Co mbinatorics Probability and Computing - PDF document

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].Ourproofisdi
erentfromthatin[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 nrŠ1Š
nrŠ1 ontainsasubsetofatleastverticessuchthateveryverticesofhaveatmmonneighbours.Proof.beasubsetofrandomverticesofhosenuniformlywithrepetitions.denotethecardinalityoflinearityofexpectation, nr=1 nrvV|N(v)|r1 nrn
vV|N(v)| nr=1 nrŠ1
2|E(G)| nr=dr wheretheinequalityfollowsfromtheconvexityofdenotetherandomvariablecountingthenumberof-tuplesinwithfewercommonneighbours.Foragiven-tupleheprobabilitythatwillbeasubsetofisprecisely( .Asthereareatmost(ofcardinalityforwhich1,itfollowsthat nr. Alon,M.KrivelevichandB.SudakovApplyinglinearityofexpectationonceagainandrecallingcondition(2.1)ofthelemma,weconcludethat nrŠ1Š
nrŠ1 HencethereexistsachoiceforsuchthatforthecorrespondingsetwegetPicksuchaset,andforevery-tuplefromwithfewerthancommonneighbours,deleteonevertexfrom.Denotetheobtainedsetby.The,andevery-tupleofverticesofhasatleastcommonneighbours.Thiscompletestheproof. Theorem2.2.B,Fbeabipartitegraphwithsidesofsizesespectively.Supposethatthedegreesofallverticesdonotexceed.LetV,Ebeagraphonverticeswithaveragedegree.If nrŠ1Š
nr+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]=aA
dG1(a,B |B|4r|A|
|E(G1)| |A|4r24rŠ1n1Š4r wherethe“rstinequalityfollowsfromtheconvexityofInordertoestimatetheexpectedvalueof,observethatfora“xed3-tupletheprobabilitythatwillbeasubsetofisprecisely Asthereareatmostofcardinality3forwhichollowsthat |B|4r

e|A| 3r3r
n0.1 |B|4r=
e|A| 3r|B|3r
n0.4 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)2r
nr 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]=aA
dG1(a,B |B|2r|A|
|E(G1)| |A|2r22rŠ2Š2r whereweusedthefactthat�n/ByJensensinequalityandthefactthattheunctionisconvex,nvex,X]2r(2h)2rnr.AsexplainedintheproofofLemma3.2, BylinearityofexpectationweconcludethatthatY]Šh2r
nr�22rh2rnrŠh2rnrŠh2r
nr�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 nrŠ1Š
nrŠ1 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 drŠ1Š
drŠ1 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 nrŠ1Š
nrŠ1 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 drŠ1Š
drŠ1 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.Ifissu
cientlylargethen 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.Ifissu
cientlylarge,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,providedissu
cientlylarge. 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 Y

k XtY

k +1)+1) +1) +1)+1) +1) +1) Thisimpliesthatthereisaparticularsubsetofsize+1).Fixsuchasetbeallthesubsetsofofsize.Weconstructasetofdistinctverticessuchthat,foreveryisadjacenttoallverticesinrrangethesetsinanon-decreasingorderofandassigntheonebyonetothesesetsinthisorder.Alwayspickthenextvertexfrom anNumbersofBipartiteGraphsandRelatedRamsey-TypeQuestionssuchthatitisdi
erentfromallpreviousverticesand.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