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Further applications of the Container Method Jo´zsefBaloghandAdamZs.Wagner 6 1 0 2 n a J 8 2 AbstractRecently,Balogh–Morris–SamotijandSaxton–Thomasonprovedthathy- pergraphssatisfyingsomenaturalconditionshaveonlyfewindependentsets.Their ] O main results already have several applications. However, the methods of proving C these theorems are even more far reaching. The general idea is to describe some . familyofevents,whosecardinalityaprioricouldbelarge,onlywithafewcertifi- h cates.Here,weshowsomeapplicationsofthemethods,includingcountingC -free t 4 a graphs, considering the size of a maximumC -free subgraph of a random graph 4 m andcountingmetricspaceswithagivennumberofpoints.Additionally,wediscuss [ someconnectionswiththeSzemere´diRegularityLemma. 1 v 9 1 Introduction 0 8 7 Recently, an importanttrend in probabilistic combinatoricsis to transfer extremal 0 results in a dense environmentinto a sparse, random environment.The first main . 1 breakthrough is due to Conlon–Gowers [8] and Schacht [23]. Not much later, 0 Balogh–Morris–Samotij[2] and Saxton–Thomason[22] had a differentapproach, 6 1 whichnotonlyprovedmostoftheresultsofConlon–Gowers[8]andSchacht[23], : but also provided counting versions of these results. Additionally, there are some v furtherapplicationsofthemaintheoremsof[2]and[22].However,theproofmeth- i X r a Jo´zsefBalogh DepartmentofMathematicalSciences,UniversityofIllinoisatUrbana-Champaign,Urbana,Illi- nois61801,USAe-mail:[email protected]. ResearchispartiallysupportedbySimons Fellowship, NSA Grant H98230-15-1-0002, NSF CAREER Grant DMS-0745185, NSF Grant DMS-1500121,ArnoldO.BeckmanResearchAward(UIUCCampusResearchBoard15006)and MarieCurieFP7-PEOPLE-2012-IIF327763. AdamZs.Wagner DepartmentofMathematicalSciences,UniversityofIllinoisatUrbana-Champaign,Urbana,Illi- nois61801,USAe-mail:[email protected]. 1 2 Jo´zsefBaloghandAdamZs.Wagner odshavethepotentialtobemoreinfluentialthanthetheoremsthemselves.Thegen- eralidea,whichcanbetracedbacktotheclassicalpaperofKleitman–Winston[13] (andmoreexplicitlyinseveralpapersofSapozhenko)istodescribesomefamilyof events,whosecardinalityaprioricouldbelarge,onlywithafewcertificates. Here,wetrytooutlinewhatliesbehindthismethod.Becauseeachoftheabove mentionedfourpapers[8, 23, 2, 22] already gavea nice surveyof the field, addi- tionallyConlon[7],Ro¨dl–Schacht[20]andSamotij[21]havesurveypapersofthe topics, we choose a different route. Each of the following four sections discusses themethodwithsomeapplications.Threeofthesectionscontainnewresults. InSection2 we state an importantcorollaryof themainresultsof [2]and [22] andwilldiscussitsconnectionwiththeSzemere´diRegularityLemma[24]. (n) In Section 3 we estimate the volumeof the convexsubsetof [0,1] 2 represen- tingmetricspaceswithnpoints,andconsideradiscretevariantofthisproblemas well. Kozma–Meyerovitch–Peled–Samotij[15] consideredthe followingquestion: chooserandomlyandindependently n numbersfromtheinterval[0,1],labelling 2 the edges of a complete graph K , what is the probability that any three of them n (cid:0) (cid:1) formingatrianglewillsatisfythetriangleinequality?Theyusedentropyestimates toderiveanupperboundonthisprobabilityandnoticedthatSzemere´diRegularity Lemmacanbeusedtocountmetricspaceswhosealldistancesbelongtoadiscrete