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OntheNumberofPentagonsinTriangle-FreeGraphsHamedHatamiJanHladkyyDanielKral'zSergueiNorinexAlexanderRazborov{AbstractUsingtheformalismof\ragalgebras,weprovethateverytriangle-freegraphGwithnverticescontainsatmost(n=5)5cyclesoflength ve.Moreover,theequalityisattainedonlywhennisdivisibleby veandGisthebalancedblow-upofthepentagon.Wealsocomputethemaximalnumberofpentagonsandcharacterizeextremalgraphsinthenon-divisiblecaseprovidednissucientlylarge.ThissettlesaconjecturemadebyErd}osin1984.1.IntroductionTriangle-freegraphsneednotbebipartite.Buthowexactlyfarfrombeingbipartitecantheybe?In1984,Erd}os[Erd84,Questions1and2]consideredthreequantitativere nementsofthisquestion.Moreprecisely,heproposedtomeasure\non-bipartiteness"by: SchoolofComputerScience,McGillUniversity,hatami@cs.mcgill.ca.yMathematicsInstituteandDIMAP,UniversityofWarwick,honzahladky@gmail.com.SupportedbyEPSRCawardEP/D063191/1.zInstituteofMathematics,DIMAPandDepartmentofComputerScience,Univer-sityofWarwick,CoventryCV47AL,UnitedKingdom.Previousaliation:InstituteofComputerScience(IUUK),FacultyofMathematicsandPhysics,CharlesUniversity,Mal-ostranskenamest25,11800Prague1,CzechRepublic.E-mail:D.Kral@warwick.ac.uk.TheworkofthisauthorleadingtothisinventionhasreceivedfundingfromtheEu-ropeanResearchCouncilundertheEuropeanUnion'sSeventhFrameworkProgramme(FP7/2007-2013)/ERCgrantagreementno.259385.xDepartmentofMathematics&Statistics,McGillUniversity,snorin@math.mcgill.ca.{UniversityofChicago,razborov@cs.uchicago.edu.PartofthisworkwasdonewhiletheauthorwasatSteklovMathematicalInstitute,supportedbytheRussianFoundationforBasicResearch,andatToyotaTechnologicalInstitute,Chicago.1 (i)theminimalpossiblenumberofedgesinasubgraphspannedbyhalfofthevertices;(ii)theminimalpossiblenumberofedgesthathavetoberemovedtomakethegraphbipartite;(iii)thenumberofcopiesofpentagons(cyclesoflength5)inthegraph.Alltheseparametersvanishonbipartitegraphs,andErd}osconjecturedthatintheclassoftriangle-freegraphseveryoneofthemismaximizedbybalancedblow-upsofthepentagon.Simonovits(referredtoin[Erd84])ob-servedthatanotherexamplewhichattainstheconjecturedextremumfor(i)isprovidedbybalancedblow-upsofthePetersengraph.For(iii),Michael[Mic11]noticedthatthecycleoflengtheightwithfourchordsjoiningtheop-positeverticeshaseightpentagons,thusmatchingthenumberofpentagonsintheeight-vertex(almostbalanced)blow-upsofthepentagon.The rsttwoofErd}os'squestionshavebeeninvestigatedin[EFPS88,Kri95,KS06].Gy}oriinvestigatedthethirdquestionin[Gy}o89].Intermsofdensities,Erd}os'sconjectureregarding(iii)statesthatthedensityofpen-tagonsinanytriangle-freegraphsisatmost5! 55.Gy}oriprovedanupperboundof335! 