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A hypergraph regularity method for linear hypergraphs

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A hypergraph regularity method for linear hypergraphs
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Khan, Shoaib Amjad
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Regularity lemma
Counting lemma
Forbidden families
F-counting algorithm
Constructive removal lemma
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ABSTRACT: Szemerédi's Regularity Lemma is powerful tool in Graph Theory, yielding many applications in areas such as Extremal Graph Theory, Combinatorial Number Theory and Theoretical Computer Science. Strong hypergraph extensions of graph regularity techniques were recently given by Nagle, Rödl, Schacht and Skokan, by W.T. Gowers, and subsequently, by T. Tao. These extensions have yielded quite a few non-trivial applications to Extremal Hypergraph Theory, Combinatorial Number Theory and Theoretical Computer Science. A main drawback to the hypergraph regularity techniques above is that they are highly technical. In this thesis, we consider a less technical version of hypergraph regularity which more directly generalizes Szemeredi's regularity lemma for graphs. The tools we discuss won't yield all applications of their stronger relatives, but yield still several applications in extremal hypergraph theory (for so-called linear or simple hypergraphs), including algorithmic ones. This thesis surveys these lighter regularity techiques, and develops three applications of them.
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Thesis (M.A.)--University of South Florida, 2009.
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by Shoaib Amjad Khan.
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AHypergraphRegularityMethodforLinearHypergraphs by ShoaibAmjadKhan Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofArts DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:BrendanNagle,Ph.D. BrianCurtin,Ph.D. StephenSuen,Ph.D. DateofApproval: May11,2009 Keywords:RegularityLemma,CountingLemma,ForbiddenFamilies, F -Counting Algorithm,ConstructiveRemovalLemma c Copyright2009,ShoaibAmjadKhan

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Dedication Tomyparentswhoselovingprayershavebeenmyanchorinturbulenttimes.

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Acknowledgements Iwouldliketothankallpeoplewhohavehelpedandinspiredmeduringmythesis research. Iespeciallywishtothankmyadvisor,Dr.BrendanNagle,forhiscontinuous guidanceandencouragementthroughoutmythesisstudy.Hisperpetualenergyand enthusiasminresearchhavebeenaconstantsourceofinspirationforme.Inaddition, hehasalwaysbeenaccessibleandwillingtohelpmeoutofanypitfallsthatIstumbled intoduringthecourseofmyresearch.Asaresult,Ineverfeltisolatedandresearch lifebecamesmoothandrewardingforme.Thisthesiscouldnothavebeenpossible withouthisperson. IamgratefultobothDr.BrianCurtinandDr.StephenSuenfortakingtheirtime togoovermythesismanuscriptandfortheirconstructivecritique. MysincerethanksareduetoProf.NatasaJonoskaforherunderstandingsupport andencouragementduringmynalsemester.Iamindebtedtoherforopeningup newvistasformebyteachingwhathasbecomeoneofthemostmemorableclasses thatIhavehadthefortunetotake. Mydeepestgratitudegoestomyfamilyandfriendsforalltheirhelp.Idedicate myworktomyparentswhoseunwaveringloveandprayershavebeenmyanchorin turbulentwaters. Foreveryoneaforementioned,andforachancetothankthem,IthankYoumy KindandLovingCompanion.MayYournamebeexalted,honoredandgloried.

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TableofContents Abstractii 1Introduction1 1.1Tools....................................3 1.2Results...................................5 2RegularityLemmaFor k -UniformHypergraphs7 3CountingLemmaForSimple k -UniformHypergraphs16 3.1MainTheorem..............................16 3.2AGeneralization.............................20 4Bounding j Forb n; F j 22 5CountingAlgorithmForSimpleHypergraphs28 6ConstructiveRemovalLemma33 References36 i

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AHypergraphRegularityMethodforLinearHypergraphs ShoaibA.Khan ABSTRACT Szemeredi'sRegularityLemmaispowerfultoolinGraphTheory,yieldingmany applicationsinareassuchasExtremalGraphTheory,CombinatorialNumberTheory andTheoreticalComputerScience.Stronghypergraphextensionsofgraphregularity techniqueswererecentlygivenbyNagle,Rodl,SchachtandSkokan,byW.T.Gowers, andsubsequently,byT.Tao.Theseextensionshaveyieldedquiteafewnon-trivial applicationstoExtremalHypergraphTheory,CombinatorialNumberTheoryand TheoreticalComputerScience. Amaindrawbacktothehypergraphregularitytechniquesaboveisthattheyare highlytechnical.Inthisthesis,weconsideralesstechnicalversionofhypergraph regularitywhichmoredirectlygeneralizesSzemeredi'sregularitylemmaforgraphs. Thetoolswediscusswon'tyieldallapplicationsoftheirstrongerrelatives,butyield stillseveralapplicationsinextremalhypergraphtheoryforso-called linear or simple hypergraphs,includingalgorithmicones.Thisthesissurveystheselighterregularity techiques,anddevelopsthreeapplicationsofthem. ii

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1Introduction Szemeredi'sregularitylemma[28,29]isoneofthemostimportanttoolsincombinatorics,withnumerousapplicationsrangingacrosscombinatorialnumbertheory, extremalgraphtheoryandtheoreticalcomputersciencesee[17,18]forexcellentsurveys.Roughlyspeaking,thelemmaassertsthateverygraphcanbedecomposedinto aboundednumberofrandom-likeparts,ormoreformally, -regularpairs,whichwe nowdene. Foragraph G = V;E and > 0,wesaytwonon-emptydisjointsubsets X;Y V are -regular ifforall X 0 X; j X 0 j > j X j and Y 0 Y; j Y 0 j > j Y j ,wehave j d G X;Y )]TJ/F22 11.9552 Tf 12.099 0 Td [(d G X 0 ;Y 0 j < ,where d G X 0 ;Y 0 = j G [ X 0 ;Y 0 ] j = j X 0 jj Y 0 j isthe density ofthebipartitesubgraph G [ X 0 ;Y 0 ]of G consistingofalledges f x;y g2 E with x 2 X 0 and y 2 Y 0 .Szemeredi'sregularitylemmaisformallystatedasfollows. Theorem1.1SzemerediRegularityLemma[28,29] Forall > 0 andallintegers t 0 ,thereexistsaninteger T 0 suchthateverygraph G = V;E oforderatleast t 0 admitsanequitableand -regularpartition V G = V 0 [ V 1 [[ V t t 0 t T 0 meaning i j V 1 j = = j V t j j V 0 j < j V G j ; iiallbut )]TJ/F23 7.9701 Tf 6.068 -4.379 Td [(t 2 pairs V i ;V j 1 i
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graphcountinglemmawouldsayforexample,thatif V i ;V j ;V k haveallpairs V i ;V j V i ;V k V j ;V k -regularwithrespectivedensities d ij ;d ik ;d jk > 0,then G [ V i ;V j ;V k ] contains f d ij d ik d jk j V i jj V j jj V k j triangles K 3 ,where f 0as 0.Joint applicationofTheorem1.1andtheCountingLemmaforgraphsisknownasthe GraphRegularityMethod [26]. ThegreatimportanceofSzemeredi'sRegularityLemmaledtoasearchforextensionsto k -uniformhypergraphs,seeforexample[5,6,11,12].Whiletheseearly generalizationsdidleadtosomeinterestingapplications,theydidnotseemtocapturethefullpowerofSzemeredi'slemmaforgraphs.Inparticular,theydidnot allowfortheembeddingofsmallsubsystemswithinaregularstructurei.e.,theydid notadmitacompanioncountinglemmaforhypergraphs.TherstgeneralizedregularitylemmathatdidhavethispropertywasthelemmaofFranklandRodl[10] for3-uniformhypergraphs.ExtendingFranklandRodl'slemma[10],regularitylemmasandcountinglemmasfor k -uniformhypergraphs,alsoallowingtheembeddingof smallsubstructures,weredevelopedlaterbyNagle,Rodl,SchachtandSkokan[20,27], Gowers[13,14],andsubsequentlyTao[30].Thecombineduseofahypergraphregularitylemmaandahypergraphcountinglemmaisknownasthe HypergraphRegularity Method Amaindrawbackinapplyingthehypergraphregularityandcountinglemmas discussedaboveistheirsignicantlytechnicalformulations.Forone,theydonot transparentlygeneralizeSzemeredi'sregularitylemma.Inparticular,Szemerediregularitylemma`regularizes'graphedgesw.r.t.vertices.The k -uniformhypergraph regularitylemmasabove`regularize' k -tuplesw.r.t. k )]TJ/F19 11.9552 Tf 12.699 0 Td [(1-tuples,whicharethen regularizedw.r.t. k )]TJ/F19 11.9552 Tf 12.66 0 Td [(2-tuples,andsoon.Theresulting`regularparts'inthese lemmasare,infact,afamilyofhypergraphswhich,inaddition,becomeincreasingly sparseovertheloweruniformities. Whilethetechnicalityofthehypergraphregularitytoolsdiscussedaboveisnecessarytoachievesomeoftheirapplications,itisnotnecessarytoachievealldesired applications.Inthisthesis,weexploreseveralhypergraphproblemsusingconsiderablysimplerregularitytools.Thesetools,inaverydirectsense,generalizethe 2

