhasatmostneighboursin,allofthemobviouslyin.LetbethesetofneighboursofThesetofimagesisasubsetofofcardinalityatmost,andhasthereforeatleastcommonneighboursinhetotalnumberofverticesembeddedsofarisstrictlylessthan,thereisavertex)connectedtoallverticesinndnotusedintheembeddingpreviously.Setfromtheabovedescriptionthatoncetheembeddingends,thefunctionproducesacopy Corollary2.3.beabipartitegraphwithmaximumdegreeononeside.Thenthereexistsaconstantsuchthatn,H tethatthelastcorollaryistightforeveryvalueof2.Indeed,bytheconstructionin[2](modifyingthatin[17]),andbythewell-knownresultsof[21],forevery“xed1)!+1theTurannumberofthecompletebipartitegraphr,sis(NotealsothattheassertionofthecorollarycanbededucedfromthemainresultofFurediin[14].AnimprovedversionofhisresultisprovedinSection6.3.TurannumbersofbipartitedegenerategraphsRecallthatagraphis-degenerateifeveryoneofitssubgraphscontainsavertexofdegreeatmosteedthefollowingeasyandwell-knownfact. anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestionsProposition3.1.U,Fbean-degenerategraphonvertices.ThenthereisanorderingoftheverticesofsuchthatforeverythevertexatmostjiThefollowinglemmaissimilartoaresultprovedin[20].Lemma3.2.everyintegerandeveryintegereverygraphV,Everticesandatleast edgescontainsdisjointsetssuchthatevery-tupleofverticesinhasatleast mmonneighboursinandevery-tupleofverticesinhasatleast mmonneighboursinProof.Note“rstthat,since 1000.PartitionthevertexsetintodisjointsetsA,Bofcardinalities 2,|B|=n suchthatatleasthalfoftheedgesofcrossbetweenTheexistenceofsuchapartitioncanbeproved,forexample,bychoosingasetofthedesiredsizeatrandomandbyestimatingtheexpectednumberofedgesbetweenanditscomplement.)Letdenotethebipartitesubgraphofconsistingofalledgesofbetw.Obviously, 2|E(G)|1 2n2Š1 Chooseatrandomasubsetconsistingof4(notnecessarilydistinct)randommembersof.De.Lettherandomvariablecountingthenumberof3-tuplesincommonneighbourhoodinhasfewerthan vertices.Weestimatetheexpectations,thatis,,X]=aAdG1(a,B |B|4r|A||E(G1)| |A|4r24rŠ1n1Š4r wherethe“rstinequalityfollowsfromtheconvexityofInordertoestimatetheexpectedvalueof,observethatfora“xed3-tupletheprobabilitythatwillbeasubsetofisprecisely Asthereareatmostofcardinality3forwhichollowsthat |B|4re|A| 3r3rn0.1 |B|4r=e|A| 3r|B|3rn0.4 BylinearityofexpectationweconcludethatthatXŠY]=E[X]ŠE[Y]n0.6.Hencethereexistsachoiceofforwhich.Choosesucha,andforeach3-tupleinwithfewerthancommonneighbours,deleteonevertexfromollowsthatthereisasetofcardinalitysuchthatevery3-tupleinhasatleastcommonneighboursinFixanasabove. Alon,M.KrivelevichandB.Sudakovseatrandomasubsetconsistingof2(notnecessarilydistinct)uniformlychosenmembersof.Notet,andestimatetheprobabilitythatcontainsan-tuplewhosecommonneighbourhoodhaslessthan vertices.AsinthecalculationofofY]above,thisprobabilityisat |A1|2r|B|r r!(nŠ0.5)2rnr Hencethereexistsachoiceofforwhichevery-tupleinhasatleastcommonneighboursinWeclaimthatthepair(ul“lstherequirementsofthelemma.Indeed,fordesiredpropertyholdsby(3.2).Toshowitfor,consideranarbitrarysubsetofcardinality.As3.1)thesethasatcommonneighboursinbserve,crucially,thatbythede“nitionofcommonneighboursofbelongtoollowsthat.As),thestatementisproved. Theorem3.3.EverygraphV,Everticeswith