Journal of Combinatorial Theory, Series A

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1 Joural of Combiatorial Theory, Series A Cotets lists available at ScieceDirect Joural of Combiatorial Theory, Series A Geeratig all subsets of a fiite set with disjoit uios David Ellis a, Bey Sudaov b,1 a St Joh s College, Cambridge, CB 1TP, Uited Kigdom b Departmet of Mathematics, UCLA, Los Ageles, CA 90095, Uited States article ifo abstract Article history: Received 3 October 010 Available olie xxxx Keywords: Geerator Disjoit uios Extremal set theory If X is a -elemet set, we call a family G P X a -geerator for X if every x X ca be expressed as a uio of at most disjoit sets i G. Frei, Lévêque ad Sebő cojectured that for >, the smallest -geerators for X are obtaied by taig a partitio of X ito classes of sizes as equal as possible, ad taig the uio of the power-sets of the classes. We prove this cojecture for all sufficietly large whe =, ad for a sufficietly large multiple of whe Elsevier Ic. All rights reserved. 1. Itroductio Let X be a -elemet set, ad let P X deote the set of all subsets of X. WecallafamilyG P X a -geerator for X if every x X cabeexpressedasauioofatmost disjoit sets i G. For example, let V i be a partitio of X ito classes of sizes as equal as possible; the F, := PV i \{ } is a -geerator for X. Wecalla-geerator of this form caoical. If = q + r, where 0 r <, the F, = r q 1 + r q+1 1 = + r q. Frei, Lévêque ad Sebő [8] cojectured that for ay, this is the smallest possible size of a -geerator for X. addresses: dce7@cam.ac.u D. Ellis, bsudaov@math.ucla.edu B. Sudaov. 1 The research of Bey Sudaov was supported i part by NSF CAREER award DMS ad by a USA-Israeli BSF grat /$ see frot matter 011 Elsevier Ic. All rights reserved. doi: /j.jcta

2 30 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A Cojecture 1 Frei, Lévêque, Sebő. If X is a -elemet set,, ad G P Xisa-geeratorforX,the G F,.If>, equality holds oly if G is a caoical -geerator for X. They proved this for 3, but their methods do ot seem to wor for larger. For =, Cojecture 1 is a weaeig of a cojecture of Erdős. We call a family G P X a -base for X if every x X cabeexpressedasauioofatmost ot ecessarily disjoit sets i G. Erdős see [9] made the followig Cojecture Erdős. If X is a -elemet set, ad G P Xisa-base for X, the G F,. I fact, Frei, Lévêque ad Sebő [8] made the aalogous cojecture for all. Cojecture 3 Frei, Lévêque, Sebő. If X is a -elemet set,, ad G P Xisa-baseforX,the G F,.If>, equality holds oly if G is a caoical -geerator for X. Agai, they were able to prove this for 3. I this paper, we study -geerators whe is large compared to. Our mai results are as follows. Theorem 4. If is sufficietly large, X is a -elemet set, ad G P X is a -geerator for X, the G F,. Equality holds oly if G is of the form F,. Theorem 5. If N, is a sufficietly large multiple of, X is a -elemet set, ad G is a -geerator for X, the G F,. Equality holds oly if G is of the form F,. I other words, we prove Cojecture 1 for all sufficietly large whe =, ad for asufficietly large multiple of whe 3. We use some ideas of Alo ad Fral [1], ad also techiques of the first author from [5], i which asymptotic results were obtaied. As oted i [8], if G P X is a -geerator or eve a -base for X, the the umber of ways of choosig at most sets from G is clearly at least the umber of subsets of X. Therefore G, which immediately gives G /. Moreover, if G =m, the m. i i=0 Crudely, we have 1 m m 1, i i=0 so m i i=0 m + m 1. Hece, if is fixed, the m 1 + O1/m, 1