5214thatiswithinafactor1.03oftheoptimal.Furedi(personalcommunication)re nedGy}ori'sapproachandobtainedanupperboundwithinafactor1.001oftheoptimal.Inthispaperwesettle(iii)inthedensitysense(Theorem3.1),whichalsoimpliestheexactsolutionwhennisdivisibleby5(Corollary3.3).Theproofofthisresultisacalculationin\ragalgebras(introducedin[Raz07]).Furthermore,weobtaintheasymptoticuniqueness(that,again,turnsintotheuniquenessintheordinarysensewhen5jn)byarelativelysimpleargu-mentinTheorem3.2.InSection4weuseamoretechnicalapproachto ndtheexactsolutionfornsucientlylarge.Weleaveitasanopenquestiontoproveanexactboundonthemaximumnumberofpentagonsinann-vertextriangle-freegraphforallvaluesofn.Weassumeacertainfamiliaritywiththetheoryof\ragalgebrasfrom[Raz07].FortheproofofthecentralTheorem3.1onlythemostbasicnotionswhichdealwithCauchy-Schwarztypecalculationsarerequired.Thus,insteadoftryingtoduplicatede nitions,weoccasionallygivepointerstorelevantplacesin[Raz07]andsomesubsequentpapers.ForourproofofTheorem3.2weneedalittlebitmorethanthesebasics.WerecallthecorrespondingbitsinSection2.2.2 C5 M4 C 5 Figure1:Models 2.Preliminaries2.1.NotationWedenotevectorswithboldfont,e.g.a=(a(1);a(2);a(3))isavectorwiththreecoordinates.Foreverypositiveintegerk,let[k]denotethesetf1;:::;kg.Following[Raz07,De nition1],fortwographsHandG,thedensityofHinGasaninducedsubgraphisdenotedbyp(H;G).Thatisp(H;G)istheprobabilitythatthesubgraphinducedonj(H)jrandomlychosenverticesofGisisomorphictoH.Exceptforthestand-aloneSection2.3,weexclusivelywork[Raz07,x2]inthetheoryTTF Graphoftriangle-freegraphs.Recallfrom[Raz07]thatforatheoryTandapositiveintegern,thesetofallmodelsofTonnelementsuptoanisomorphismisdenotedbyMn[T].Weworkwiththenotionoftypes,\rags,and\ragalgebras,andusethesamenotationasin[Raz07,x2.1]wherethisterminologyisintroduced.Letuslistthosemodels,typesand\ragsthatwillbeneededinthispaper.Let2M2[TTF Graph]andC52M5[TTF Graph]respectivelydenotetheedgeandthepentagon.ThesetwographsalongwithtwoothergraphsthatwillbeneededforprovingtheuniquenessandtheexactresultareillustratedinFigure1.Wedenotethetrivialtypeofsize0by0.LetPdenotethetypeofsize5basedonC5(seeFigure2).Fori=0;1;2,letidenotethetypeofsize3withiedgeswherethelabelingischoseninsuchawaythatthepermutationof1and2isanautomorphism(seeFigure2).Foratypeofsizekandanindependentsetofvertices[k]in,letFVdenotethe\rag(G;)2Fk+1inwhichtheonlyunlabeledvertexisconnectedtothesetf(i)ji2g.Notethatsinceweareworkinginthetheoryoftriangle-freegraphs,wehaveFk+1=fFVj[k]isanindependentseting: 3 01 2 111222333 P 12345 Figure2:Types 2.2.Operator,andextensionmeasuresAsusual,thealgebrageneratedby-\ragsisdenotedA.Wethende