edgescontainsevery-degeneratebipartitegraphB,FProof..Ordertheverticesofinsuchawaythat,forevery1avertex)hasatmostneighboursprecedingit.Suchanorderingispossiblebyposition3.1.applyLemma3.2todisjointsubsets)suchthatevery-tupleofverticesinhasatleastcommonneighboursin,andevery-tupleinatleastcommonneighboursinstructanembedding)byplacingimagesofverticesfrominto,andimagesofverticesofintoconstructthedesiredembedding,weproceedaccordingtothechosenorder(oftheverticesof.Ifthecurrentvertex)isavertexfrom,we“rstlocatetheimagesji,ofthealreadyembeddedneighboursof.Theji,isasubsetofofcardinalityatmost.Itthereforehasatleastcommonneighboursin,andobviouslnotallofthemhavealreadybeenusedintheembedding.Wepickoneunusedvertexandset.If,wecanrepeattheaboveargument,interchangingtherolesof Corollary3.4.every-degeneratebipartitegraphverticesandforeveryn,H Infacttheconstant10inthiscorollarycanbeimprovedto4asstatedinthefollowingtheorem,whoseproofissimilartothatofoneofthelemmasin[3].Thistheoremalso anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestionsimprovestheestimateinTheorem3.3,butwebelieveitisinstructivetoincludethesomewhatsimplerproofofthattheoremaswell,andpresentthenextproofseparately.Theorem3.5.beabipartite-degenerategraphoforder.Then,foralln,H Proof.Theclaimistrivialfor=1andwethusassume2.Letbeagraphofordewithatleast edges.AsdescribedintheproofofLemma3.2,thereisabipartitesubgraphwithpartsofsizessuch 2|E(G)|1 2h1/2rn2Š1 Chooseatrandomanorderedsubsetconsistingof2(notnecessarilydistinct)randommembersof.De.Let.Lettherandomvariablecountingthenumberofordered3-tuplesofverticesincommonneighbourhoodinhasfewerthanvertices.Wenextestimatetheexpectationsandof.Usingtheconvexityof,wegettX]=aAdG1(a,B |B|2r|A||E(G1)| |A|2r22rŠ2Š2r whereweusedthefactthat�n/ByJensensinequalityandthefactthattheunctionisconvex,nvex,X]2r(2h)2rnr.AsexplainedintheproofofLemma3.2, BylinearityofexpectationweconcludethatthatY]Šh2rnr�22rh2rnrŠh2rnrŠh2rnr�0.Hencewecan“xachoiceofsuchthatCallanorderedsubsetof2(notnecessarilydistinct)elementsof(i)allelementsofarecontainedinthecommonneighbourhoodofasetofsizeforwhich,or(ii)thereexistsanorderedsubsetof3elementsofwhose“rst2membersformtheorderedset,suchtOtherwiseitiscalledgood.Toreisgoodif:(i)foreveryofsizeforwhich),wehaveand(ii)forallsubsetsofsize Alon,M.KrivelevichandB.SudakovEveryssatisfyingcreatesatmost(ordered2-tuplesin,andeveryorderedsubsetofsize3generatesexactlyonebadordered2-tuple.Therefore,thetotalnumberofbadordered-tuplesisatmostollowsthatthereissomeordered2-tuplewhichisgood.Fixsuchagoodandde“ne.AsinthederivationofTheorem3.3,tocompletetheproofitsucestoshowthatevery-tupleofverticesinhasatleastcommonneighboursin,andevery-tupleofverticesinhasatleastcommonneighboursin.Forthedesiredpropertyfollowsdirectlyfromthefactthatisgood,andfrom(3.3).Toshowitforconsideranarbitraryofcardinality.Letdenotetheordered3-tupleofelementsofstartingwiththe2membersofandcontinuingwiththemembersof.ByThecrucialobservationisnowthat,bythede“nitionof,allcommonneighboursofbelongto.HenThiscompletestheproof. ubstituting,forexample,inthelasttheorem,weobtainthefollowingstrength-eningofTheorem3.3.Theorem3.6.EverygraphV,Everticeswith