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originalSzemeredigraphregularitylemmaandgraphcountinglemma. 1.1Tools Webegindiscussingourworkbystatingahypergraphregularitylemmafor k -uniform hypergraphswhichgeneralizesSzemerediregularitylemmaforgraphs.Tothatend, weneedthefollowingconcepts.Fora k -uniformhypergraph H onan n vertexset V and > 0,wecalla k -tupleofnonemptypairwisedisjointsets X 1 ;X 2 ;:::;X k V regularifforall X 0 i X i i 2 [ k ],satisfying j X 0 i j j X i j ,wehave j d H X 0 1 ;X 0 2 ;:::;X 0 k )]TJ/F22 11.9552 Tf -426.157 -20.922 Td [(d H X 1 ;X 2 ;:::;X k j ,where d H X 0 1 ;X 0 2 ;:::;X 0 k = jH [ X 0 1 ;X 0 2 ;:::;X 0 k ] j j X 0 1 jj X 0 1 j ::: j X 0 k j isthe density ofthe k -partitesubhypergraph H [ X 0 1 ;X 0 2 ;:::;X 0 k ]of H consistingof allhyperedges f x 1 ;:::;x k g2H with x i 2 X 0 i .Sometimeswewrite k X 0 1 ;X 0 2 ;:::;X 0 k k todenote H [ X 0 1 ;X 0 2 ;:::;X 0 k ].Further,wesayapartition V H = V 0 [ V 1 [ :::V t is equitable if j V 1 j = = j V t j and j V 0 j 0 ,everyinteger k 2 and every t 0 1 ,thereexistsaninteger T 0 s.t.every k -uniformhypergraph H oforderat least t 0 admitsanequitableand -regularpartition f V 0 ;V 1 ;:::;V t g with t 0 t T 0 meaning i j V 1 j = = j V t j j V 0 j < j V H j ; iiallbutatmost )]TJ/F23 7.9701 Tf 6.261 -4.379 Td [(t k ofthe k -tuples V i 1 ;V i 2 ;:::;V i k 1 i 1
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Theorem1.2,duetoCzygrinowandRodl,doesjustthis.Forgraphs,i.e., k =2,this wasdonebyAlonetal.[2]. Theorem1.3Czygrinow,Rodl[6] Forallintegers k 2 and t 0 1 andall > 0 ,thereexistsaninteger T 0 sothatforevery k -uniformhypergraph H on n vertices,onemayconstructintime O n 2 k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 log 2 n ,an -regularand t -equitablepartition V H = V 1 [[ V t t 0 t T 0 ,whoseevery -irregular k -tuple V i 1 ;:::;V i k 1 i 1 < d 0 foreach F 2F .Inthiscontext,wewrite F H forthecollectionof copiesof F in H whichcross"thepartition V 1 [[ V f ,i.e.,haveavertexineach class V i ,1 i f Theorem1.4CountingLemmaforSimple k -UniformHypergraphs Forall integers f k 2 ,forallconstants d 0 ; 2 ; 1] ,andforallsimple k -uniformhypergraphs F onvertexset [ f ] ,thereexists > 0 sothatwhenever F andhypergraph H areasintheprecedingsetupwithconstants f;k;d 0 ;;n ,where n n 0 f;k;d 0 ;; issucientlylarge, jF H j = n f Y F 2F d F : 4

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Theorem1.4isproveninChapter3.WealsodiscussaslightextensionofTheorem1.4 inChapter3.WeremarkthatTheorem1.4doesnotholdwheneverthesubhypergraph F isnotsimplesee[16]fordetails.Wenowproceedtoournewresults. 1.2Results Forourrstproblem,let F = fF i g i 2 I beapossiblyinnitefamilyofsimple k uniformhypergraphs.LetForb n; F denotethecollectionof k -uniformhypergraphs H onvertexset f 1 ;:::;n g whichcontainnocopyof F2 F asasubhypergraph. Letex n; F =max fjHj : H2 Forb n; F g denotetheclassical Turannumber almost noneofwhichareknownfor k 3.Observethatlog 2 j Forb n; F j ex n; F because allsubhypergraphsofamaximal H2 Forb n; F alsobelongtoForb n; F .Weshow that,inasense,thislowerboundisbestpossible. Theorem1.5 Let F = fF i g i 2 I beapossiblyinnitefamilyofsimple k -uniformhypergraphs.Forall > 0 ,thereexists n 0 = n 0 2 N sothatforall n>n 0 j Forb n; F j 2 ex n; F + n k : Theorem1.5,andvariousversionsofit,werestudiedbyawidevarietyofauthors [1,3,4,8,9,15,19,21{25].WementionthatTheorem1.5holdsevenwhen F consistsof notnecessarilysimplehypergraphs F see[21],butthisproofisquitetechnicaland reliesonthehypergraphregularitytechniquesof[13,14,20,27,30]mentionedearlier. Inthecasewhereall F2 F aresimple,weareabletogiveaneasierproof,anddoso inChapter4. Forournextresult,weconsidertheproblemofestimatingthefrequency jF G j of asubhypergraph F on f verticesinahosthypergraph G on n vertices.Weslightly abusethenotation jF G j here,sincethereisnopartitiontocross".Clearly,one cancompute jF G j preciselyintime O n f .Weshowthatwhen F issimple, jF G j maybeaccuratelyapproximatedinconsiderablyshortertime. 5

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Theorem1.6 Let F beasimple k -uniformhypergraphon f k 2 vertices,and let > 0 begiven.Thereexistsanalgorithmwhich,foragiven k -uniformhypergraph G on n vertices,computesavalue G intime O n 2 k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 log 2 n forwhich jjF G j)]TJ/F19 11.9552 Tf -417.824 -20.921 Td [( G j 0 andsimple k -uniformhypergraphs F on f vertices,thereexists > 0 andinteger n 0 = n 0 f;k;; F ; sothatthefollowingholds: Givena k -uniformhypergraph H on n n 0 verticeswhichcontainsfewerthan n f copiesof F ,onemaydelete,intime O n 2 k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 log 2 n ,atmost n k edgesfrom H tomakeit F -free. Weprovetheorem1.7inChapter6. 6