nethefollowing\upwardoperator":A0!Aasintroducedin[Raz07,x2.3,x2.3.1].Foratriangle-freegraphG2F0k(=Mk),welet(G)bethesumofallthoseF=(H;)2Fk+jjforwhichtheunlabeledverticesformacopyofG,i.e.,H im()isisomorphictoG.WethenextendlinearlytoA0.Itturnsout[Raz07,Theorem2.6]that:A0!Aisanalgebrahomomorphism.Thenextnotionweshallneedisthatofextensionmeasure,asintroducedin[Raz07,De nition8].Supposethat 2Hom+(A0[TTF Graph];R)issuchthat ()�0(1)foratype(in(1)weviewasanunlabeledgraph).Then,thereexistsauniqueprobabilitymeasurePonBorelsetsofHom+(A[TTF Graph];R)withthepropertythatZHom+(A[TTF Graph];R) (f)P(d )= (JfK) (J1K);(2)foranyf2A.See[Raz07,Theorem3.5].WesaythatPextends .S( )isthesupportofthismeasure,i.e.theminimalclosedsubsetAsuchthatP[A]=1[Raz11,Section2.1.1].Notethattheintegrationin(2)canberestrictedtoS( ).Observethat canbereconstructedfromitsextensionPsimplybypickinganarbitrary 2S( )andletting (g)= ((g))(g2A0)(3)(cf.[Raz07,Corollary3.19]). 4 2.3.In niteblow-upsandHom+(A00TGraphR)Inordertoconverttheasymptoticresultintotheexactone,weneedtoexplorealittlebitmoretheconnectionbetweenblow-upsofagraphandthecorrespondingalgebrahomomorphismfromHom+(A0[TGraph];R)alreadyusedinasimilarcontextin[Raz08,Theorem4.1].Fora nitegraphGandapositiveintegervectork=(k()j2(G)),wede netheblow-upG(k)ofGasthegraphwith(G(k))def=[v2V(G)fg[k()]E(G(k))def=f((v;i);(w;j))j=^(v;w)2E(G)g:Whenallk()areequaltosomepositiveintegerk,thecorrespondingblow-upiscalledbalancedanddenotedsimplybyG(k).ForeverygraphH,itiseasytoseethatthesequencefp(H;G(k))gk2Nisconvergent.Itfollows[Raz07,x3]thatthereexistsahomomorphism G2Hom+(A0[TGraph];R)suchthatforeverygraphH,wehavelimk!1p(H;G(k))= G(H).Notethatsincetheblow-upofatriangle-freegraphisalsotriangle-free,ifGistriangle-free,thenactually G2Hom+(A0[TTF Graph];R).Letusnowgiveacombinatorialdescriptionof G.Fora nitegraphH,letusdenotebys(H;G)thenumberofstronghomomorphismsfromHtoGthatwede neasthosemappings :(H) !(G)forwhich( (); ())2E(G)ifandonlyif(v;w)2E(H).Thisnotionisanaturalhybridofinducedembeddingsandgraphhomomorphisms,anditisaveryspecialcaseofthenotionoftrigraphhomomorphisms(seee.g.[HN04]).Itiseasytocheckthat G(H)=m! jAut(H)js(H;G) nm;(4)wheremandnrespectivelydenotej(H)jandj(G)j,andAut(H)isthegroupofautomorphismsofH.LetussaythatHistwin-freeifnotwoverticesinHhavethesamesetofneighbors.Everystronghomomorphismofatwin-freegraphintoanyothergraphisnecessarilyaninducedembedding.Therefore,fortwin-freeH,wehaves(H;G)=p(H;G) nmjAut(H)jand(4)considerablysimpli esto G(H)=p(H;G)n(n 1):::(n m+1) nm:(5) 5 ForthespecialcaseH=Kr,thisformulawasalreadyusedin[Raz08,Section4.1]),andinthispaperweareinterestedinanothercase, C5(C5)=5! 