edgescontainsevery-degeneratebipartitegraphB,Fwithatmostvertices.AsmentionedintheIntroduction,anoldconjectureofErdos([9],seealso[7]),assertsthat,forevery“xed-degeneratebipartitegraph,ex(n,HMoreover,for=2Erdosconjectured(see[13],[12],[7])that,forany“xedbipartitegraphn,H)ifandnlyifis2-degenerate.Thelasttheoremsdonotproveanyoftheseconjectures,butdosupplyanestimateofasimilarform,andhenceprovideevidencesupportthem.Theproblemofreducingtheconstant4inTheorem3.5,allthewayto1,remainsachallengingopenquestionwhoseresolutionseemstorequiresomeadditional4.RamseynumbersofgraphswithgivenmaximumdegreeInthissectionwedescribeanapplicationofLemma2.1intheproofofthefollowingRamsey-typeresult.Theorem4.1.beagraphwithverticesandchromaticnumberupposethatthereisaproper-colouringofinwhichthedegreesofallvertices,besidespossiblythoseinthe“rstcolourclass,areatmost,where.De“nek,rtobe�krotherwise.Then,foreveryintegerH,K log(logk,rNotethatintheabovetheoremisalwaysatmost+1,sincethegraph-degenerateandisthus()-colourable.Toprovethistheoremwewillneedthe anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestionsollowingwell-knownboundontheindependencenumberofagraphcontainingfewtriangles(see,emma12.16in[5],andsee[1]foramoregeneralresult).Proposition4.2.beagraphonverticeswithmaximumdegreeatmost,suchthattheneighbourhoodofeveryvertexinspansatmostedges.Thenontainsanindependentsetoforderatleast (log2)logProofofTheorem4.1.applyinductionon.Startingwith=2and=1,andconsiderared…blueedgecolouringof.Notethatinthiscaseisjustdisjointunionofstars.Iftheredgraphhasaveragedegreeatleast4thenitcontainsubgraphwithminimumdegree2hissubgraphonecan“ndanyunionofstarsofordejustgreedily.Otherwise,theaveragedegreeoftheredgraphisatmost4,sobyTuranstheoremitcontainsablueindependentsetofsize100+1)Nowletandconsiderared…blueedgecolouringof logIfthenumberofrededgesisatleast 2 log,thenweclaimthattheredgraphcontainsasetofatleastvertices,suchthateveryofthemhaveatleastcommonneighboursintheredgraph.Indeed,byLemma2.1itsucestocheckthat log nrŠ1ŠnrŠ1 Thisindeedholds,since2.BythereasoningdescribedinSection2,thisimpliesthattheredgraphcontainsacopyofNextsupposethattheredgraphhasatmost 2 logedges.Then,byrepeatedlydeletingverticesofdegreelargerthan log,wecanobtainaredswithmaximumdegreeatmostandatleast2vertices.Iftheneighbourhoodofeveryvertexinspansatmost logedges,thenbyProposition4.2itcontainsblueindependentsetofsizeatleast (log2log log))log(100logHereweusedthat14andthatlog(100log5)logforallOtherwise,thereisasubsetofverticesofofsizeatmostwhichspansatleastedges.ThentheconditionsofLemma2.1aresatis“edagain,since drŠ1ŠdrŠ1 Thereforetheredgraphcontainsasetofatleastvertices,suchthateveryofthemhaveatleastcommonneighboursintheredgraph.Aswasexplainedearlier,thisimpliesthattheredgraphcontainsacopyofhowingthatindeedtheresultholdsfor=2.Assumingtheresultfor1,weproveitfor3.Givenasinthetheorem,“xaproper-colouringofitwithcolourclassesinwhichthedegreesofallverticesbesidespossiblythoseinareatmost.Moreover,takesuchacolouringinwhichthecardinalityofisaslargeaspossible.Clearlyeveryvertexin Alon,M.KrivelevichandB.Sudakovhasaneighbourin(sinceotherwisewecanshiftitto,contradictingthemaximality).Putandlettheinducedsubgraphof.Theis(chromatic,andithasaproper(1)-colouringinwhichthedegreesofallverticesbesidesthoseinthe“rstcolourclassareatmostPut log(logk,randconsiderared…blueedgecolouringofefore,ifthenumberofrededgesisat 2 log(logk,rthenweclaimthattheredgraphcontainsasetofatleast log(logvertices,sothatanyofthemhaveatleastcommonneighboursintheredgraph.Note1)=k,r).Hence,byLemma2.1,toprovethisclaimitsucestocheck