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2RegularityLemmaFor k -UniformHypergraphs InthischapterwestateandprovetheHypergraphRegularityLemma.Fordenitions oftechnicaltermsandnotation,refertothediscussiononTheorem1.2giveninthe introduction. Theorem2.1 Forevery > 0 ,forallintegers k andevery t 0 1 ,thereexistsan integer T 0 s.t.every k -uniformhypergraph H oforderatleast t 0 admitsan -regular partition f V 0 ;V 1 ;:::;V t g with t 0 t T 0 TheproofofTheorem2.1followstheoriginalargumentofSzemerediforsimplegraphs [28,29].Inparticular,ifanequitablepartition f V 0 ;V 1 ;:::;V t g of V = V H isnot regular,thenweshallrenetheclasses V 1 ;V 1 ;:::;V t toformanewequitablepartition f V 0 0 ;V 0 1 ;:::;V 0 t 0 g of V = V H ,where t 0 t 4 t )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 andwherethelatterpartitionis `closer'tobeing -regularthantheformer.Moreprecisely,forpairwisedisjointsets X 1 ;X 2 ;:::;X k V ,wedeneameasureofregularityasfollows: q X 1 ;X 2 ;:::;X k := j X 1 jj X 2 j ::: j X k j n k d 2 X 1 ;X 2 ;:::;X k = k X 1 ;X 2 ;:::;X k k 2 j X 1 jj X 2 j ::: j X k j n k ; .1 call q X 1 ;X 2 ;:::;X k the index ofthe k -tuple X 1 ;X 2 ;:::;X k .Forpartitions i of X i ,let q 1 ; 2 ;:::; k = X f q Y 0 1 ;Y 0 2 ;:::;Y 0 k : Y 0 1 2 1 ;Y 0 2 2 2 ;:::;Y 0 k 2 k g : .2 7

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Forapartition P = f C 1 ;C 2 ;:::;C t g of V ,let q P := X i i
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ProofofTheorem2.1. Let > 0and t 0 1begiven,andwithoutlossofgenerality, let 1 = 4.Let s :=2 k = k +3 .Thisnumber s isanupperboundonthenumberof iterationsofLemma2.2thatcanbeappliedtoapartitionofahypergraphbeforeit becomes -regular.Recallthat q P 1forallpartitions P Thereisoneformalrequirementwhichapartition f C 0 ;C 1 ;:::;C t g with j C 1 j = j C 2 j = = j C t j hastosatisfybeforeLemma2.2canbere-applied,viz.,thesize j C 0 j ofitsexceptionalsetmustnotexceed n .Witheachiterationofthelemma, however,thesizeoftheexceptionalsetcangrowbyupto n= 2 t )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 .Wethuswantto choose t largeenoughsothateven s incrementsof n= 2 t )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 adduptoatmost 1 2 n and n largeenoughthat,foranyinitialvalueof j C 0 j T 0 ,let C 0 V beminimalsuchthat t divides j V n C 0 j ,andlet f C 1 ;:::;C t g beanypartitionof V n C 0 intosetsofequalsize.Then j C 0 j
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Inwhatfollows,weprovetwofurtherlemmaswhichareeventuallyneededinthe proofofLemma2.2.Webeginbyshowingthatwhenwereneapartition,thevalue of q willnotdecrease. Lemma2.3 i Let X 1 ;X 2 ;:::;X k V bedisjoint.If i isapartitionof X i ,then q 1 ; 2 ;:::; k q X 1 ;X 2 ;:::;X k ii If P P 0 arepartitionsof V and P 0 renes P ,then q P 0 q P Proof.i Let i =: f Y i 1 ;Y i 2 ;:::;Y il i g ,where Y ij X i ,forall i 2 [ k ].Then, q 1 ; 2 ;:::; k = X j i 2 [ l i ] i 2 [ k ] q Y 1 j 1 ;Y 2 j 2 ;:::;Y kj k = 1 n k X j i 2 [ l i ] i 2 [ k ] k Y 1 j 1 ;Y 2 j 2 ;:::;Y kj k k 2 j Y 1 j 1 jj Y 2 j 2 j ::: j Y kj k j C:S 1 n k P j i 2 [ l i ] i 2 [ k ] k Y 1 j 1 ;Y 2 j 2 ;:::;Y kj k k 2 P j i 2 [ l i ] i 2 [ k ] j Y 1 j 1 jj Y 2 j 2 j ::: j Y kj k j = 1 n k k X 1 ;X 2 ;:::;X k k 2 P j 1 2 [ l 1 ] j Y 1 j 1 j P j 2 2 [ l 2 ] j Y 2 j 2 j ::: P j k 2 [ l k ] j Y kj k j = 1 n k k X 1 ;X 2 ;:::;X k k 2 j X 1 jj X 2 j ::: j X k j = q X 1 ;X 2 ;:::;X k ; wheretheinequality C:S followsfromtheCauchy-Schwarzinequality. 10

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ii Let P =: f C 1 ;C 2 ;:::;C t g andfor i 2 [ t ],let C i bethepartitionof C i induced by P 0 .Then, q P = X i 1 0 ,andlet X 1 ;X 2 ;:::X k V bedisjoint.If X 1 ;X 2 ;:::X k is not -regular,thentherearepartitions i = Y i 1 ;Y i 2 of X i suchthat q 1 ; 2 ;:::; k q X 1 ;X 2 ;:::;X k + k +2 j X 1 jj X 2 j ::: j X k j n k : Proof. Suppose X 1 ;X 2 ;:::X k isnot -regular.Thentherearesets Y i 1 X i ,with: j Y i 1 j j X i j suchthat j j > for := d Y 11 ;Y 21 ;:::Y k 1 )]TJ/F22 11.9552 Tf 12.945 0 Td [(d X 1 ;X 2 ;:::X k .Let i := f Y i 1 ;Y i 2 g where Y i 2 := X i n Y i 1 .Wenowshowthat 1 ; 2 ;:::; k satisfytheconclusionofthelemma.For brevity,weshallwrite y ij := j Y ij j ; e i 1 ;:::;i k := k Y 1 i 1 ;Y 2 i 2 ;:::;Y ki k k ; x i := j X i j and e := k X 1 ;X 2 ;:::;X k k .AsintheproofofLemma2.3: 11

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q 1 ; 2 ;:::; k = 1 n k X i 1 ;:::;i k 2f 1 ; 2 g e i 1 ;:::;i k 2 y 1 i 1 y 2 i 2 y ki k = 1 n k e 1 ; 1 ;:::; 1 2 y 11 y 21 y k 1 + P i 1 ;:::;i k 2f 1 ; 2 g i 1 ;i 2 ;:::;i k 6 = ; 1 ;:::; 1 e i 1 ;:::;i k 2 y 1 i 1 y 2 i 2 y ki k C:S 1 n k e 1 ; 1 ;:::; 1 2 y 11 y 21 y k 1 + e )]TJ/F20 7.9701 Tf 6.586 0 Td [( e 1 ; 1 ;:::; 1 2 x 1 x 2 x k )]TJ/F20 7.9701 Tf 6.587 0 Td [( y 11 y 21 y k 1 ; wherethelastinequalityfollowsfromCauchy-Schwarzinequality.Bydenitionof wehave e 1 ; 1 ; ; 1 = y 11 y 21 y k 1 e x 1 x 2 x k + y 11 y 21 y k 1 .So, n k q 1 ; 2 ;:::; k 1 Q i y i 1 Q i y i 1 e Q i x i + Q i y i 1 2 + 1 Q i x i )]TJ/F19 11.9552 Tf 11.955 0 Td [( Q i y i 1 Q i x i )]TJ/F20 7.9701 Tf 6.586 0 Td [( Q i y i 1 Q i x i e )]TJ/F22 11.9552 Tf 11.955 0 Td [( Q i y i 1 2 = Q i y i 1 e 2 Q i x i 2 + 2 e Q i y i 1 Q i x i + 2 Y i y i 1 + Q i x i )]TJ/F28 11.9552 Tf 11.955 8.967 Td [(Q i y i 1 Q i x i 2 e 2 )]TJ/F19 11.9552 Tf 13.151 8.791 Td [(2 e Q i y i 1 Q i x i + 2 Q i y 2 i 1 Q i x i )]TJ/F28 11.9552 Tf 11.956 8.967 Td [(Q i y i 1 e 2 Q i x i + 2 Y i y i 1 j j > e 2 Q i x i + k +2 Y i x i since y i 1 x i bythechoiceof Y i 1 Finally,weshowthatifapartitionhasenoughirregular k -tuplesofpartitionsets tofallshortofthedenitionofan -regularpartition,thensub-partitioningallthose k -tuplesatonceresultsinanincreaseof q byaconstant. ProofofLemma2.2. Forall1 i 1
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respectivelywith jC i 1 ;i 2 ;:::;i k j = = jC i k ;i 1 ;:::;i k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 j =2and, q C i 1 ;i 2 ;:::;i k ;:::; C i k ;i 1 ;:::;i k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 q C i 1 ;C i 2 ;:::;C i k + k +2 j C i 1 jj C i 2 j ::: j C i k j n k = q C i 1 ;C i 2 ;:::;C i k + k +2 c k n k : .6 Foreach i =1 ;:::;t ,let C i betheuniqueminimalpartitionof C i thatrenes everypartition C i 1 ;:::;i k with i 1 = i ; i 1 6 = i j 8 j 6 =1.Inotherwords,ifweconsidertwo elementsof C i asequivalentwhenevertheylieinthesamepartitionsetof C i 1 ;:::;i k with i 1 = i ; i 1 6 = i j 8 j 6 =1;then C i isthesetofequivalenceclasses.Thus, jC i j 2 t )]TJ/F21 5.9776 Tf 5.757 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 Nowconsiderthefollowingpartitionof V : C := C 0 [ t [ i =1 C i with C 0 asexceptionalset.Then C renes P and: t jCj t 2 t )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 : .7 Let C 0 := ff v g : v 2 C 0 g .Now,if P isnot -regular,thenformorethan )]TJ/F23 7.9701 Tf 6.261 -4.379 Td [(t k ofthe k -tuples C i 1 ;:::;C i k with1 i 1 <
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Hence,byourdenitionof q forpartitionswithexceptionalset,andLemma2.3i: q C = X 1 i 1
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fromeachset C 6 = C 0 of C .Hence, j C 0 0 jj C 0 j + d jCj : 7 j C 0 j + c 4 t )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 t 2 t )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 = j C 0 j + ct= 2 t )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 j C 0 j + n= 2 t )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 : 15