setofafixedsize. Mubayi and Terry [19] more recently pushed the discrete case into 0,1 -law { } typeresults.Usingthegeneralresultsof[2]and[22],weimprovetheresultsimplied bytheregularitylemma,andobtaingoodresultsforthecontinuouscase.Parallelto ourwork,Kozma,Meyerovitch,Morris,Peled andSamotij[16] practicallysolved thecontinuousversionoftheproblemusingmoreadvancedversionsofthemethods of[2]and[22].Inordertoavoidduplicatework,ourgoalinthissectionisclarity overpushingthemethodtoitslimit. It was probably Babai–Simonovits–Spencer [1] who first considered extremal problemsinrandomgraphs.Mostoftheirproposedproblemsareresolved,butthere wasnoprogressonthefollowing:Whatisthemaximumnumberoftheedgesofa C -freesubgraphoftherandomgraphG(n,p)when p=1/2?Inthecase p=o(1), 4 strongboundsweregivenbyKohayakawa–Kreuter–Steger[14].Here,inSection4, we improve on the trivial upper bound, noting that an n-vertexC -free graph can 4 haveatmost(1/2+o(1))n3/2edges. Theorem1.1 For every p (0,1), there is a c>0 that the largestC -free 4 ∈ subgraph of G(n,p) has at most (1/2 c)n3/2 edges w.h.p. In particular, if − p=0.5wecantakec=0.028. Kleitman–Winston[13]authoredoneofthefirstpapersinthefieldwhosemain ideawastofindsmallcertificatesoffamiliesofsetsinordertoprovethatthereare notmanyofthem.TheyprovedthatthenumberofC -freegraphsonnverticesisat 4 most2cn3/2 forc 1.081919.InSection5,weimprovetheconstantintheexponent ≈ FurtherapplicationsoftheContainerMethod 3 by using ideas found in Section 3. The improvement is tiny; the main aim is to demonstratethatthemethodhaspotentialapplicationsforotherproblemsaswell. 2 Connections to Szemere´di’s RegularityLemma In this section we describe some connections between the Szemere´di Regularity Lemma and the new counting method. Originally, without using the regularity lemma,Erdo˝s–Kleitman–Rothschild[10]estimatedthenumberofK -freegraphs. k Theorem2.1 Thereare2(1+o(1))·ex(n,Kk) Kk-freegraphsonnvertices,where ex(n,K )isthemaximumnumberoftheedgesofaK -freegraphonnvertices. k k Here, we sketch a well-knownproofof Erdo˝s–Frankl–Ro¨dl[9], using the Sze- mere´diRegularityLemma.WeapplytheRegularityLemmaforaK -freegraphG , k n which outputs a cluster graph R, wheret =O(1). Then we clean G , i.e., we re- t n move edges inside the clusters, between the sparse and the irregularpairs. Define C to be the blow-up of R to n vertices(getting back the same preimage vertices n t ofG ).ObservethatC containsallbuto(n2)edgesofG andC isK -free,hence n n n n k e(C ) ex(n,K ).NowwecancountthenumberofchoicesforG :Thenumberof n k n ≤ choices forC is O(1) nn, the numberof choices for E(G ) E(C ) givenC , is n n n n · ∩ atmost2ex(n,Kk),andthenumberofchoicesforE(Gn) E(Cn)givenCn,is2o(n2), − completingtheproofoftheresult. An important corollary of the main results of Balogh–Morris–Samotij [2] and Saxton–Thomason[22]isthefollowingcharacterizatonofK -freegraphs. k Proposition2.2 There is a t 2O(logn·n2−1/(k−1)) and a set G1,...,Gt of ≤ { } graphs,eachcontainingo(nk)copiesofK ,suchthatforeveryK -freegraph k k H thereisani [t]suchthatH G. i ∈ ⊆ Notethatitfollowsfromstandardresultsinextremalgraphtheorythateachsuch graphG necessarilycontainsatmost(1+o(1)) ex(n,K )edges. i k · Thisprovidesan evenshorterproofofTheorem2.1:Foreach K -freegraphH k thereisani [t]suchthatH Gi.Thenumberofchoicesforiis2O(logn·n2−1/(k−1)), andthenum∈berofsubgraphs⊆ofGiisatmost2(1+o(1))ex(n,Kk). TheproofusingtheRegularityLemmayieldstwovariantsofProposition2.2,one whichisweaker,asitgivest=2o(n2),thoughitisstrongenoughforthisparticular application,andonethatwecalltheSzemere´diApproximateContainerLemma. 