3 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A so G! 1/ / 1 o1. Observe that if = q + r, where 0 r <, the where F, = + r q < + r q = / 1 + r/ r/ < c 0 /, 3 c 0 := 1/ = log to 3 d.p.. log Now for some prelimiaries, we use the followig stadard otatio. For N, [] will deote the set {1,,...,}. Ifx ad y are disjoit sets, we will sometimes write their uio as x y, rather tha x y, to emphasize the fact that the sets are disjoit. If N, ad G is a graph, K G will deote the umber of -cliques i G. Let T s deote the s-partite Turá graph the complete s-partite graph o vertices with parts of sizes as equal as possible, ad let t s = et s. Forl N, C l will deote the cycle of legth l. If F is a labelled graph o f vertices, with vertex-set {v 1,...,v f } say, ad t = t 1,...,t f N f, we defie the t-blow-up of F, F t, to be the graph obtaied by replacig v i with a idepedet set V i of size t i, ad joiig each vertex of V i to each vertex of V j wheever v i v j is a edge of F. With slight abuse of otatio, we will write F t for the symmetric blow-up F t,...,t. If F ad G are graphs, we write c F G for the umber of ijective graph homomorphisms from F to G, meaig ijectios from V F to V G which tae edges of F to edges of G. Thedesity of F i G is defied to be d F G = c F G G G 1 G F +1, i.e. the probability that a uiform radom ijective map from V F to V G is a graph homomorphism from F to G. Hece, whe F = K, the desity of K s i a -vertex graph G is simply K G/. Although we will be iterested i the desity d F G, it will sometimes be more coveiet to wor with the followig closely related quatity, which behaves very icely whe we tae blow-ups. We write Hom F G for the umber of homomorphisms from F to G, ad we defie the homomorphism desity of F i G to be h F G = Hom F G G F, i.e. the probability that a uiform radom map from V F to V G is a graph homomorphism from F to G. Observe that if F is a graph o f vertices, ad G is a graph o vertices, the the umber of homomorphisms from F to G which are ot ijectios is clearly at most f f 1. Hece, d G F h GF f f f 1 1 f + 1 h GF O1/, 4 if f is fixed. I the other directio, f d F G 1 f + 1 h F G 1 + O1/ h F G 5 if f is fixed. Hece, whe worig iside large graphs, we ca pass freely betwee the desity of a fixed graph F ad its homomorphism desity, with a error of oly O1/.

4 3 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A Fially, we will mae frequet use of the AM/GM iequality: Theorem 6. If x 1,...,x 0,the 1/ x i 1 x i.. The case via extremal graph theory For a sufficietly large multiple of, it turs out to be possible to prove Cojecture 1 usig stability versios of Turá-type results. We will prove the followig Theorem 5. If N, is a sufficietly large multiple of, X is a -elemet set, ad G is a -geerator for X, the G F,. Equality holds oly if G is of the form F,. We eed a few more defiitios. Let H deote the graph with vertex-set P X, where we joi two subsets x, y X if they are disjoit. With slight abuse of termiology, we call H the Keser graph o P X although this usually meas the aalogous graph o X r. If F, G P X, we say that G -geerates F if every set i F is a disjoit uio of at most sets i G. The mai steps of the proof. First, we will show that for ay A P X with A Ω /, the desity of K +1 s i the iduced subgraph H[A] is o1. Secodly, we will observe that if is a sufficietly large multiple of, ad G P X has size close to F, ad -geerates almost all subsets of X, the K H[G] is very close to K T G, the umber of K s i the -partite Turá graph o G vertices. We will the prove that if G is ay graph with small K +1 -desity, ad with K G close to K T G, theg ca be made -partite by removig a small umber of edges. This ca be see as a stregtheed variat of the Simoovits Stability Theorem [7], which states that ay K +1 -free graph G with eg close to the maximum et G, ca be made -partite by removig a small umber of edges. This will eable us to coclude that H[G] ca be made -partite by the removal of a small umber of edges, ad therefore the structure of H[G] is close to that of the Turá graph T G. This i tur will eable us to show that the structure of G is close to that of a caoical -geerator F, Propositio 9. Fially, we will use a perturbatio argumet to show that if is sufficietly large, ad G F,, the G = F,, completig the proof. I fact, we will first show that if A P X with A Ω /, the the homomorphism desity of K +1 t i H[A] is o1, provided t is sufficietly large depedig o. Hece, we will eed the followig relatively well-ow lemma relatig the homomorphism desity of a graph to that of its blow-up. Lemma 7. Let F be a graph o f vertices, let t = t 1, t,...,t f N f,adletf t deote the t-blow-up of F. If the homomorphism desity of F i G is p, the the homomorphism desity of F t i G is at least p t 1t t f. Proof. This is a simple covexity argumet, essetially that of [7]. It will suffice to prove the statemet of the lemma whe t = 1,...,1, r for some r N. We thi of F as a labelled graph o vertex set [ f ] ={1,,..., f }, ad G as a labelled graph o vertex set []. Defie the fuctio χ :[] f {0, 1} by { 1 if i vi is a homomorphism from F to G, χv 1,...,v f = 0 otherwise.