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3CountingLemmaForSimple k -UniformHypergraphs InthischapterwestateandprovetheCountingLemmaforsimple k -uniformhypergraphs.WeaskthereadertorecallthehypothesisforCountingLemmafromthe discussionprecedingTheorem1.4intheintroduction. 3.1MainTheorem Theorem3.1CountingLemmaforSimple k -UniformHypergraphs Forall integers f k 2 ,forallconstants d; 2 ; 1] ,andforallsimple k -uniformhypergraphs F onvertexset [ f ] ,thereexists > 0 sothatwhenever F andhypergraph H are asinthehypothesisofTheorem1.4,withconstants f;k;d;;n ,where n n 0 f;d;k; issucientlylarge,then jF H j = d jFj n f : Wesayafewwordsabouttheproof.First,notethattheCountingLemmastated herepromisesbothanupperandalowerbound: )]TJ/F22 11.9552 Tf 12.379 0 Td [( n f Q d F jF H j + n f Q d F .Inwhatfollowsweshowthelowerboundonlysincethecorresponding upperboundfollowsbysymmetricarguments.Second,fornotationalsimplicity,we shallassumethatall d F = d 0 = d recallthehypothesisofTheorem1.4.Theproof allowingthedensities d F d 0 F 2F ,isphilosophicallythesame.Ourproofof Theorem3.1isalreadyfairlyheavyinnotation. Proof. Letintegers f k 2and d = d 0 ; 2 ; 1]begivenalongwithsimple k -uniformhypergraph F onvertexset[ f ].Todene > 0andgiveourproofof 3.1,weinducton jFj .If jFj =0,thenany > 0willdoandtheresultistrivial. 16

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Indeed,inthiscase jHj =0andtherefore jF H j = n f )]TJ/F22 11.9552 Tf 11.563 0 Td [( d 0 n f .If jFj =1,set = d andtheresultistrivial.Indeed,inthiscase,suppose F = f i 1 ;:::;i k g2 )]TJ/F20 7.9701 Tf 5.479 -4.379 Td [([ f ] k Then H = H [ V i 1 ;:::;V i k ].Byhypothesis, jHj = d n k ,inwhichcase jF H j d )]TJ/F22 11.9552 Tf 11.955 0 Td [( n k n f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k = )]TJ/F22 11.9552 Tf 11.955 0 Td [( d 1 n f Now,let F begivenasinthehypothesisofTheorem1.4with jFj 1edges. Deleteanyedge F 1 2F from F w.l.o.g, F 1 = f f )]TJ/F22 11.9552 Tf 11.656 0 Td [(k +1 ;:::;f g .Set e = 4 andlet 1 = Thm: 3 : 1 f;k;d; e ;F n F 1 betheconstantguaranteedbytheinductionhypothesis. Set = min f e d f k ; 1 g .1 Weprovethatwiththischoiceof > 0, jF H j )]TJ/F22 11.9552 Tf 11.854 0 Td [( d jFj n f .Weconsiderthe followingsubhypergraphsof F .Set F )]TJ/F19 11.9552 Tf 10.515 -4.339 Td [(= Fn F 1 and F = F [ f 1 ;:::;f )]TJ/F22 11.9552 Tf 11.998 0 Td [(k g ].Then F )]TJ/F19 11.9552 Tf 11.905 -4.338 Td [(isasimple k -uniformhypergraphon f verticesand jFj)]TJ/F19 11.9552 Tf 32.284 0 Td [(1edgesand F isa simple k -uniformhypergraphon f )]TJ/F22 11.9552 Tf 11.884 0 Td [(k verticesandatmost jFj)]TJ/F19 11.9552 Tf 30.891 0 Td [(1edges.Similarly, dene H )]TJ/F19 11.9552 Tf 11.413 -4.338 Td [(= HnH [ V f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k +1 ;:::;V f ]and H = H [ V 1 ;:::;V f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k ].Notethat F )]TJ/F22 11.9552 Tf 7.085 -4.338 Td [(; H )]TJ/F19 11.9552 Tf 7.085 -4.338 Td [( and F ; H areeachasinhypothesisofTheorem1.4. Wenowdenesomerelatedconcepts.For F 0 2F H and F )]TJ/F20 7.9701 Tf -1.187 -7.88 Td [(0 2F )]TJ/F19 11.9552 Tf 7.085 -4.339 Td [( H )]TJ/F19 11.9552 Tf 7.085 -4.339 Td [( F 0 2F H ,wesaythat F )]TJ/F20 7.9701 Tf -1.187 -7.879 Td [(0 ,resp. F 0 extends F 0 if F 0 F )]TJ/F20 7.9701 Tf -1.187 -7.879 Td [(0 ,resp. F 0 F 0 .Let F 0 2F H F )]TJ/F20 7.9701 Tf -1.188 -7.88 Td [(0 2F )]TJ/F19 11.9552 Tf 7.085 -4.339 Td [( H )]TJ/F19 11.9552 Tf 7.084 -4.339 Td [(and F 0 2F H begiven.For F 0 2F H ,let ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 = fF )]TJ/F20 7.9701 Tf -1.187 -7.892 Td [(0 2F )]TJ/F19 11.9552 Tf 7.084 -4.936 Td [( H )]TJ/F19 11.9552 Tf 7.085 -4.936 Td [(: F )]TJ/F20 7.9701 Tf -1.188 -7.892 Td [(0 extends F 0 g ; ext F F 0 = fF 0 2F H : F 0 extends F 0 g : Observethat: jF H j = X F 0 2F H j ext F F 0 j ; jF )]TJ/F19 11.9552 Tf 7.084 -4.937 Td [( H )]TJ/F19 11.9552 Tf 7.085 -4.937 Td [( j = X F 0 2F H j ext F )]TJ/F19 11.9552 Tf 6.753 -0.298 Td [( F 0 j : .2 17