4 Jo´zsefBaloghandAdamZs.Wagner Proposition2.3 Thereisat =2o(n2) andaset G ,...,G ofgraphs,each 1 t containing at most (1+o(1))ex(n,K ) edges, a{nd o(nk) co}pies of K , such k k thatforeveryK -freegraphH thereisani [t]suchthatH G. k i ∈ ⊆ Proposition2.4 There isat =O(1)anda set G ,...,G ofgraphs,each 1 t { } K -free,suchthatforeveryK -free graphH there isani [t]andapermu- k k ∈ tationofV(H),givinganisomorphicgraphH ,suchthat E(H ) E(G) = ′ ′ i | − | o(n2). Most of the tools fromthis section are likely to be usefulin countingmaximal K -freegraphs,wheretherearestillnosatisfactoryboundswhenr 4. r ≥ Problem2.5 WhatisthenumberofmaximalK -freegraphswithvertexset r [n]?Forr=3thiswasaquestionofErdo˝s,whichwassettledin[4]and[3]. 3 The number ofmetricspaces Ourgoalistoestimatethenumberofmetricspacesonnpoints,wherethedistance betweenanytwopointsliesin 1,...,r forsomer=r(n).Thisproblemwascon- { } sideredfirstbyKozma–Meyerovitch–Peled–Samotij[15],who,usingtheregularity lemmagaveanasymptoticboundonthenumberofsuchmetricspacesfora fixed constantr. Recently,Mubayi–Terry[19] provideda characterisationofthe typical structureofsuchmetricspacesforafixedconstantr,whilen ¥ .Wewillbemore → interestedinwhathappensifrisallowedtogrowasafunctionofn.Ourmainresult isthefollowing: Theorem3.1 Fixanarbitrarysmallconstante >0.If n1/3 r=O , log43+e n! thenthenumberofsuchmetricspacesis FurtherapplicationsoftheContainerMethod 5 r+1 (n2)+o(n2) . 2 (cid:24) (cid:25) Kozma–Meyerovitch–Peled–Samotij[15] pointed out that the discrete and the continuousproblemsarerelated.Theyconsideredthesamequestioninthecontinu- ouscasewithdistancesin[0,1].TheirentropybasedapproachyieldsTheorem3.1 forr<n1/8. At the end of this section we show how our results translate to the continuous setting. For a positive integer r define m(r)= r+1 . We will use an easy corollary of ⌈ 2 ⌉ Mubayi–Terry([19]Lemma4.9): Lemma3.2 LetA,B,C [r], allnon-empty.Supposethetriple A,B,C doesnot ⊂ { } contain a non-metric triangle – that is, every triple a,b,c:a A,b B,c C { ∈ ∈ ∈ } satisfies the triangle-inequality.Then if r is evenwe have A + B + C 3m(r), | | | | | |≤ andifrisoddwehave A + B + C 3m(r)+1. | | | | | |≤ LetH bethe3-uniformhypergraphwithvertexsetr rows,oneforeachcolor, and n columns, one for each edge of K . A vertex (i,f) of H corresponds to 2 n theeventthatthegraphedge f hascolori.ThreeverticesofH formahyperedge (cid:0) (cid:1) when the graph edge coordinates of the vertices form a triangle in K while the n ‘colors’ do not satisfy the triangle inequality. With other words, the hyperedges correspondtonon-metrictriangles,andindependentsetshavingexactlyonevertex fromeachcolumncorrespondtopointsofthemetricpolytope.Ourplanistoprove asupersaturationstatement,butfirstweneedtwolemmas. The first lemma we use is due to Fu¨redi [11]. For a graph G, write G2 for the “propersquare”ofG,i.e.,wherexyisanedgeifandonlyifthereisazsuchthatxz andzyareedgesinG.Writee(G)forthenumberofedgesinG. Lemma3.3 ForanygraphGwithnvertices,wehave e(G2) e(G) n/2 . ≥ −⌊ ⌋ ThesecondlemmaboundsthesizeofthelargestindependentsetinH. Lemma3.4 Let S V(H) have no empty columnsand containno edges in H. Thenifrisevenwe⊂have S m(r) n ,andifrisoddwehave S m(r) n +rn. | |≤ 2 | |≤ 2 Proof. Theevencasefollowsdirectl(cid:0)y(cid:1)fromLemma3.2,andwenotethatth(cid:0)is(cid:1)bound istight–letScontaintheinterval[r/2,r]fromeachcolumn.Theboundintheodd case is slightly more difficult, and we make no effort to establish a tight bound, whichshouldprobablybe S m(r) n +n/2. | |≤ 2 Letrbeodd,andletA,B,CbethreecolumnsthatformatriangleofS.Notethatif (cid:0) (cid:1) forsomek 1wehave A m(r)+kand B m(r)+kthen C m(r) 2k+1 ≥ | |≥ | |≥ | |≤ − ≤ m(r) kbyLemma3.2.WriteB forthesetofcolumnsinSoforderatleastm(r)+k k − 6 Jo´zsefBaloghandAdamZs.Wagner andwriteS forthesetofcolumnsinSoforderatmostm(r) k.LetG bethegraph k k − on[n]withedgesB .ThenbyLemma3.3weget k S e(G2) e(G ) n/2 B n. | k|≥ k ≥ k −⌊ ⌋≥| k|− Hence n (cid:229) r (cid:229) r (cid:229) r S m(r) = k(B B ) k(S S )= (B S ) nr, | |− 2 | k|−| k+1| − | k|−| k+1| | k|−| k| ≤ (cid:18) (cid:19) k=1 k=1 k=1 andtheresultfollows. ⊓⊔ Nowwearereadytoproveasupersaturation-likeresult. Lemma3.5 Lete >0andletS V(H)withnoemptycolumns. 1. Ifrisevenand S (1+e )⊂n m(r),thenScontainsatleast e n hyperedges. | |≥ 2 10 3 2. Ifr is odd,n>ne sufficientlylargeand S (1+e ) n m(r), thenScontains (cid:0) (cid:1) | |≥ 2 (cid:0) (cid:1) atleast e4 n hyperedges. 40000 3 (cid:0) (cid:1) Proof. Supposefi(cid:0)rs(cid:1)tthatriseven.Thenthereareatleast e n trianglesinGsuch 10 3 that the correspondingcolumns contain at least (1+e /10)3m(r) vertices from S. (cid:0) (cid:1) Indeed,ifthiswasnotthecase,then e n 3r+ n 3m(r)(1+e /10) n e (r+m(r)) 10 3 3 = m(r)+ > S, n 2 2 10 | | (cid:0) (cid:1) (cid:0) (cid:1)− (cid:18) (cid:19)(cid:18) (cid:19) whichisacontradiction.Hencepart1ofthelemmafollowsfromLemma3.2. Now suppose r is odd. Given T [n], write f (T) for the set of vertices of S S ⊂ containedinthe T columnsofH correspondingtotheedgesspannedbyT.Set |2| n =20/e , so thatby Lemma3.4, wheneverT [n] with T =n and f (T) 0 0 S m(r) n0 (1+e )(cid:0)then(cid:1)H[f (T)]containsahyper⊂edge. | | | |≥ 2 3 S First,weclaimthatthereareatleast e n choicesofT [n]with T =n and f (T(cid:0)) (cid:1) m(r) n0 (1+e ). 4 n0 ⊂ | | 0 | S |≥ 2 3 (cid:0) (cid:1) Indeed,ifthiswasnotthecase,thenwewouldhave (cid:0) (cid:1) e n n0 r+ n m(r) n0 (1+e ) n e r e m(r) S 4 n0 2 n0 2 3 = m(r)+ + | |≤ (cid:0) (cid:1)(cid:0) (cid:1) (cid:0)nn0−(cid:1)−22 (cid:0) (cid:1) (cid:18)2(cid:19)(cid:18) 4 3 (cid:19) (1) < n m(r)(1+(cid:0)e ), (cid:1) 2 (cid:18) (cid:19) whichisnotpossible.SothenumberofhyperedgescontainedinSisatleast e n n 3 e e 3 n e(H[S]) / − , ≥ 4 n n 3 ≥ 4203 3 (cid:18) 0(cid:19) (cid:18) 0− (cid:19) (cid:18) (cid:19) andtheresultfollows. ⊓⊔ FurtherapplicationsoftheContainerMethod 7 Write d¯fortheaveragedegreeof H, andfor j [3] definethe j-th maximum ∈ co-degree D =max e E(H):s e :s V(H)and s = j . j {|{ ∈ ⊂ }| ⊂ | | } Below,wewillmakeuseofaversionofthecontainertheoremof[2,22],thewayit wasformulatedbyMousset–Nenadov–Steger[18]. Theorem3.6 There exists a positive integerc such thatthe following holds for every positive integer N. Let H be a 3-uniform hypergraphof order N. Let0 p 1/(36c)and0<a <1besuchthatD (H,p) a /(27c),where ≤ ≤ ≤ 4D 2D D (H ,p)= 2 + 3. d¯p d¯p2 ThenthereexistsacollectionofcontainersC