nrŠ1ŠnrŠ1 nr log(logk,r log(logThus,thereisasetasclaimed.Bytheinductionhypothesis,eithertheinducedbluesubgraphoncontainsacopyofhichcasethedesiredresultfollows,ortheinducedredgraphoncontainsacopyof.Inthelattercase,thiscopycanbecompletedtoformaredcopyofinceeverysetofverticesofhasatleastcommonneighboursintheredgraph.Nextsupposethattheredgraphhasatmost 2 log(logk,redges.Then,byrepeatedlydeletingverticesofdegreelargerthan log(logk,rwecanobtainaredsubgraphwithmaximumdegreeandatleast2vertices.Iftheneighbourhoodofeveryvertexinspansatmost log(logk,r anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestionsedges,thenbyProposition4.2itcontainsablueindependentsetofsizeatleast (log2)log log))log(100loginally,wecanassumethatthereisasubsetofverticesofofsizeatmostwhichspansatleastrededges.ThentheconditionsofLemma2.1forgettingasetasbeforeareagainsatis“ed,since drŠ1ŠdrŠ1 dr log(logk,r log(logThereforetheredgraphcontainsasetofatleast log(logvertices,suchthateveryofthemhaveatleastcommonneighboursintheredgraph.Aswasexplainedearlier,usingthiswecaneither“ndintheredgraphacopyoforinthebluegraphacopyofhiscompletestheproofofthetheorem. Aneasyprobabilisticargumentshowsthattheabovetheoremisnearlytightwhenislargeandthe“xedgraphr,smuchbiggerthan.Infact,forevery0,andevery“xed,ifr,),thenr,sforall),whereasbyTheorem4.1,r,slog.Seealso[22]forsomerelatedresults.5.OnaRamsey-typeproblemofErdAsmentionedintheIntroduction,thefollowingconjecturewasraisedbyErdos(see[7]).Conjecture5.1.Thereexistsanabsoluteconstantsuchthat,foreverygraphedgesandnoisolatedvertices, Herewe“rstdescribeaveryshortproofoftheconjectureforbipartitegraphsTheorem5.2.beabipartitegraphwithedgesandnoisolatedvertices.Then Theorderoftheexponentinthisestimateisasymptoticallytight.Indeed,letthecompletebipartitegraph .Thenitcontaedgesanditiseasytocheckthatalmosteverytwo-edge-colouringofthecompletegraphoforder2 ,wherecolourofeveryedgeischosenrandomlyandindependentlywithprobability12,doesnotcontainamonochromaticcopyof . Alon,M.KrivelevichandB.SudakovProofofTheorem5.2.rstweprovethat -degenerate.Otherwise,byde“nition,containsasubgraphwithminimaldegreelargerthan .Let(U,Whebipartition.Clearly,everyvertexinhasatmostneighboursin.Therefore andweobtainacontradiction,sincethenumberofedgesin m|W||E(G)|. andsupposethattheedgesofthecompletegraphare2-coloured.Thenclearlyatleast 2(n2)n2Š1 8 edgeshavethesamecolour.Theseedgesformamonochromaticgraphwhichsatis“estheconditionsofTheorem3.6with thisgraphcontainsevery -degeneratebipartitegraphoforder .Inparticular,sincetheorderofisobviouslyboundedby2,itcontainsacopyhiscompletestheproof. Theorem5.3.beagraphwithedgesandnoisolatedvertices.Ifissucientlylargethen logToprovethistheoremweneedtwolemmasofGraham,RodlandRucinski[16].Westartwithsomenotation.LetbeagraphwithvertexsetandletbeasubsetofThenweletletU]denotethesubgraphofinducedby,andtsnumberofedges.Theedgedensity)ofisde“nedby Similarly,ifaretwodisjointsubsetsof,theX,Yhenumberofedgesadjacenttoexactlyonevertexfromandonefrom,andthedensityofthepairX,Y)isde“nedbyX,YX,Y Wesaytis(,ifforall,wehwesaythat,ifallpairs(X,YdisjointsubsetsofsatisfyX,Yhefollowingtwolemmasareprovedin[16].Lemma5.4.Letthenumberss,,,satisfy,,log.Thenifisa-densegraphonvertices,thenthereexistssizeatleastsuchthatthatU]isbi-,/Lemma5.5.betwointegersandleta,,bepositivenumberssuchthat,for anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestionsAlso,letbeagraphonverticeswithmaximumdegreeatmost.Ifisagraphoforderatleast(+1)whichisbi- -dense,thenontainsacopyofingthesetwolemmaswenextprovethefollowingstatement.Proposition5.6.beanintegerandletbeagraphwithvertexset logsuchthateverysubsetofofsizeatleasthasdensityatleast.Ifissucientlylarge,thenontainsacopyofeverygraphverticeswithmaximumdegreeatmost m logProof.=log)=lo(8log)and issucientlylarge,itiseasytocheckthat 8log2log Also,byassumption,everysubsetofofsize log +1)(log+3) 8loglog log log loghasdensityatleast).Therefore,byLemma5.4,containsaninducedsubgraphoforderatleast (log+2) 8loglog log log log logsuchthatisbi-(2 beagraphoforder2withmaximumdegreeatmost= m log.Set =(+1)2 .Thenitiseasytocheckthat2(+1) m2 log,andthatforevery0weh 4m m log m log m log m23 m= 2Š mŠ2Š m 23 m Inaddition,wehavethatisbi-( )-dense.ThussatisfyalltheconditionsofLemma5.5andthereforeitfollowsthatcontainsacopyof Having“nishedallthenecessarypreparationswearenowreadytoproveTheorem5.3.ProofofTheorem5.3.V,E)beagraphwithedgesandnoisolatedvertices.Then,clearly,thenumberofverticesofisatmost2.Letbethesubsetof logverticesofoflargestdegrees.Letdenotethesubgraphofinducedby Alon,M.KrivelevichandB.Sudakovtheset,andlet(tsmaximumdegree.Notethat 2vV0d(v)1 2)|V0| logThereforethemaximumdegreeofisboundedby m log logandlet-colouringoftheedgesofthecompletegraphDe“ne,for1 log,setsofvandelementsasfollows.isthesetofallverticesofHavingchosen,selarbitrarily.Havingselected,detobethelargestofthesetse“nition,2.Also,byinduction,itiseasytoshowthat 2iŠ1ŠiŠ1tŠt�n 2iŠ1Š1�n Inparticularweobtainthatfor log log logDe“neanewcolouringbyse(1or2)ifforSincethiscolouringsplitstheabove21verticesintotwoparts,thereisaallhavingthesamecolour.Withoutlossofgeneralitywecanassumethatthiscolouris1.Notethattheverticesofformamonochromaticcliqueofsize2 logwhichhascolour1andthatalltheedgesbetweenarealsocoloured1.bethegraphconsistingofalltheedgeswithinthesetwithcolour1.Firstsupposethatthedensityofeverysubsetofofsizeatleast4isatleast1.Theposition5.6,containsacopyofthegraph.Itiseasytoseethatsuchacopyoftogetherwiththesetofverticesformsamonochromaticsubgraphofcontaining.Ontheotherhandifisasubsetofofsizeatleast4anddensitylessthan1thenaneasycomputationshowsthatspansatleast 2mX|2�1Š1 2mŠ1|X|2 edgesofthesecondcolour.Therefore,byTuranstheoremthereisamonochromaticcliqueofsize2ofthesecondcolour.Thiscliquecontainseverygraphon2andinparticularacopyofhiscompletestheproofofthetheorem. tethattheproofactuallyshowsthefollowing,whichisobviouslystrongerthantheassertionofTheorem5.3.Theorem5.7.everygraphedgesandnoisolatedvertices,G,K log,providedissucientlylarge. anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestions6.ImprovedboundsonaTuran-typeproblemGivenintegersk,t2and1de“nethegraphk,sasfollows.Thevertexsetofthisgraphconsistsoftwodisjointsetsofsizes)+1and,respectively.unsthroughall-elementsubsetsofand1,andForevery1wejoinandalsotoevery.Inparticular,for=1andthisgraphistheinducedsubgraphonthe“rstthreelayersoftheBoolean-cube.Inthissectionweprovethefollowingresult.Theorem6.1.k,s+1)ThisimprovesaresultofFuredi[14],whoprovedthatk,s+1)tethatthisresult,aswellasthatof[14],suppliesanalternativeproofforCorollary2.3,k,kcontainseverybipartitegraphwithmaximumdegreeononesideandatvertices.Thedependenceoninourestimateaboveisessentiallyoptimalforandthedependenceonisessentiallyoptimalforall!,asshownbytheexamplesin[2].Proof.V,E)beagraphon2verticeswithatleast2+1)edges.AsinSection3,startwithapartitionofthevertexsetintodisjointsets,eachofcardinality,suchthatatleasthalfoftheedgesofcrossbetween.Letdenotethebipartitesubgraphofconsistingofalledgesofbetw.Byde“nition,+1).Withoutlossofgeneralityassumethatbeasequenceofnotnecessarilydistinctverticesofhosenuniformlyandindependentlyatrandom,anddenote.Letbethesetofallthecommonneighboursofverticesfrom,thatis,andletdenotethesizeoflinearityofexpectationandJensensinequality, |V1|t=vV2 dG1(v) t ntnvV2dG1(v) nt nt=n(|E(G1)|t =2(+1)Foreverysubsetofverticesofsize,de“nea)by .Lettherandomvariablewhichsumsthetotalweightofsubsetsofofsizewithatmost(+1)(ommonneighboursin.Notethat,foragivensubsetofsizetheprobabilitythatitbelongstoisprecisely .Thereforewecanobtainthe Alon,M.KrivelevichandB.SudakovollowingboundontheexpectationoffY]=SV2,|S|=t,|NG1(S)|(s+1) +1) +1) +1) t!vV1 dG1(v) t +1) t!vV2 dG1(v) t +1) X].Thisimpliesthat+1) +1)Hencethereexistsachoiceofsuchthattherandomvariablesforthecopondingsetsatisfy(6.1).Picksuchasethen,by(6.1),wehave+1)+1)andomsubsetofofsizepreciselyandlettherandomvariablewhichcountsthetotalweightofsubsetsofsizewithatmost(+1)commonneighboursin.Notethat,foreverysuchheprobabilitythatitliesinequalshusitiseasytoseethat Xk Yk XtYk +1)+1) +1) +1)+1) +1) +1) Thisimpliesthatthereisaparticularsubsetofsize+1).Fixsuchasetbeallthesubsetsofofsize.Weconstructasetofdistinctverticessuchthat,foreveryisadjacenttoallverticesinrrangethesetsinanon-decreasingorderofandassigntheonebyonetothesesetsinthisorder.Alwayspickthenextvertexfrom anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestionssuchthatitisdierentfromallpreviousverticesand.Notethatifthisgreedyprocedurefailsatstep,thenclearlythesets+1).Also,byde“nition,wehavethatforallhusweobtainacontradiction, +1) |NG1(Si)|r |NG1(Sr)|�r 1 inally,recallthatisadjacenttoalltheverticesinandhencealsotoalltheverticesintogetherwiththeverticesinandtheverticesformk,shiscompletestheproofofthetheorem. 7.Concludingremarkspologicalcopyofagraphisanygraphobtainedfrombyreplacingeachedgebyasimplepath,whereallthesepathsareinternallyvertex-disjoint.A1-subdivisionisthetopologicalcopyofobtainedbyreplacingeachedgeofbyapathoflength2.In[10]Erdosaskedwhetheranygraphonverticeswithedgescontainsa1-subdivisionof forsomepositivedependingon.Wenotethattheresultsin[6],aswellasthosein[18],implythatanysuchgraphcontainsatopologicalopyof butthiscopyisnotnecessarilya1-subdivision.ever,theexistenceofa1-subdivisionoftherequiredsizefollowsimmediatelyfromTheorem6.1with=2,=1and=( ).Asimilarresultcanalsobederivedfromthemainresultof[8],andcanalsobeproveddirectlyfromthereasoningintheproofofLemma2.1here.Infact,itisnotdiculttoshowthat,forany“xedpositive,thereisapositivec,),suchthatanygraphonverticeswithatleastverticesofdegreeatleasteachcontainsasetofatleast