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Wenowprovideevaluationsofext F F 0 andext F )]TJ/F19 11.9552 Tf 6.753 -0.299 Td [( F 0 foraxedterm F 0 2 F H .Tothatend,x F 0 2F H .Set: V F 0 = [ fF )]TJ/F20 7.9701 Tf -1.187 -7.892 Td [(0 : F )]TJ/F20 7.9701 Tf -1.187 -7.892 Td [(0 2 ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 g : .3 Notethat V F 0 V .Foreach f )]TJ/F22 11.9552 Tf 11.955 0 Td [(k +1 i f ,set: V F 0 i = V i V F 0 : Wenowmakethefollowingclaim. Claim3.2 For F 0 2F H xed, j ext F )]TJ/F19 11.9552 Tf 6.753 -0.299 Td [( F 0 j = j V F 0 f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 j ::: j V F 0 f j ; j ext F F 0 j = jH [ V F 0 f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k +1 ;:::;V F 0 f ] j : WedefertheproofofClaim3.2totheendofthissectioninfavorofcontinuing theproofofTheorem3.1.Call F 0 2F H big if j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 j >n k and small otherwise.Write F big H F small H forthecollectionofall big small elements F 0 2F H .ItfollowsfromClaim3.2thatforeach F 0 2F big H ,wehave: j ext F F 0 j = d j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 j : .4 Indeed,bythe d; -regularityof H [ V F 0 f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 ;:::;V F 0 f ], j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 j = j V F 0 f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 jj V F 0 f j >n k j V F 0 f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 jj V F 0 f j >n j ext F F 0 j = jH [ V F 0 f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 ;:::;V F 0 f ] j = d j V F 0 f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 jj V F 0 f j = d ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 : .5 WemaynowconcludetheproofofTheorem3.1usingthefollowingclaim. 18

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Claim3.3 X F 0 2F big H j ext F )]TJ/F19 11.9552 Tf 6.753 -0.299 Td [( F 0 j )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 e d jFj)]TJ/F20 7.9701 Tf 18.133 0 Td [(1 n f : Indeed,by.2,.4andClaim3.3,wehave: jF H j : 2 X F 0 2F big H j ext F F 0 j : 4 X F 0 2F big H d )]TJ/F22 11.9552 Tf 11.955 0 Td [( j ext F )]TJ/F19 11.9552 Tf 6.753 -0.299 Td [( F 0 j Claim: 3 : 3 d )]TJ/F22 11.9552 Tf 11.955 0 Td [( )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 e d jFj)]TJ/F20 7.9701 Tf 18.134 0 Td [(1 n f )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 e d jFj n f : ItremainstoproveClaims3.2and3.3. ProofofClaim3.2. Werstestablishtherstidentity.That j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 jj V F 0 f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 j ::: j V F 0 f j isclear,soweestablishthelowerbound.Tothatend,let v f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 2 V F 0 f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 ;:::;v f 2 V F 0 f begiven.Weclaimthat F 0 [f v f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 ;:::;v f g2F )]TJ/F19 11.9552 Tf 7.085 -4.339 Td [( H )]TJ/F19 11.9552 Tf 7.085 -4.339 Td [(.Assume,onthe contrary,that F 0 [f v f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 ;:::;v f g = 2F )]TJ/F19 11.9552 Tf 7.085 -4.338 Td [( H )]TJ/F19 11.9552 Tf 7.085 -4.338 Td [(andwrite,forsimplicity, F 0 = f v 1 ;:::;v f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k g ,where v 1 2 V 1 ;:::;v f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k 2 V f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k .Since f v 1 ;:::;v f g2F )]TJ/F19 11.9552 Tf 7.085 -4.339 Td [( H )]TJ/F19 11.9552 Tf 7.084 -4.339 Td [(,there exists F 2 = f i 1 ;:::;i k g2F )]TJ/F19 11.9552 Tf 7.085 -4.338 Td [(,1 i 1 <
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each f v f )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 ;:::;v f g2H [ V F 0 f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k +1 ;:::;V F 0 f ]satises F 0 [f v f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k +1 ;:::;v f g2F H ThisprovesClaim3.2. ProofofClaim3.3. Byourinductionhypothesis, jF )]TJ/F19 11.9552 Tf 7.085 -4.937 Td [( H )]TJ/F19 11.9552 Tf 7.085 -4.937 Td [( j )]TJ/F28 11.9552 Tf 11.995 0 Td [(e d jFj)]TJ/F20 7.9701 Tf 18.134 0 Td [(1 n f : .6 Ontheotherhand,by.2,wehave jF )]TJ/F19 11.9552 Tf 7.084 -4.936 Td [( H )]TJ/F19 11.9552 Tf 7.084 -4.936 Td [( j = X F 0 2F big H j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 j + X F 0 2F small H j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 j X F 0 2F big H j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 j + jF small H j n k X F 0 2F big H j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 j + n f : Returningto,wesee: X F 0 2F big H j ext F )]TJ/F19 11.9552 Tf 6.752 -0.299 Td [( F 0 j )]TJ/F28 11.9552 Tf 11.996 0 Td [(e d jFj)]TJ/F20 7.9701 Tf 18.133 0 Td [(1 n f )]TJ/F22 11.9552 Tf 11.955 0 Td [(n f )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 e d jFj)]TJ/F20 7.9701 Tf 18.133 0 Td [(1 n f AndthiscompletesourproofoftheCountingLemma. 3.2AGeneralization TheproofofTheorem3.1canbeeasilymodiedtoproveaslightgeneralizationof theCountingLemma.WeusethisgeneralizationinChapter6,andsowestateit now. Setup ExtendedCountingEnvironmentLet d 0 ;> 0begivenandlet F bea p -partite 20

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simple k -uniformhypergraphwithvertexpartition V F = W 1 [[ W p .Let H bea p -partite k -uniformhypergraphwithvertexpartition V H = V 1 [[ V p j V 1 j = = j V p j = n ,satisfyingthefollowingproperty:forall K = f i 1 ;:::;i k g2 )]TJ/F20 7.9701 Tf 5.479 -4.378 Td [([ p ] k H [ V i 1 ;:::;V i k ]is d K ; -regular,where d K d 0 ,if F hasanedgecrossing W i 1 [[ W i k ,and H [ V i 1 ;:::;V i k ]= ; otherwise. For F and H asgivenabove,vertices v 1 ;:::;v f V H aresaidtospana partiteisomorphic copyof F in H ifthereexistsabijection : V F !f v 1 ;:::;v f g where v j 2 V i ,1 j f ,1 i p ,ifandonlyif )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 v j 2 W i andwhere,foreach F 2F F 2H .Inthiscontext,weshallwrite F H againabusingnotationfortheset ofallpartite-isomorphiccopiesof F in H Theorem3.4ExtendedCountingLemma Forallintegers f p k 2 constants d 0 ; 2 ; 1] ,andall p -partitesimplehypergraphs F on f vertices,there exists > 0 sothatwhenever F and k -uniformhypergraph H areasinthepreceding Setupwiththeseconstantsand n sucientlylarge,then jF H j = n f Y F 2F n d K F : K F = f i 1 ;:::;i k g2 )]TJ/F20 7.9701 Tf 5.48 -4.379 Td [([ p ] k s.t. F W i 1 [[ W i k o : 21