P(V(H))suchthat (i)everyindependentsetinH iscontained⊂insomeC C, (ii)forallC C wehavee(H[C]) a e(H ),and ∈ ∈ ≤ (iii)thenumberofcontainerssatisfies log C 39c(1+log(1/a ))Nplog(1/p). | |≤ Proof of Theorem 3.1. Let H be the hypergraph defined earlier, i.e., the 3- uniform hypergraph with vertex set formed by pairs of the r colors and the n 2 edgesofK ,with3-edgescorrespondingtonon-metrictriangles.Lete ,d >0bear- n bitrarilysmallconstantsandset p= 1 anda = 1010clog4+2d n.InH weha(cid:0)v(cid:1)e rlog2+d n n D nr2, D r, D =1, d¯ r2n/64and 1 2 3 ≤ ≤ ≥ 64r2log2+d n 64r2log4+2d n a D (H,p) 4 + . ≤ r2n 2r2n !≤ 27c ThenTheorem3.6providescontainerswith e(H[C]) a e(H) 104cr3n2log4+2d n, ≤ ≤ andthenumberofcontainersis c310rn2 logn logr loglogn log C · · · =o(n2). | |≤ rlog2+d n Nowassume n1/3 r=o . log(4+2d )/3n! 8 Jo´zsefBaloghandAdamZs.Wagner Thenthemaximumnumberofedgesinacontaineriso(n3),hencebyLemma3.5, andthefactthatausefulcontainerdoesnothaveanemptycolumn,wehaveforn largeenough, n V(C) <(1+e )m(r) . | | 2 (cid:18) (cid:19) Hence, the number of colourings in a container is at most (1+e )(n2)m(r)(n2) = m(r)(n2)+o(n2).Thelogarithmofthenumberofcontainersiso(n2).Thetotalnumber ofgoodcolouringsisatmostthenumberofcontainerstimesthemaximumnumber ofcolouringsinacontainer.Hencethetotalnumberofgoodcolouringsis m(r)(n2)+o(n2), asrequired. ⊓⊔ Now we turn our attention to the continuous setting. The set-up in [15] is as follows. Given a metric space with n points and all distances being in [0,1], we (n) regardthesetofdistancesasavectorin[0,1] 2 .Wewillcalltheunionofallsuchn (n) pointsin[0,1] 2 forallfinitemetricspacesthemetricpolytopeMn.Moreprecisely, the metric polytopeMn is the convexpolytopein R(n2) defined by the inequalities 0<d 1andd d +d . ij ij ik jk ≤ ≤ Notethatifa+b cthen ≥ a + b c . (2) ⌈ ⌉ ⌈ ⌉≥⌈ ⌉ Theorem3.7 Fixd >0constant.Thenforn>nd sufficientlylarge,wehave (vol(Mn))1/(n2)≤ 21+n611−d . Proof. First consider the discrete setting, colouring with r colours, where r is the evenintegerclosestton16−d2.W.l.o.g.d <1/4andset 1/p=n31−d4, a =300cnd −32, wherecistheconstantfromTheorem3.6.Then 1 1 n1/3 d /4 n2/3 d /2 D (H,p)<300 + 300 − + − a , (cid:18)rnp p2r2n(cid:19)≤ n7/6−d /2 n4/3−d !≤ andwegetcontainerswith e(H[C]) a n3r3 n3 1/6 d /4, − − ≤ ≤ FurtherapplicationsoftheContainerMethod 9 wherethenumberofcontainerssatisfies log C n2rplog3n n2 1/6 d /5. − − | |≤ ≤ Hence,byLemma3.5,thenumberofverticesinacontainerisatmost n V(C) (1+n 1/6 d /6) m(r). − − | |≤ 2 (cid:18) (cid:19) Thisimpliesthatthenumberofcolouringscontainedinacontainerisatmost (n) col(C)≤ V(n2C)! 2 ≤ (1+n−1/6−d /6)m(r) (n2)≤m(r)(n2)en2−1/6−d/7. (cid:16) (cid:17) (cid:0) (cid:1) Thatis,thetotalnumberX ofcolouringsisatmost X m(r)(n2)en2−1/6−d/7en2−1/6−d/5 m(r)(n2)en2−1/6−d/8. ≤ ≤ Now consider colouringsin the continuoussetting. Cut up each edge of the cube intorpieces.Wegetby(2)that (vol(Mn))1/(n2)≤ (m(r)+1)r((n2n2))en2−1/6−d/8!1/(n2) (n) 1/(n2) (n) 4 2 1 1 1 ≤ 2− 2 (cid:18)1+r(cid:19) ! (cid:18)1+n1/6+d /9(cid:19)≤ 2+n61−d , asrequired. ⊓⊔ Remark.Kozma,Meyerovitch,Morris,PeledandSamotij[16]usingastonger supersaturation result and a somewhat different container type of theorem, in- dependently, parallel to our work, improved the error term in Theorem 3.7 to (log2n)n 1/2,wherethe 1/2isbestpossibleasitwaspointedoutin[15]. − − It would be interesting to extend the above ideas to generalised metric spaces (note that there are several different definitions of these), hence we propose the followingpurposelyvaguequestion: Problem3.8 Itisanaturalquestiontoask:isthereaninterestingextension of the problem discussed in this section, when instead of requiring metric triangles,onewantsmetricd-dimensionalsimplicesforafixedd? 