5 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A The we have h F G = 1 f v 1,...,v f [] f χv 1,...,v f = p. The homomorphism desity h F 1,...,1,r G of F 1,...,1, r i G is h F 1,...,1,r G = 1 f 1+r = 1 f 1 1 f 1 1 = f = p r. v 1,...,v f 1,v 1,v,...,v r f f f r [] f 1+r 1 χv 1,...,v f 1, v f v 1,...,v f 1 [] f 1 v f [] 1 v 1,...,v f 1 [] f 1 v f [] v 1,...,v f 1,v f [] f χv 1,...,v f 1, v f χ v 1,...,v f 1, v i f r r χv 1,...,v f 1, v f Here, the iequality follows from applyig Jese s Iequality to the covex fuctio x x r. This proves the lemma for t = 1,...,1, r. By symmetry, the statemet of the lemma holds for all vectors of the form 1,...,1, r, 1,...,1. Clearly,wemayobtaiF t from F by a sequece of blow-ups by these vectors, provig the lemma. The followig lemma a rephrasig of Lemma 4. i Alo ad Fral [1] gives a upper boud o the homomorphism desity of K +1 t i large iduced subgraphs of the Keser graph H. Lemma 8. If A P Xwith A =m = δ+1/+1,the h K+1 t H[A] + 1 δt 1. Proof. We follow the proof of Alo ad Fral cited above. Choose + 1t members of A uiformly at radom with replacemet, A j i 1 i +1, 1 j t. The homomorphism desity of K +1 t i H[A] is precisely the probability that the uios t U i = j=1 A j i are pairwise disjoit. If this evet occurs, the U i / + 1 for some i. Foreachi [], wehave { Pr U i / + 1 } t { j = Pr A } S i S X: S /+1 j=1 t { j Pr A } S i S /+1 j=1 = S t /m S /+1 /+1 /m t = δt 1. r

6 34 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A Hece, { Pr U i / + 1 } Therefore, as required. { Pr U i / + 1 } + 1 δt 1. h K+1 t H[A] + 1 δt 1, From the trivial boud above, ay -geerator G has G /,soδ 1/ + 1, ad therefore, choosig t = t := + 1, we see that h K+1 t H[G] + 1. Hece, by Lemma 7, h K+1 H[G] O /t. Therefore, by 5, d K+1 H[G] O /t a provided is sufficietly large depedig o, where a > 0 depeds oly o. Assume ow that is a multiple of, so that F, = /. We will prove the followig stability result. Propositio 9. Let N befixed.ifisamultipleof,adg P Xhas G 1 + η F, ad -geerates at least 1 ɛ subsets of X, the there exists a equipartitio S i of X such that G P S i 1 C ɛ 1/ D η 1/ ξ F,, where C, D,ξ > 0 deped oly o. We first collect some results used i the proof. We will eed the followig theorem of Erdős [6]. 6 Theorem 10 Erdős. If r, ad G is a K +1 -free graph o vertices, the K r G K r T. We will also eed the followig well-ow lemma, which states that a dese -partite graph has a iduced subgraph with high miimum degree. Lemma 11. Let G be a -vertex, -partite graph with eg 1 1/ δ /. The there exists a iduced subgraph G Gwith G = 1 δ ad miimum degree δg 1 1/ δ 1. Proof. We perform the followig algorithm to produce G.LetG 1 = G. Suppose that at stage i, we have a graph G i o i + 1 vertices. If there is a vertex v of G i with dv<1 1/ η i, let G i+1 = G i v; otherwise, stop ad set G = G i. Suppose the process termiates after j = α steps. The we have removed at most