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4Bounding j Forb n; F j Wearenowreadytopresentaproofforourrstapplicationofthehypergraph regularitymethod.RecallthediscussionprecedingTheorem1.5intheintroduction forthesemanticsoftechnicalnotation. Theorem4.1 Let F = fF i g i 2 I beapossiblyinnitefamilyofsimple k -uniformhypergraphs.Forall > 0 ,thereexists n 0 = n 0 2 N sothatforall n>n 0 j Forb n; F j 2 ex n; F + n k : Proof. Letfamily F = fF i g i 2 I and > 0begivenasinTheorem4.1.Ourproof ofTheorem4.1beginswithadescriptionofauxiliaryconstantsweuseintheproof. First,let d 0 2 ; 1besmallenoughsothat 20 d 0 log e 4 d 0 <: .1 Itiswellknownthatthesequence )]TJ/F23 7.9701 Tf 5.833 -4.379 Td [(s k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ex s; F 1 s =1 isnon-increasing,andtherefore,thelimits lim s !1 ex s; F =lim s !1 ex s; F )]TJ/F23 7.9701 Tf 5.832 -4.379 Td [(s k andlim s !1 e ex s; F =lim s !1 ex s; F s k .2 exist.Let s 0 2 N belargeenough,sothatforall s 1 ;s 2 s 0 max fj ex s 1 ; F )]TJETq1 0 0 1 272.291 137.896 cm[]0 d 0 J 0.478 w 0 0 m 11.381 0 l SQBT/F19 11.9552 Tf 272.291 131.075 Td [(ex s 2 ; F j ; j e ex s 1 ; F )]TJ/F28 11.9552 Tf 14.324 0 Td [(e ex s 2 ; F jg < 20 ; .3 22

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andlet t 0 = max f s 0 ; 1 d 0 g : .4 Finally,forasimple k -unifromhypergraph F on s 0 vertices,let F = Thm: 3 : 1 F ;d 0 ; with =1 = 2,bethepositiveconstantguaranteedbyTheorem3.1,theCounting LemmaforSimpleHypergraphs.Set, = min F f F g d 0 ; .5 wheretheminimumistakenoverallsimple k -uniformhypergraphs F on s 0 vertices. Notethat d 0 followsfromTheorem3.1.Inallthatfollowswetaketheinteger n sucientlylargewheneverneeded.Thisconcludesourdiscussionoftheconstants. Toeach G2 Forb n; F ,associatean -regular t G -equitablepartition P G : V G = [ n ]= V 0 [ V 1 [[ V t G t 0 t G T 0 .Every G2 Forb n; F admitsatleastonesuch partitionbyTheorem2.1,andifsome G2 Forb n; F admitsmultiplesuchpartitions, wechooseonearbitrarily.Deneanequivalencerelation onForb n; F asfollows. Foreach G 1 ; G 2 2 Forb n; F G 1 G 2 P G 1 = P G 2 : Let= 1 [[ N bethecorrespondingpartitionofForb n; F .Notethat: j j = N T 0 X i = t 0 n b n=i c i =2 O n : .6 Nowxanequivalenceclass j ,1 j N ,i.e.xa t -equitablevertexpartition P :[ n ]= V 0 [ V 1 [[ V t ,where t 0 t T 0 ,withrespecttowhichevery G2 j is -regular.Wenowpartition j asfollows.Forafunction 2f 0 ; 1 g [ t ] k ,let j; = n G2 j : 8f i 1 ;::;i k g2 [ t ] k ; G [ V i 1 ;::;V i k ]is )]TJ/F19 11.9552 Tf 9.298 0 Td [(regand d G V i 1 ;::;V i k d 0 f i 1 ;::;i k g =1 o : .7 23

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Then j = S f j; : 2f 0 ; 1 g [ t ] k g isapartitionwithatmost 2 t k 2 T 0 k =2 O .8 parts.TheproofofTheorem1restsonthefollowingproposition. Proposition4.2 For 1 j N and 2f 0 ; 1 g [ t ] k xed, j j; j 2 ex n; F + 2 n k IndeedbyProposition4.2and.6-.8,wehave: j Forb n; F j = N X j =1 j j j = N X j =1 X fj j; j : 2f 0 ; 1 g [ t ] k g ; 2 ex n; F + 2 n k + O n + O 2 ex n; F + n k : .9 ItremainstoproveProposition4.2. ProofofProposition4.2. Fix1 j N and 2f 0 ; 1 g [ t ] k .Notethatevery G2 j; canbewrittenastheunion, G = G V 0 [G V 1 [[G V t [G 0 [G 1 where G V 0 = f K 2G : K V 0 6 = ;g G V a = f K 2G : j K V a j 2 g for1 a t ,and for i =0 ; 1, G i = [ fG [ V i 1 ;::;V i k ]: f i 1 ;::;i k g2 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i g : .10 Since P isa t -equitablepartitionsharedbyallof j; jG V 0 [G V 1 [[G V t jj V 0 j n k )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + t b n=t c 2 n k )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 tn k )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + 1 2 t n k T 0 n + 1 2 t 0 n k o + 1 2 t 0 n k : .11 24

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Aswell,wehave jG 0 j + d 0 n k : .12 Indeed,the -regularityof P ensuresatmost t k k -tuples f i 1 ;:::;i k g2 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 could have G [ V i 1 ;:::;V i k ]being -irregulargivingrisetoatmost t k b n=t c k k -tuples K 2G Otherwise,when d G V i 1 ;:::;V i k d 0 ,wehave jG [ V i 1 ;:::;V i k ] j d 0 b n=t c k overat most )]TJ/F23 7.9701 Tf 6.262 -4.379 Td [(t k k -tuples f i 1 ;:::;i k g2 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 .Inotherwords,combining.11and.12, wemaywriteevery G2 j; asadisjointunion: G = G [G 1 ; .13 where, jG j o + 1 2 t 0 + + d 0 n k 4 d 0 n k : .14 and G 1 isgivenin.10.Wenowusethedecompositionin.13tocountallof j j; j .Indeednotethatthereareatmost 4 d 0 n k X i =0 )]TJ/F23 7.9701 Tf 5.48 -4.379 Td [(n k i n k n k 4 d 0 n k n k e 4 d 0 4 d 0 n k =2 4 d 0 n k log e 4 d 0 + k log n 2 5 d 0 n k log e 4 d 0 : 1 2 4 n k .15 k -graphs G oftheformin.13and.14.Similarly,thereareatmost 2 b n=t c k j )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 j 2 n=t k j )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 j .16 k -graphsoftheformin.13.Weusethefollowingclaim. 25

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Claim4.3 j )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 j ex t; F + 5 t k : Using.13,.14,.15andClaim4.3,weseethatcf..2 log 2 j j; j n t k ex t; F + 5 t k + 4 n k = n k e ex t; F + 9 20 n k : 3 n k e ex n; F + 1 20 + 9 20 n k =ex n; F + 2 n k ; .17 aspromisedbyProposition4.2.ItremaintoproveClaim4.3. ProofofClaim4.3. Assume,onthecontrary,that j )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 j > ex t; F + 5 t k : .18 Forconsistencyofnotation,weshallwritethe k -uniformhypergraph )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 onvertex set[ t ]as J ,whereourassumptionaboveisthat jJj > ex t; F + 5 )]TJ/F23 7.9701 Tf 6.261 -4.379 Td [(t k .Nowwith s 0 : 3 t 0 t givenin.2,notethattheremustalsoexist S 0 2 )]TJ/F20 7.9701 Tf 5.632 -4.379 Td [([ t ] s 0 forwhich jJ [ S 0 ] j ex s 0 ; F +1.Forifnotwewouldhave ex t; F + 5 t k t )]TJ/F22 11.9552 Tf 11.955 0 Td [(k s 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(k < jJj t )]TJ/F22 11.9552 Tf 11.956 0 Td [(k s 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(k t s 0 ex s; F ; .19 orequivalently, ex t; F + 5 < ex s 0 ; F ; .20 contradicting.3.Now,x S 0 2 )]TJ/F20 7.9701 Tf 5.631 -4.379 Td [([ t ] s 0 and,forsimplicityofnotationandw.l.o.g, suppose S 0 = f 1 ;:::;s 0 g .Thehypergraph J [ S 0 ]hasmorethanex s 0 ; F manyedges, andtherefore,mustcontainacopyofsome F 0 2 F .Weshowthesamecopymust 26