10 Jo´zsefBaloghandAdamZs.Wagner 4 The largestC -free subgraph ofarandom graph 4 NowweturnourattentiontoprovingTheorem1.1.Asbefore,foragraphGwrite G2 forthe“propersquare”ofG,i.e.,wherexyisanedgeifandonlyifthereisaz suchthatxzandzyareedgesinG. To simplify the technicality of the proof, we present here only the case when p=1/2;forlarger p we can use a properlychosensmaller c and the argumentis thesame. AssumethatH isthelargeC -freegraphthatweaimtofindinG(n,1/2).Anat- 4 uralapproachwouldbetodowhatFu¨redi[12]did,workingonaRamseytypeprob- lem.UsingtheKleitman–Winstonmethod,hegaveanupperboundonthenumber ofC -freegraphswithmedges.If pissufficientlysmall,thentheexpectednumber 4 of copies of such graphs in G(n,p) is o(1), proving the desired result. His upper boundwas, assumingthat m>2n4/3(log2n), that the numberof such graphsis at most(4n3/m2)m,hencetheexpectednumberofthemis(4pn3/m2)m=o(1),aslong asm>(1/2 a)n3/2and p<1/16 a,wherea>0issomesmallconstant. − − Theideaofourproofisthattheunionboundintheaboveargumentistoowaste- ful;hence,insteadofconsideringeachH separately,wewanttoshowthatforevery C -freesubgraphHwithmanyedges,thereexistssomecertificateinthegraph.Each 4 of these certificates will say that in a certain part of the graphone needs to select (1 o(1))d n3/2 edges of G(n,p) out of possible (1+o(1))d n3/2 pairs. Since the − numberof certificatesis muchsmaller than 1/(probabilitythatthe aboveunlikely eventholds),whichisoforder2cn3/2,wecannowshowusingtheunionboundthat w.h.p.thereisnosuchsubgraph. Soourfirsttaskistobuildsuchacertificate.FixaC -freegraphH withatleast 4 m>(1/2 c)n3/2 edges, where for the p=1/2 case we can set c=10 5. First − fixalinear−orderp ofthevertexsetofH,whichwewilluseasatiebreakeramong vertices.Wewillalsoneedasecondlinearordering,asfollows: Definition4.1 GivenagraphG,amin-degreeorderingisanordering v ,...,v 1 n { } ofV(G)suchthatforeachi [n],thevertexv isofminimumdegreeinG[v,...,v ]. i i n ∈ Whentherearemultiplesuchvertices,thenweletv bethefirstamongtheminthe i orderingp . Fix a min-degree ordering of H and let Y := v ,...,v and X =V(H) Y, 1 s { } − where we will choose s such that the minimum degree of H[X] is large. Now we fix F H[X] with the following properties. The first is that F is sparse, i.e., e(F) n3⊂/2/log2n.ThesecondisthattheindependentsetsinF2 approximatethe ≤ independentsets in H2[X] rather well. In particular, we choose F such that every largeindependentsetofH2[X]isina‘container’determinedbyF,wherethenum- ber of containers is at most r:=2n2/5log20n. The key observation is that for every v Y itsneighborhoodinX spansanindependentsetinH2[X],asH isC -free. j 4 ∈The certificate for H will be the vector [Y,F, d s , r s ], where d := { j}j=1 { j}j=1 j N(v ) X , and r r is the index of the container containing N(v ) X, and j j j | ∩ | ≤ ∩ s= Y .Thenumberofcertificatesisatmost2n n2/2 nn rn 22n3/2/logn. | | · n3/2/log2n · · ≤ (cid:0) (cid:1)