7 1 1/ η D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A j j i = 1 1/ η edges, ad the remaiig graph has at most j = 1 α 1 1/ / edges. But our origial graph had at least 1 1/ δ / edges, ad therefore so 1 1/ η 1 1 α / + 1 α 1 1/ / 1 1/ δ /, η1 α η δ. Choosig η = δ,weobtai η1 α η1 η, ad therefore so 1 α 1 η, α 1 1 η 1/ η. Hece, our iduced subgraph G has order G = 1 δ, ad miimum degree δ G 1 1/ δ 1. We will also eed Shearer s Etropy Lemma. Lemma 1 Shearer s Etropy Lemma. See [4]. Let S be a fiite set, ad let A be a r-cover of S, meaig a collectio of subsets of S such that every elemet of S is cotaied i at least r sets i A. LetF be a collectio of subsets of S. For A S, let F A ={F A: F F} deote the projectio of F oto the set A. The F r A A F A. I additio, we require two stability versios of Turá-type results i extremal graph theory. The first states that a graph with a very small K +1 -desity caot have K r -desity much higher tha the -partite Turá graph o the same umber of vertices, for ay r. Lemma 13. Let r be itegers. The there exist C, D > 0 such that for ay α 0, ay-vertexgraphg with K +1 -desity at most α has K r -desity at most 1 r + 1 r 1 + Cα 1/+ + D/.

8 36 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A Proof. We use a straightforward samplig argumet. Let G be as i the statemet of the lemma. Let ζ l be the umber of l-subsets U V G such that G[U] cotais a copy of K+1, so that ζ is simply the probability that a uiform radom l-subset of V G cotais a K +1. Simple coutig or the uio boud gives l ζ α. + 1 By Theorem 10, each K +1 -free G[U] cotais at most r l r K r s. Therefore, the desity of K r s i each such G[U] satisfies 1 r + 1 l r d Kr G[U] r ll 1 l r r O1/l. 7 r Note that oe ca choose a radom r-set i graph G by first choosig a radom l-set U, ad the choosig a radom r-subset of U. The desity of K r s i G is simply the probability that a uiform radom r-subset of V G iduces a K r, ad therefore d Kr G = E U [ dk G[U] ], where the expectatio is tae over a uiform radom choice of U.IfU is K +1 -free, which happes with probability 1 ζ, we use the upper boud 7; if U cotais a K +1, which happes with probability ζ,weusethetrivialboudd K G[U] 1.WeseethatthedesityofK r s i G satisfies 1 r + 1 d Kr G 1 ζ 1 + O1/l + ζ r 1 r + 1 l + O1/l + α r r O1/l + r l +1 α. Choosig l = mi{ α 1/+,} proves the lemma. The secod result states that a -vertex graph with a small K +1 -desity, a K -desity ot too much less tha that of T, ad a K 1 -desity ot too much more tha that of T, ca be made ito a -partite graph by the removal of oly a small umber of edges. Theorem 14. Let G be a -vertex graph with K +1 -desity at most α, K 1 -desity at most 1 + β! 1, ad K -desity at least 1 γ!, where γ 1/. TheGcabemadeitoa-partitegraphG 0 by removig at most β + γ α + /!