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alsoappearinevery G2 j; Forb n; F ,aclearcontradiction,establishingthat .18wasfalse. Indeed,x G 0 2 j; andconsider G 0 [ V 1 ;:::;V s 0 ].Bydenitionof ,wehave,for each f i 1 ;:::;i k g2F 0 J = )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ,that G 0 [ V i 1 ;:::;V i k ]is -regularwithdensity d G 0 [ V i 1 ;:::;V i k ] d 0 .Byourchoiceof > 0in.5,Theorem3.1CountingLemma implies: jF 0 G 0 j )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 d jF 0 j 0 n t s 0 1 2 d s 0 k 0 n s 0 T s 0 0 = n s 0 > 0 : .21 aspromised. 27

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5CountingAlgorithmForSimpleHypergraphs Inwhatfollows,weconsidertheproblemofestimatingthefrequency jF G j ofa subhypergraph F on f verticesinahosthypergraph G on n vertices.Clearly,one cancompute jF G j preciselyintime O n f .Weshowthatwhen F issimple, jF G j maybeaccuratelyapproximatedinconsiderablyshortertime. Theorem5.1 Let F beasimple k -uniformhypergraphon f k 2 vertices,and let > 0 begiven.Thereexistsanalgorithmwhich,foragiven k -uniformhypergraph G on n vertices,computesavalue G intime O n 2 k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 log 2 n forwhich jjF G j)]TJ/F19 11.9552 Tf -417.629 -20.922 Td [( G j 0begivenasinthehypothesisofTheorem5.1.Deneauxiliary constants d 0 > 0and t 0 2 N by d 0 = = = 6and d 3 = e = t 0 : .1 Let 0 = Thm: 3 : 1 F ;d 0 ; > 0betheconstantguaranteedbyTheorem3.1Counting Lemma,andset =min f = 6 ; 0 g : .2 Let T 0 = T 0 t 0 ; .3 betheconstantguaranteedbyTheorem1.3AlgorithmicRegularityLemma.This concludesourdiscussionoftheconstants. 28

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WenowlistthestepsofthealgorithmofTheorem5.1.Afterwards,wenotethe correspondingcomplexity,andestablishtheaccuracyoftheparameter G Step1. Withtheconstants > 0and t 0 chosenabove,applyTheorem1.3 AlgorithmicRegularityLemmato G toconstructan -regular, t -equitable partition V G = V 0 [ V 1 [[ V t where t 0 t T 0 Step2. Constructthefollowingweighted ClusterHypergraph G ;! on vertexset[ t ]= f 1 ;:::;t g whoseedgesareweightedbydensity". I: Dene : )]TJ/F20 7.9701 Tf 5.479 -4.378 Td [([ t ] k [0 ; 1]by: f i 1 ;::;i k g = 8 > < > : d G V i 1 ;::;V i k if d G V i 1 ;::;V i k d 0 0otherwise where d 0 > 0waschosenin.1. II: Set G = )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; 1]tobethecollectionof f i 1 ;::;i k g2 )]TJ/F20 7.9701 Tf 5.479 -4.379 Td [([ t ] k forwhich d G V i 1 ;::;V i k d 0 Step3. Constructthefamily F G ofdistinctcopies F 0 of F in G Step4. Computeandreturn G = b n=t c f X F 0 2F G Y F 2F 0 F : .4 Thisconcludesthealgorithm.Weproceedtoananlysisofitscomplexityandaccuracy. NotethatthecomplexityofthealgorithmaboveisdeterminedbyStep1.Indeed, Theorem2.1constructsthepartition V G = V 0 [ V 1 [[ V t intime O n 2 k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 log 2 n 29

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Theweightedhypergraph G ;! isconstructedintime O n k ,sincetheredensities arecomputedandrecordedand t T 0 = O .Step3isgreedilycompletedin constanttimesince,again, t T 0 = O Wenowprove: jjF G j)]TJ/F19 11.9552 Tf 21.253 0 Td [( G j
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Claim5.2 jjF + G j)]TJ/F19 11.9552 Tf 21.253 0 Td [( G j < 2 n f : ProofofClaim5.2. Considerthefollowingsubhypergraph G 0 G oftheearlier clusterhypergraph G :foreach f i 1 ;:::;i k g2G ,let f i 1 ;:::;i k g2G 0 if,andonlyif, G [ V i 1 ;:::;V i k ]is -regular.Then G 0 isasubhypergraphof G withallbut )]TJ/F23 7.9701 Tf 6.262 -4.379 Td [(t k fewer edges.Now,foreach F 0 2F G 0 ,denethefollowingsubhypergraph H F 0 G of G : Foreach f i 1 ;:::; k g2 )]TJ/F20 7.9701 Tf 5.479 -4.379 Td [([ t ] k ,let H F 0 [ V i 1 ;::;V i k ]= 8 > < > : G [ V i 1 ;::;V i k ]if f i 1 ;::;i k g2F 0 ; otherwise. Then, jF + G j = X F 0 2F G 0 jF H F 0 j : Moreover,applyingTheorem3.1CountingLemmatoeachterminthesumabove, wehave: jF + G j = b n=t c f X F 0 2F G 0 Y F 2F 0 d F = b n=t c f X F 0 2F G 0 Y F 2F 0 F whereforeach F = f i 1 ;:::;i k g2F 0 2F G 0 d F = d G V i 1 ;:::;V i k = F .We thereforehave jF + G j)-222(b n=t c f X F 0 2F G 0 Y F 2F 0 F n f : .10 31

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Recalling.4,wesee: b n=t c f X F 0 2F G 0 Y F 2F 0 F )]TJ/F19 11.9552 Tf 12.619 0 Td [( G b n=t c f X f Y F 2F 0 F : F 0 2F G nF G 0 g b n=t c f jF G nF G 0 j b n=t c f jG nG 0 j t f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k b n=t c f t k t f )]TJ/F23 7.9701 Tf 6.586 0 Td [(k
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6ConstructiveRemovalLemma Forournalresult,weconsiderawellknownTheoremofNagle,Rodl,Shcacht,Skokan [20,27]andGowers[13,14]knownastheRemovalLemma.Thistheoremroughly assertsthatifa`large'hypergraph H contains`few'copiesofaxedsubhypergraph F ,thenonemaydelete`few'edgesfrom H todestroyallthesecopies.Theoriginal proofreliesonthehypergraphregularityandcountinglemmasof[13,14,20,27]and ishighlytechnical.Inthecasethat F issimple,wegiveaneasierproofwhichis,in fact,constructive,whereastheoriginalmoregeneralproofwasnot. Theorem6.1ConstructiveRemovalLemma Forallintegers f k 2 > 0 andsimple k -uniformhypergraphs F on f vertices,thereexists > 0 andinteger n 0 = n 0 f;k;; F ; sothatthefollowingholds. Givena k -uniformhypergraph H on n n 0 verticeswhichcontainsfewerthan n f copiesof F ,onemaydelete,intime O n 2 k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 log 2 n ,atmost n k edgesfrom H tomakeit F -free. ProofofTheorem6.1. Letintegers f k 2, > 0andsimple k -uniformhypergraph F on f verticesbegiven.Todenethepromisedconstants,werstconsiderseveralauxiliaryconstants.Set =1 = 2, d 0 = = 3and t 0 = d 4 = e .Wenow appealtoTheorem3.4GeneralizedCountingLemma.Tothatend,let F be p partiteforsome f p k 2.Let Thm : 3 : 4 p = Thm : 3 : 4 f;p;k;;d 0 ; F > 0be theconstantguaranteedbyTheorem3.4,andlet Thm : 3 : 4 =min p Thm : 3 : 4 p where theminimumistakenoverallintegers k p f forwhich F is p -partite.Let =min f = 3 ; Thm : 3 : 4 g .Let T 0 = T Thm : 1 : 3 t 0 ; betheintegerguaranteedbyThe33