55:Ourapproachtoextractingexactresultsfromasymptoticonesheavilyreliesonthefactthat Gisagraphinvariant.Thiswas rstprovenbyLovasz[Lov67].Asimpleproofofasimilarstatementinthecontextofgraphlimitsisgivenin[Lov12,Theorem5.32]. Theorem2.1 LetG1andG2be nitegraphswiththesamenumberofverticesandsuchthat G1= G2.ThenG1andG2areisomorphic.3.MainresultsRecall[Raz07,De nitions5and6]thatforanon-degeneratetypeinatheoryT,andf;g2A[T],theinequalityfqmeansthat (f) (g)forevery 2Hom+(A[T];R):thisistheclassofallinequalitiesthatholdasymptoticallyon\ragsofthegiventheory[Raz07,Corollary3.4].Weabbreviatefgtofgwhenisclearfromthecontext.Our rsttheorem,whichanswersthequestionofErd}os,saysthatinthetheoryoftriangle-freegraphs,wehaveC55! 55.Notethatwhileinthetheoryofgeneralgraphs,the\ragC5correspondstoinducedpentagons,inthetheoryoftriangle-freegraphs,everypentagonisinduced. Theorem3.1 InthetheoryTTF Graph,wehaveC55! 55:Proof.Theproofisbyadirectcomputationinthe\ragalgebraA0[TTF Graph](cf.[Raz10,HKN09]and[Raz11,Section4.1]).Weclaimthat62500C5+1097 12M4+68 3C 5+ 2Xi=0JQ+i(g+i)Ki!+200 2 52+JQ 1(g 1)K1+158266J(F2f1g F2f2g)2K22400: (6) ThegraphsM4andC 5areillustratedonFigure1.Forthede nitionofthealgebraoperationssee[Raz07,Eq.(5)],andforthede nitionofthe 6 averagingoperatorJKsee[Raz07,x2.2].Letusnowde nethenotationsg+= iandQ+= iin(6).Foratypeofsizekandaninteger0jk,weletfjdef=XfFVj[k]anindependentsetofsizejing;whereFVareasde nedinSection2.1.Thevectorsg+= iarethefollowingtuplesofelementsfromAi4:g+0def=(f01 f02;f01 2f00+3f03);g+1def=(2f10 f11;f11 f12;F1f3g);g+2def=(6f20+f21 4f22;2f20 2f22+F2f3g);g 1def=(F1f1g F1f2g;F1f2;3g F1f1;3g);andQ+= iarepositive-de nitequadraticformsrepresentedbythefollowingpositive-de nitematrices:M+0def= 976022522252592!M+1def=0BB@13900 671 12807 67131334 51136 12807 51136981571CCAM+2def= 22708 40788 4078878132!M 1def= 1416 16452 16452256488!:Theinequality(6)canbecheckedbyexpandingtheleft-handsideasalinearcombinationofelementsfromM5(thatis,triangle-freegraphson5vertices{thereare14ofthem)andcheckingthatallcoecientsarelessorequalthan2400.Weveri edtheinequalityusingaMaplesheetandaCprogramwhichwereindependentlyprepared.TheCprogramisavailableasanancillary leonthearXiv(arXiv:1102.1634).Itfollowsfrom[Raz07,Theorem3.14]thatallthesummandsontheleft-handsideof(6)arenonnegativeaselementsofA0[TTF Graph]whichinturnimpliesthatC52400 62500=5!=55. Remark1 Anexplanationofourusageofthe+= superscriptsintheproofofTheorem3.1canbefoundin[Raz10,Section4].Here,theparticularchoiceofsubspacesspannedbythevectorsg+= iisdictatedbythesameprinciplesasin[Raz10,Section4]. 