9 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A edges, which removes at most β + γ α + /! K s. Proof. If N, ad G is a graph, let K G = { S V G : G[S] is a clique } deote the set of all -sets that iduce a clique i G. IfS V G, letns deote the set of vertices of G joied to all vertices i S, i.e. the itersectio of the eighbourhoods of the vertices i S, ad let ds = NS. ForS K G, let f G S = dt. T S, T = 1 We begi by setchig the proof. The fact that the ratio betwee the K -desity of G ad the K 1 - desity of G is very close to 1/ will imply that the average E f G S over all sets S K G is ot too far below. The fact that the K +1 -desity of G is small will mea that for most sets S K G, every 1-subset T S has NT spaig few edges of G, ad ay two distict 1-subsets T, T S have NT NT small. Hece, if we pic such a set S which has f G S ot too far below the average, the sets {NT : T S, T = 1} will be almost pairwise disjoit, will cover most of the vertices of G, ad will each spa few edges of G. Small alteratios will produce a -partitio of V G with few edges of G withi each class, provig the theorem. We ow proceed with the proof. Observe that E f G = S K G T S, T = 1 dt K G T K = 1 G dt K G T K 1 G dt K 1 GK G = K G K 1 GK G = K G K 1 G 1 γ! β! = 1 γ β 1 The first iequality follows from Cauchy Schwarz, ad the secod from our assumptios o the K - desity ad the K 1 -desity of G. We call a set T K 1 G dagerous if it is cotaied i at least α +1 K+1 s. Let D deote the umber of dagerous 1-sets. Double-coutig the umber of times a 1-set is cotaied i a K +1,weobtai D α α, + 1

10 38 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A sice there are at most α +1 K+1 s i G. Hece, D α. 1 Similarly, we call a set S K G treacherous if it is cotaied i at least α K +1 s. Doublecoutig the umber of times a -set is cotaied i a K +1, we see that there are at most α treacherous -sets. Call a set S K G bad if it is treacherous, or cotais at least oe dagerous 1-set; otherwise, call S good. The the umber of bad -sets is at most α α = + 1 α, 1 so the fractio of sets i K G which are bad is at most Suppose that + 1 α 1 γ! = + 1 α. 1 γ! max { f G S : S is good } <1 ψ + 1. Observe that for ay S K G, wehave f G S + 1, sice dt + 1foreachT S 1.Hece, E f G < α 1 ψ+ + 1 α γ! 1 γ! 1 ψ α + 1, 1 γ! a cotradictio if ψ = ψ 0 := 1 1 γ 1 + β α 1 γ! γ + β α.! Let S K G be a good -set such that f G S 1 ψ Write S ={v 1,...,v }, let T i = S \{v i } for each i, ad let N i = NT i for each i. Observe that N i N j = NS for each i j, ad NS =ds α. LetW i = N i \ NS for each i; observe that the W i s are pairwise disjoit. Let R = V G \ W i be the set of leftover vertices. Observe that Ni \ NS = f G S NS 1 ψ + 1 α, ad therefore the umber of leftover vertices satisfies R <ψ+ α +.

11 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A We ow produce a -partitio V i of V G by extedig the partitio W i of V G \ R arbitrarily to R, i.e., we partitio the leftover vertices arbitrarily. Now delete all edges of G withi V i for each i. The umber of edges withi N i is precisely the umber of K +1 s cotaiig T i, which is at most α +1. The umber of edges icidet with R is trivially at most ψ + α Hece, the umber of edges deleted was at most ψ + α α β + γ α + /.! Removig a edge removes at most K s, ad therefore the total umber of K s removed is at most β + γ α + / = completig the proof.! β + γ ! α + /, Note that the two results above together imply the followig Corollary 15. For ay N, there exist costats A, B > 0 such that the followig holds. For ay α 0, if Gisa-vertexgraphwithK +1 -desity at most α,adk -desity at least 1 γ!, where γ 1/,theGcabemadeitoa-partitegraphG 0 by removig at most γ + A α 1/+ + B / edges, which removes at most γ + A α 1/+ + B / K s. Proof of Propositio 9. Suppose G P X has G =m 1 + η F,, ad -geerates at least 1 ɛ subsets of X. Our aim is to show that G is close to a caoical -geerator. We may assume that ɛ 1/C ad η 1/D, so by choosig C ad D appropriately large, we may assume throughout that ɛ ad η are small. By choosig ξ appropriately small, we may assume that 0, where 0 is ay fuctio of. We first apply Lemma 13 ad Theorem 14 with G = H[G], where H is the Keser graph o P X, G P X with G =m 1 + η F,, ad G -geerates at least 1 ɛ subsets of X. By 6, we have d K+1 H[G] a, ad therefore we may tae α = a. Applyig Lemma 13 with r = 1, we may tae β = b for some b > 0.