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orem1.3.Weset = d f k 0 = T f 0 andtake n 0 = n 0 f;k;;d 0 ; F ; sucientlylarge wheneverneeded. Let k -uniformhypergraph H on n n 0 verticesbegiven,where H containsfewer than n f copiesofthexedhypergraph F .Weshowthatintime O n 2 k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 log 2 n wemaylocatefewerthan n k edgesin H whoseremovalmakes HF -free.Tothat end,withconstants > 0and t 0 above,applyTheorem1.3constructiveregularity lemmato H toobtain,intime O n 2 k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 log 2 n ,an -regularand t -equitablepartition V H = V 1 [[ V t t 0 t T 0 ,whoseevery -irregular k -tuple V i 1 ;:::;V i k is identied. Wenowdeletethefollowingedges H 2H : 1.all H 2H forwhich H V 0 6 = ; or j H V i j 2forsome1 i t ; 2.all H 2H forwhichthereexist1 i 1 <
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O b n=t c k .Finally,theedges H 2H satisfyingtotalatmost t k b n=t c k n k 3 n k andcanbeidentiedintime O t k n=t k = O n k sinceTheorem1.3alreadyidentied all -irregular k -tuples V i 1 ;:::;V i k ,1 i 1 < 1 2 d f k 0 n T 0 f = n f ; whichcontradictsourhypothesisthat jF H j n f 35

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References [1]V.E.Alekseev, Ontheentropyvaluesofhereditaryclassesofgraphs ,Discrete MathematicsandApplications 3 ,191{199. [2]N.Alon,R.A.Duke,H.Lefmann,V.Rodl,R.Yuster, Thealgorithmicaspects oftheregularitylemma ,J.Algorithms 16 ,no.1,80{109. [3]B.Bollobas,A.Thomason, Projectionsofbodiesandhereditarypropertiesof hypergraphs ,Bull.London.Math.Soc. 27 ,417{424. [4]B.Bollobas,A.Thomason, Hereditaryandmonotonepropertiesofgraphs ,The MathematicsofPaulErd}os,II,Springer,Berlin997,70{78. [5]F.Chung. Regularitylemmasforhypergraphsandquasirandomness .Random StructuresandAlgorithms,2:241{252,1991. [6]A.Czygrinow,V.Rodl, Analgorithmicregularitylemmaforhypergraphs ,SIAM J.Comput. 30 ,no.3{4,293{332. [7]R.Duke,H.Lefmann,V.Rodl, Afastapproximationalgorithmforcomputing thefrequenciesofsubgraphsofagivengraph ,SIAMJournalofComputingVol. 24,pp.598{620,1995 [8]P.Erd}os,P.Frankl,V.Rodl, Asymptoticnumberofgraphsnotcontainingaxed subgraphandaproblemforhypergraphshavingnoexponent ,GraphsComb. 2 ,113{121. 36

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[9]P.Erd}os,D.J.Kleitman,B.L.Rothschild, Asymptoticenumerationof K n -free graphs ,ColloquioInternazionalesulleTeorieCombinatorieRome,1973,Tomo II,pp.19{27.AttideiConvergniLincei,No.17,Accad.Naz.Lincei,Rome,1976. [10]P.Frankl,V.Rodl, Extremalproblemsonsetsystems ,RandomStructuresAlgorithms 20 ,no.2,131{164. [11]P.Frankl,V.Rodl. Theuniformitylemmaforhypergraphs .GraphsandCombinatorics,8:309{312,1992. [12]A.Frieze,R.Kannan. Theregularitylemmaandapproximationschemesfordense problems .InProceedingsoftheSymposiumonFoundationsofComputerScience, pp.12{20.IEEE,1996. [13]W.T.Gowers, Quasirandomness,countingandregularityfor3-uniformhypergraphs ,Combin.Probab.Comput. 15 ,no.1{2,143{184. [14]W.T.Gowers, HypergraphregularityandthemultidimensionalSzemerediTheorem ,Ann.of.Math. 166 ,no.3{897{946. [15]Y.Kohayakawa,B.Nagle,V.Rodl, Hereditarypropertiesoftriplesystems ,Combin.Probab.Comput. 12 ,155{189. [16]Y.Kohayakawa,B.Nagle,V.Rodl,M.Schacht, Weakregularityandlinearhypergraphs ,JournalofCombinatorialTheoryB.toappear [17]J.Komlos,A.Shokoufandeh,M.Simonovits,E.Szemeredi, Theregularitylemma anditsapplicationsingraphtheory ,Theoreticalaspectsofcomputerscience Tehran,2000,LectureNotesinComput.Sci.,vol.2292,Springer{Berlin,2002, pp.84{112. [18]J.Komlos,M.Simonovits, Szemeredi'sRegularityLemmaanditsapplicationsin graphtheory ,Combinatorics,PaulErd}osiseighty,Vol.2,pp.295-352,1996. [19]B.Nagle,V.Rodl, Theasymptoticnumberoftriplesystemsnotcontaininga xedone ,Disc.Math. 235 ,271{290. 37

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[20]B.Nagle,V.Rodl,M.Schacht, Thecountinglemmaforregular k -uniformhypergraphs ,RandomStructuresAlgorithms 28 ,no.2,113{179. [21]B.Nagle,V.Rodl,M.Schacht, Extremalhypergraphproblemsandtheregularity method ,Topicsindiscretemathematics,AlgorithmsCombin.,vol.26,Springer, Berin,2006,pp.247{278. [22]H.J.Promel,A.Steger, Excludinginducedsubgraphs:quadrilaterals ,Random StructuresAlgorithms 2 ,55{71. [23]H.J.Promel,A.Steger, Theasymptoticnumberofgraphsnotcontainingaxed color-criticalsubgraph ,Combinatorica 12 ,463{473. [24]H.J.Promel,A.Steger, ExcludinginducedsubgraphsIII.Ageneralasymptotic RandomStructuresAlgorithms 3 ,no.1,19{31. [25]H.J.Promel,A.Steger, Theasymptoticstructureof H -freegraphs ,GraphstructuretheorySeattle,WA,1991,Contemp.Math.,vol.147,Amer.Math.Soc., Providence,RI,1993,pp.167{178. [26]V.Rodl,B.Nagle,J.Skokan,M.Schacht,Y.Kohayakawa, Thehypergraphregularitymethodanditsapplications ,Proc.Natl.Acad.Sci.USA 102 ,no.23, 8109{8113. [27]V.Rodl,J.Skokan, Regularitylemmafor k -uniformhypergraphs ,RandomStructuresAlgorithms 25 ,no.1,1{42. [28]E.Szemeredi, Onsetsofintegerscontainingno k elementsinarithmeticprogression ,ActaArithmetica 27 ,199{245,Collectionofarticlesinmemoryof JuriVladimirovicLinnik. [29]E.Szemeredi, Regularpartitionsofgraphs ,Problemesencombinatoiresettheorie desgraphes,Colloq.Internat.CNRS,Univ.Orsay,Orsay,1976,Colloq.Internat.CNRS,vol.260,CNRS,Paris,1978,pp.399{401. 38

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[30]T.Tao, Avariantofthehypergraphremovallemma ,J.Combin.TheorySer.A 113 ,no.7,1257{1280. 39


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A hypergraph regularity method for linear hypergraphs
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ABSTRACT: Szemerdi's Regularity Lemma is powerful tool in Graph Theory, yielding many applications in areas such as Extremal Graph Theory, Combinatorial Number Theory and Theoretical Computer Science. Strong hypergraph extensions of graph regularity techniques were recently given by Nagle, Rdl, Schacht and Skokan, by W.T. Gowers, and subsequently, by T. Tao. These extensions have yielded quite a few non-trivial applications to Extremal Hypergraph Theory, Combinatorial Number Theory and Theoretical Computer Science. A main drawback to the hypergraph regularity techniques above is that they are highly technical. In this thesis, we consider a less technical version of hypergraph regularity which more directly generalizes Szemeredi's regularity lemma for graphs. The tools we discuss won't yield all applications of their stronger relatives, but yield still several applications in extremal hypergraph theory (for so-called linear or simple hypergraphs), including algorithmic ones. This thesis surveys these lighter regularity techiques, and develops three applications of them.
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