7 Therearetwopossibleandratherstraightforwardgeneralizationsoftheaboveproblem,neitherofwhichwecould ndintheliterature.OnecanaskforthemaximumnumberofcopiesofoddcyclesC2`+1,orforthemaximumnumberofinducedcopiesofoddcyclesC2`+1inatriangle-freegraphofa xedorder.Inourpentagoncase`=2thetwoquestionsarethesame.EmilVaughancommunicatedtousthatusingthe\ragalgebrasmethodhecanprovethattheblow-upsofthepentagonandoftheheptagon,respectivelyareasymptoticallyextremalforthetwoproblemsfor`=3.Letusnowturntothequestionofuniquenessoftheoriginalproblem. Theorem3.2 Thehomomorphism C5istheuniqueelementinHom+(A0[TTF Graph];R)thatful lls (C5)=5! 55:(7)Proof.Fix 2Hom+(A0[TTF Graph];R)suchthat(7)holds.RecallthatPisthetypeofsize5basedonC5(seeFigure2).Insteadofprovingdirectlythat = C5,wewillargueabouttheextensionPPof andthenutilize(3).Observethat(6)impliesthat (M4)= (C 5)=0:(8)Trivially,JFP;KPC 5.AsM4isaninducedsubgraphofeachofthegraphsFPf1g;:::;FPf5g(viewedasunlabelledgraphs),thereexistsaconstant �0suchthatJFPfigKP M4foreveryi2Z5.Hence(8)impliesthat (JFP;KP)= (JFPfigKP)=0:(9)LetYbethesetofthoseelements 2Hom+(AP[TTF Graph];R)forwhich (FP;)=0,or (FPfig)=0forsomei2Z5;notethatYisopen.WeclaimthatY\SP( )=;:(10)Indeed,letusconsidertheP-\ragf=FP;+PiFPfig.By(9)wehave (JfKP)=0.Pluggingthisin(2),wehaveZSP( ) (f)PP(d )=0:Further,asfP0,theintegrand (f)isnon-negative.Therefore, (f)=0forPP-almostallelements and,sinceYisopen,(10)follows. 8 Pickanarbitrary P2SP( ).Letusexamine P(FPV)for\ragsFPV2FP6.SinceisanindependentsetinC5,wehavejj2,andmoreover,ifjj=2,then=fi 1;i+1gforsomei2Z5.As P62Y,wehave P(FP;)= P(FPfig)=0.Inotherwords, P(FPV)canbenon-zeroonlywhen=fi 1;i+1gforsomei2Z5.De neHPidef=FPfi 1;i+1g.WehavePi2Z5 (HPi)=1,andthustheinequalityofaritmeticandgeometricmeansgivesthatYi2Z5 (HPi)5 5;(11)withequalityonlywhen (HP1)=:::= (HP5)=1=5.RecallthatP(C5)canberepresentedasthesumofthoseF=(G;)2FP10forwhichtheunlabeledverticesformacopyofC5,say(G)nim()=f1;v2;:::;v5gwherejisadjacenttoj 1andj+1.Bytheabovediscus-sion,non-zerocontributionsto P(P(C5))canbemadeonlybythoseFforwhicheveryjisadjacentto(i(j) 1)and(i(j)+1)forsomechoiceofi(j)2Z5.SinceGistriangle-free,themappingj7!i(j)de nesagraphhomomorphismofthepentagonintoitself,andsincetherearenosuchgraphhomomorphismsotherthanisomorphisms,wemayassumewithoutlossofgeneralitythateveryjisadjacentto(j 1)and(j+1).Inotherwords, P(P(C5))= P((C(2)5)P),where(C(2)5)Pistheuniquelyde nedP-\ragbasedonC(2)5,theblow-upofthepentagon.Observethat(C(2)5)PP5!Yi2Z5HPi;(12)andthusby(3)and(11)wehavethefollowingchainofinequalities5! 55= (C5)= P(P(C5))= P((C(2)5)P)5!Yi2Z5 P(HPi)5! 