12 330 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A We have G =m 1 + η /, so m m < 1 + η.!! Notice that 1 m m η / 1 <1 + η 1 1 1/. i i=0 Sice G -geerates at least 1 ɛ subsets of X, wehave Hece, K H[G] 1 ɛ 1 + η 1 1 1/. K H[G] d K H[G] = m 1 ɛ 1 + η 1 1 1/ 1+η / 1 ɛ 1 + η 1 /! 1 + η 1 ɛ η /!, where the last iequality follows from 1 ɛ 1 + η 1 ɛ1 η 1 ɛ1 η 1 ɛ η. Therefore, the K -desity of H[G] satisfies where d K H[G] 1 γ!, γ = ɛ + η + /. Let ψ = β + γ α + /.! By Theorem 14, there exists a -partite subgraph G 0 of H[G] with Writig m K G 0 K H[G] ψ m 1 ɛ 1 + η 1 1 1/ ψ 1 ɛ 1 + η ψ 1 + η 1 /.! φ = ɛ η ψ η 1 /,!

13 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A we have K G 0 1 φ. Let V 1,...,V be the vertex-classes of G 0. By the AM/GM iequality, K G 0 V i V i = m/, ad therefore G =m K G 0 1/ 1 φ 1/ /, 8 recoverig the asymptotic result of [5]. Moreover, ay -partite graph G 0 satisfies K eg 0 G 0 /. To see this, simply apply Shearer s Etropy Lemma with S = V G 0, F = K G 0, ad A = {V i V j : i j}. TheA is a 1-cover of V G 0.NotethatF V i V j E G0 V i, V j, ad therefore K G 0 1 {i, j} [] e G0 V i, V j. Applyig the AM/GM iequality gives K G 0 1 {i, j} ad therefore K eg 0 G 0 /, e G0 V i, V j as required. It follows that eg 0 1 φ / / 1 φ / m 1 + η 1 η 1 φ / 1 1/m / 1 η φ / 1 1/m / = 1 δ1 1/m /, {i, j} e G 0 V i, V j = eg0, where δ = η + φ /. Hece, G 0 is a -partite subgraph of H[G] with G 0 = G =m, ad eg 0 1 δ 1/m /. Applyig Lemma 11 to G 0, we see that there exists a iduced subgraph H of G 0 with H 1 δ G, 9 ad δ H 1 1/ δ H 1.

14 33 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A Let Y 1,...,Y be the vertex-classes of H ;otethatthesearefamiliesofsubsetsofx. Clearly, for each i [], Y i H δ H 1/ + δ H Hece, for each i [], Y i H 1 1/ + δ H + 1 1/ 1 δ H For each i [], let S i = y Yi y be the uio of all sets i Y i. We claim that the S i s are pairwise disjoit. Suppose for a cotradictio that S 1 S. The there exist y 1 Y 1 ad y Y which both cotai some elemet p X. Sice δ H 1 1/ δ H 1, at least 1 1/ δ H 1 sets i i 1 Y i do ot cotai p. By 10, Y i = Y i 1 1/ + 1 δ H + 1, i 1 i 1 ad therefore the umber of sets i i 1 Y i cotaiig p is at most 1 1/ + 1 δ H / δ H 1 δ H +. The same holds for the umber of sets i i Y i cotaiig p, sothetotalumberofsetsih cotaiig p is at most δ H +. Hece, the total umber of sets i G cotaiig p is at most + 1 δm +. But the the umber of ways of choosig at most disjoit sets i G with oe cotaiig p is at most 1 + m δm + = O δ + O 1 1/ < 1 ɛ, cotradictig the fact that G -geerates all but ɛ of the sets cotaiig p. Hece, we may coclude that the S i s are pairwise disjoit. By defiitio, Y i P S i, ad therefore Y i S i. But from 11, Y i 1 1 δ H / δ 1 δ G / δ 1 δ1 φ 1/ / δ φ 1/ / + 1 > 1 δ φ 1/ / > / 1, usig 9 ad 8 for the secod ad third iequalities respectively. Hece, we must have S i / for each i, ad therefore S i =/ for each i, i.e.s i is a equipartitio of X. Puttig everythig together ad recallig that δ = η + φ / ad φ = O ɛ + η + c,wehave