verticessothateachpairhasatleastcommonneighbours.Thisclearlyimpliesthatanysuchgraphcontainsa1-subdivisionofforanysatisfyingAsmentionedinSection4,ourestimateinTheorem4.1isnearlytightforsomebipartitegraphs.Itseems,however,thatthisestimateisfarfrombeingtightforgraphswithalargechromaticnumber.WeconjecturethatH,Kforevery“xedwithmaximumdegreeandallsucientlylarge.NotethatTheorem4.1impliesmerelyH,Kforthiscase.TheassertionofTheorem5.2canbeextendedtographswithboundedchromaticnumber,combiningourideasherewiththetechniquesin[20].Weomitthedetails.TheproofofTheorem6.1impliesasimilarestimatefortheTurannumberofthek,sobtainedfromk,sbyreplacingthevertexanindependentsetofwiththesameneighbours.Thisisbecausethesetintheproofcanbechosenwithnorepetitions. Alon,M.KrivelevichandB.SudakovReferences[1]Alon,N.,Krivelevich,M.andSudakov,B.(1999)Coloringgraphswithsparseneighborhoods.J.Combin.TheorySer.B[2]Alon,N.,Ronyai,L.andSzabo,T.(1999)Norm-graphs:Variationsandapplications.J.Combin.TheorySer.B[3]Alon,N.andShapira,A.(2003)Testingsubgraphsindirectedgraphs.InProc.thACMCMPress,pp.700…709.[4]Bollobas,B.(1978)ExtremalGraphTheory,AcademicPress,London.[5]Bollobas,B.(2001)RandomGraphs,2ndedn,CambridgeUniversityPress,Cambridge.[6]Bollobas,B.andThomason,A.(1998)ProofofaconjectureofMader,ErdosandHajnalonpologicalcompletesubgraphs.Europ.J.Combin.[7]Chung,F.andGraham,R.(1998)ErdosonGraphs:HisLegacyofUnsolvedProblemsA.K.PetersLtd.,Wellesley,MA.[8]Duke,R.A.,Erdos,P.andRodl,V.Intersectionresultsforsmallfamilies.Submitted.[9]Erdos,P.(1967)Somerecentresultsonextremalproblemsingraphtheory.InTheoryofGraphs(Rome,1966),GordonandBreach,NewYork,pp.117…123.[10]Erdos,P.(1979)Problemsandresultsingraphtheoryandcombinatorialanalysis.InGraphTheoryandRelatedTopics(Proc.Conf.Waterloo,1977),AcademicPress,NewYork,pp.153…[11]Erdos,P.(1984)Onsomeproblemsingraphtheory,combinatorialanalysisandcombinatorialnumbertheory.InGraphTheoryandCombinatorics(Cambridge,1983),AcademicPress,ndon,pp.1…17.[12]Erdos,P.(1984)Extremalproblemsinnumbertheory,combinatoricsandgeometry.InProc.InternationalCongressofMathematicians(Warsaw,1983),Vol.1,PWN,Warsaw,pp.51…70.[13]Erdos,P.(1990)Someofmyoldandnewcombinatorialproblems.InPaths,Flows,andVLSI-Layout,Vol.9ofAlgorithmsandCombinatoricsSpringer,Berlin,pp.35…45.[14]Furedi,Z.(1991)OnaTurantypeproblemofErdCombinatorica[15]Gowers,W.T.(1998)AnewproofofSzemeredistheoremforarithmeticprogressionsoflengthfour.Geom.Funct.Analysis[16]Graham,R.,Rodl,V.andRucinski,A.(2000)OngraphswithlinearRamseynumbers.J.GraphTheory[17]Kollar,J.,Ronyai,L.andSzabo,T.(1996)Norm-graphsandbipartiteTurannumbers.Combinatorica[18]Komlos,J.andSzemeredi,E.(1996)Topologicalcliquesingraphs,II.mbin.Probab.Comput.[19]Kostochka,A.andRodl,V.(2001)OngraphswithsmallRamseynumbers.J.GraphTheory[20]Kostochka,A.andSudakov,B.(2003)OnRamseynumbersofsparsegraphs.mbin.Probab.Com[21]Kovari,T.,Sos,V.T.andTuran,P.(1954)OnaproblemofK.Zarankiewicz.lloquiumMath.[22]Krivelevich,M.(1995)BoundingRamseynumbersthroughlargedeviationinequalities.Struct.Alg.[23]Sudakov,B.(2003)AfewremarksonRamsey…Turan-typeproblems.J.Combin.TheorySer.B mbinatorics,ProbabilityandComputing(2003)12,477…494.2003CambridgeUniversityPressDOI:10.1017/S0963548303005741PrintedintheUnitedKingdom TuranNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestions NOGAALON,MICHAELKRIVELEVICHBENNYSUDAKOV .Hereweprovethisconjectureforbipartitegraphs,andprovethatforgeneralgraphsedges,G,G logforsomeabsolutepositiveconstant

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