55:Consequently,(11)mustactuallybeanequality,andtherefore P(HP1)=:::= P(HP5)=1=5.Further,thereisnoslacknessin(12),i.e., P0@5!Yi2Z5HPi (C(2)5)P1A=0:(13)Thisequalityallowsustocompletelydescribethebehaviorof PalsoonFP7.Fori;j2Z5,letHPij2FP7bede nedbyaddingtwounlabelednon-adjacentverticestoPandconnectingoneofthemto(i 1)and(i+1)and 9 theotherto(j 1)and(j+1).Notethatifi=jor(i;j)62E(P),thentheproductHPiHPjisequaltoqijHPij,whereqij=1ifi=j,andqij=1=2otherwise(sinceaddinganedgebetweenthetwounlabeledverticeswouldhavecreatedatriangle).Hence,inthiscase P(HPij)=1 25qij.Ontheotherhand,if(i;j)2E(P),then5!HPijYk2Z5nfi;jgHPkP5!Yi2Z5HPi (C(2)5)P;whichtogetherwith(13)impliesthat P(HPij)=0.Itfollowsthat P(GPij)=1 25qij,whereGPijisde nedsimilartoHPijwiththedi erencethatnowthereisanedgebetweentheunlabeledvertices.AsP P(HPij)+P P(GPij)=1,wehave P(F)=0foreachF2FP7nSi;j2Z5fGPij;HPijg.(14)LetHbenowanarbitrarytriangle-freegraphonnvertices.SimilarlytotheabovewecanexpandP(H)inAP5+n.Forahomomorphism :H!Z5ofHtoC5(withitsverticeslabeledcyclically),wewriteFP fortheP-\rag(G;)2FP5+nwheretheunlabeledvertices0def=(G)nim()induceacopyofH,andeachvertex20isadjacentonlyto( () 1)and( ()+1).Weclaimthat PevaluatestozeroatanytermF2FP5+nintheexpansionofP(H),unlessF=FP forsomehomomorphism :H!P.Indeed,theparticularcasewhenHisanedgeisshownin(14),andthegeneralcasefollowsbythesamereasoning.Furthermore,as P(HPij)=0for(i;j)2E(P),weactuallyhavethat mustbeastronghomomorphisminthiscase.Observethat P(FP ) c = P0@Yv2V(H)HPh(v)1A=5 n(15)foreachstronghomomorphism :H!P,wherec isthemultinomialcoecient,c def= njh 1(1)j;jh 1(2)j;jh 1(3)j;jh 1(4)j;jh 1(5)j!:Now, (H)= P(P(H))=P P(FP )=5 nP c (thesummationistakenoverallstronghomomorphismsfromHtoC5)thatcanbeeasilyseentocoincidewiththevalue C5(H)asgivenby(4).This nishestheproof. 10 TheupperboundonthenumberofpentagonscanbederivedfromThe-orem3.1onthein niteblow-up G2Hom+(A0[TTF Graph];R).Further-more,Theorems2.1and3.2showthattheequalityinthestatementcanbeobtainedonlywhennisdivisibleby veandGisthebalancedblow-upofthepentagon. Corollary3.3 Everyn-vertextriangle-freegraphGcontainsatmost(n=5)5pentagons.Moreover,theequalityisattainedonlywhennisdivisibleby veandGisthebalancedblow-upofthepentagon.TheboundattainedbyCorollary3.3isnottightwhennisnotdivisibleby5.Morespeci cally,letn=5`+(04),thenthenumberofpentagonsinanalmostbalancedblow-upofC5withnvertices1isequalto (n)def=`5 a(`+1)a. Conjecture1 Everytriangle-freegraphonnverticescontainsatmost (n)pentagons.TheoriginalversionofthispaperclaimedtoresolveConjecture1,buttheproofcontainedamistakethatwewerenotableto x.Aswenotedinintroduction,Michael[Mic11]observedthatforn=8thereexistsasporadicexamplewith (8)=8pentagons.Infact,[Mic11]alsoconjecturedthatthisistheonlysporadicexample,which,inparticular,wouldimplyConjecture1.WeuseastabilityargumenttosettleConjecture1forsucientlylargeninTheorem4.2below.4.ExactboundWede nethecutnormofannnmatrixAbykAk:=1 n2maxS;T[n] Xi2S;j2TAij :FortwographsG1andG2onthesamesetofvertices[n],wede netheircutdistanceasd(G1;G2)=kAG1 AG2k; 