15 G P S i D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A Y i 1 δ φ 1/ / 1 C ɛ 1/ D η 1/ ξ / provided is sufficietly large depedig o, where C, D,ξ > 0 deped oly o. Thisproves Propositio 9. We ow prove the followig Propositio 16. Let ν = o1.ifg is a -geerator for X with G F,,ad G P S i 1 ν F,, where S i is a partitio of X ito classes of sizes as equal as possible, the provided is sufficietly large depedig o, we have G = F, ad G = P S i \{ }. Note that is o loger assumed to be a multiple of ; thecase = ad odd will be eeded i Sectio 3. Proof. Let G ad S i be as i the statemet of the propositio. For each i [], let F i = P S i \{ } \ G be the collectio of all oempty subsets of S i which are ot i G. By our assumptio o G, we ow that F i o S i for each i []. Let E = G \ PS i be the collectio of extra sets i G; let E =M. By relabelig the S i s, we may assume that F 1 F F. By our assumptio o G, M F 1. Let R ={y 1 s s : y 1 F 1, s i S i, i }; observe that the sets y 1 s s are all distict, so R = F 1 S1. By cosiderig the umber of sets i E eeded for G to -geerate R, we will show that M > F 1 uless F 1 =.Ifact,our argumet would also show that M > p F 1 uless F 1 =,forayp > 0 depedig oly o. Let N be the umber of sets i R which may be expressed as a disjoit uio of two sets i E ad at most othersetsig. The M m N i i=0 1 F 1 1 c 0 /! 4c 0 F1 F S 1 S 1 1

16 334 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A = o1 F 1 S 1 = o R, 1 where we have used G F, c 0 / see 3, S 1 /, ad F 1 =o S 1 i the secod, third ad fourth lies respectively. Now fix x 1 F 1.For j 1, let A j x 1 be the collectio of 1-tuples s,...,s P S P S such that x 1 s s may be expressed as a disjoit uio y 1 y y with y j E but y i S i, i j. LetA x 1 be the collectio of 1-tuples s,...,s P S P S such that x 1 s s may be expressed as a disjoit uio of two sets i E ad at most othersetsig. Now fix j 1. For each s,...,s A j x 1,wemaywrite x 1 s s = s 1 s s j 1 y j s j+1 s, where y j = s j x 1 \ s 1 E. Sicey j S j = s j,differets j s correspod to differet y j s E, ad so there are at most E =M choices for s j. Therefore, A j x 1 S 1 S j M S 1 S j F1 F1 S S1, 1 the last iequality followig from the fact that S j S 1 1. Hece, A j x 1 F1 1 S1 = o1 S1. 13 j= Observe that for each x 1 F 1, A x 1 ad therefore S 1 A j x 1 = P S P S 3 P S, j=1 A x 1 + A1 x 1 + A j x 1 S1, j= so by 13, A x 1 + A1 x 1 1 o1 S 1. Call x 1 F 1 bad if A x 1 + S 1 ; otherwise, call x 1 good. By 1, at most a o1- fractio of the sets i F 1 are bad, so at least a 1 o1 fractio are good. For each good set x 1 F 1, otice that A1 x o1 S 1. Now perform the followig process. Choose ay s,...,s A 1 x 1 ;wemaywrite