1whena=2;3therearetwonon-isomorphicalmostbalancedblow-upsofC5withnvertices11 whereAG1andAG2denoterespectivelytheadjacencymatricesofG1andG2.IfG1andG2areunlabeledgraphsondi erentvertexsetsofthesamecardinalityn,thenwede netheirdistancebyb(G1;G2)=min~G1;~G2d(~G1;~G2);where~G1and~G2rangeoveralllabellingsofG1andG2by[n]respectively.Finally,letG1andG2begraphswithn1andn2vertices,respectively.Notethatforeverypositiveintegerk,theblow-upgraphsG(n2k)1andG(n1k)2havethesamenumberofvertices.Sowecande ne(G1;G2)=limk!1b(G(n2k)1;G(n1k)2):Thefunctionisonlyapseudometric,notatruemetric,because(G;G0)maybezerofordi erentgraphsGandG0.Infact,itisaneasyconsequenceofTheorem2.1andTheorem4.1belowthat(G;G0)=0ifandonlyifG(k)=G0(k0)forsomeintegersk;k0�0.Thefollowingtheoremshowstherelevanceofthisdistancetothecontextofthispaper. Theorem4.1([BCL+08,Theorem2.6]) Asequenceofgraphs(Gn)1n=1convergesinthedistanceifandonlyifthesequence( Gn)ni=1converges.WewillalsoneedthefollowingfactprovenbyAlon(see[Lov12,Theo-rem9.24]).IfG1andG2aretwographsonnvertices,thenb(G1;G2)(G1;G2)+17 p logn:(16) Theorem4.2 Thereexistsanintegern0suchthatanytriangle-freegraphwithnn0verticesandatleast (n)pentagonsmustbeanalmostbalancedblow-upofC5.Proof.Let1�0bea xedconstantchosentobesucientlysmalltosatisfytheinequalitiesthroughouttheproof.LetGbeatriangle-freegraphonnverticescontainingthemaximumnumberofpentagons.Throughouttheproof,wewillalwaysassumethatnissucientlylarge.Wewillusesomeadditionalnotationintheproofbelow.ForU;V(G),wewilldenotebyG[]thesubgraphofGinducedbyandwewilldenotebyG[U;V]theinducedbipartitesubgraphofGwithpartsand.For2(G)wedenotebydegU()thenumberofneighborsofin. 12 ByTheorem4.1andTheorem3.2,wehave(G;C5)1=4.LetkbesothatC(k)5isanalmostbalancedblow-upofC5onnvertices.Notethat(G;C(k)5)(G;C(5bn=5c)5)+(C(k)5;C(5bn=5c)5)=(G;C5)+O(1=n):Combiningthiswith(16),weconcludethatifnissucientlylarge,thenb(G;C(k)5)1=2:InparticularthereexistsapartitionA1;A2;:::;A5of(G)suchthatforeveryi2[5],itholdsthatjjAij n=5j1andjE(G[Ai;Ai+1])jjAijjAi+1j 1n2=2:(17)Set2=3p 1,andforeveryi2[5],letBibethesetofvertices2AisuchthatdegAi 1[Ai+1()jAi 1[Ai+1j 2n.By(17)wehavejE(G[Ai;Ai 1[Ai+1])jjAijjAi 1[Ai+1j 1n2;whichimpliesjBij1 2n2 5n.LetA0i=AinBi.Weclaimthat (a) jjA0ij n=5j2 5n+12n. (b) degA0i 1[A0i+1()jAi 1[Ai+1j 7 52n2n 5 22nforevery2A0i. (c) degA0i1()jAi1()j 7 52nn 5 22nforevery2A0i. (d) E(G[A0i])=E(G[A0i;A0i+2])=;.Conditions(a)and(b)areimmediateconsequencesoftheboundonthesizeofBi.By(b),for2A0i,wehavedegA0i1()jAi1j 7 52n,whichveri es(c).Toprove(d)notethatfor2sucientlysmall,everypairofverticesinA0ihave(many)commonneighbors.Hence,asGistriangle-free,E(G[A0i])=;.ItfollowssimilarlythatE(G[A0i;A0i+2])=;.InthenextstepweeliminatetheverticesinBdef=S5i=1Bi.Consideranarbitrary2B.WewouldeitheraddtooneofthepartsofA01;:::;A05maintainingtheconditionsonA0iestablishedabove(possiblywithaworseconstant)orshowthatisinfewpentagonsandcanbereplacedbyanothervertex,whilethenumberofpentagonsisincreased.Letp()denotethenumberofpentagonsinGcontainingv:Consider rstdeletingthevertexandaddinganewvertextoGjoinedtoeveryvertexinA01andA03.By(d)thegraphremainstriangle-free.By(a)and(c),thenewvertexisinatleastn 5 2nn 5 22n3n4 54 72n4=:p1 13 pentagonsfor2sucientlysmall.AsthenumberofpentagonsinGismaximumamongallgraphsonnvertices,weconcludethatp()p1.Considerapentagoncontainingandfourverticesin(G) B,sothatatleasttwoofthemlieinthesameA0i.Itisnothardtoverifythatsuchapentagonmustcontainapairofnon-adjacentverticesoneinA0iandanotherinA0i+1forsomei2[5].Thusthereareatmost47 52n462n4suchpentagonsby(b).AlsosincejBj2n,thereareatmost82n4pentagonscontainingandanothervertexinB.Letxi=degA0i().Notethatifxi�22nforsomei,thenxi+1=xi 1=0,asavertexinA0i1canhaveatmost22nnon-neighborsinA0iby(c)andGistriangle-free.Therefore,ifforeveryi2[5],wehavexi22norxi+222n,thenwegetacontradiction:p()142n4+102nn 5+2n3p1:Weconcludexi;xi+2&#x-279;22nforsomei2[5],andwehavexi 1=xi+1=xi+3=0bytheobservationabove.Let3=53222.Supposethatxin 5 3norxi+2n 5 3n.Repeatingthecalculationabovewehavep()142n4+n 5 3nn 5+2n3142n4+n 5 3n n3 53+2n3!n4 54 3 53 152n4=p1;wheretheintermediateinequalitiesholdfor2sucientlysmallandthelastidentityisbythechoiceof3.Thusxi;xi+2n 5 3nforsomei2[5].WeaddtoA0i+1.WerepeatthesameprocedureforeveryvertexofB.Asaresultoftheproceduredescribedintheprecedingparagraph,weobtainapartitionA001;A002;:::;A005of(G)suchthatjjA00ij n=5j22n3n,degA00i 1()n=5 3nanddegA00i+1()n=5 3nforeveryi2[5]andevery2A00i.Asin(d)itfollowsthatE(G[A00i])=E(G[A00i;A00i+2])=;foreveryi2[5],if3issucientlysmall.ThuseverypentagoninGmustcontainavertexfromeachofthesetsA001;A002;:::;A005.Consequently,thenumberofpentagonsinGisupperboundedbyQ5i=1jA00ijandtheequalityholdsifandonlyifeveryvertexofA00iisjoinedtoeveryvertexinA00i+1foreveryi2[5].ItiseasytoseethattheaboveproductismaximizedwhenjjA00ij jA00jjj1foreveryi;j2[5].ThusGmustbeanalmostbalancedblow-upofC5,asdesired. 14 AcknowledgementWhenpreparingthe nalversionofthispaperwelearnedthatAndrzejGrzesik[Grz12]independentlyprovedTheorem3.1.WethankOlegPikhurkoforpointingouttheworkofZoltanFureditous.WealsothankT.S.MichaelandHumbertoNavesforpointingoutthe\rawintheoriginalversionofthemanuscript,andrefereesfortheircomments.References [BCL+08] ChristianBorgs,JenniferT.Chayes,LaszloLovasz,VeraT.Sos,andKatalinVesztergombi.Convergentsequencesofdensegraphs.I.Subgraphfrequencies,metricpropertiesandtesting.Adv.Math.,219(6):1801{1851,2008. 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