17 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A x 1 s s = z 1 s s with s,...,s P S P S, z 1 E, z 1 S 1 = x 1, ad z 1 \ S 1. Picp 1 z 1 \ S 1.At most 1 S1 of the members of A 1 x 1 have uio cotaiig p 1, so there are at least o1 S 1 remaiig members of A 1 x 1. Choose oe of these, t,...,t say. By defiitio, we may write x 1 t t = z t t with t,...,t P S P S, z E, z S 1 = x 1, ad z \ S 1. Sicep 1 / z,wemust have z z 1.Picp z \ S 1, ad repeat. At most 3 4 S1 of the members of A 1 x 1 have uio cotaiig p 1 or p ; there are at least o1 S 1 members remaiig. Choose oe of these, u,...,u say. By defiitio, we may write x 1 u u = z 3 u u with u,...,u P S P S, z 3 E, z 3 S 1 = x 1, ad z 3 \ S 1. Note that agai z 3 is distict from z 1, z,sicep 1, p / z 3. Cotiuig this process for + 1 steps, we ed up with a collectio of + 1distictsetsz 1,...,z +1 E such that z l S 1 = x 1, l [ + 1]. Dothisfor each good set x 1 F 1 ; the collectios produced are clearly pairwise disjoit. Therefore, E o1 F 1. This is a cotradictio, uless F 1 =.Hece,wemusthaveF = =F =, ad therefore G = PS i \{ }, provig Propositio 16, ad completig the proof of Theorem The case = via bipartite subgraphs of H Our aim i this sectio is to prove the = case of Cojecture 1 for all sufficietly large odd, which together with the = case of Theorem 5 will imply Theorem 4. If is sufficietly large, X is a -elemet set, ad G P X is a -geerator for X, the G F,. Equality holds oly if G is of the form F,. Recall that { F, = / if is eve; 3 1/ if is odd. Suppose that X is a -elemet set, ad G P X is a -geerator for X with G =m F,. The coutig argumet i the Itroductio gives m 1 + m +,

18 336 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A which implies that G 1 o1 /. For odd, we wish to improve this boud by a factor of approximately 1.5. Our first aim is to prove that iduced subgraphs of the Keser graph H which have order Ω / are o1-close to beig bipartite Propositio 18. Recall that a graph G = V, E is said to be ɛ-close to beig bipartite if it ca be made bipartite by the removal of at most ɛ V edges, ad ɛ-far from beig bipartite if it requires the removal of at least ɛ V edges to mae it bipartite. Usig Szemerédi s Regularity Lemma, Bollobás, Erdős, Simoovits ad Szemerédi [3] proved the followig Theorem 17 Bollobás, Erdős, Simoovits, Szemerédi. For ay ɛ > 0, there exists gɛ N depedig o ɛ aloe such that for ay graph G which is ɛ-far from beig bipartite, the probability that a uiform radom iduced subgraph of G of order gɛ is o-bipartite is at least 1/. Buildig o methods of Goldreich, Goldwasser ad Ro [10], Alo ad Krivelevich [] proved without usig the Regularity Lemma that i fact, oe may tae gɛ log1/ɛb ɛ where b > 0isaabsolutecostat.Asobservedi[],thisistightuptothepoly-logarithmicfactor, sice ecessarily, gɛ 1 6ɛ. We will first show that for ay fixed c > 0 ad l N, ifa P X with A c /, the the desity of C l+1 s i H[A] is at most o1. To prove this, we will show that for ay l N, thereexists t N such that for ay fixed c > 0, if A P X with A c /, the the homomorphism desity of C l+1 t i H[A] is o1. Usig Lemma 7, we will deduce that the homomorphism desity of C l+1 i H[A] is o1, implyig that the desity of C l+1 s i H[A] is o1. This will show that H[A] is o1-close to beig bipartite Propositio 18. To obtai a sharper estimate for the o1 term i Propositio 18, we will use 14, although to prove Theorem 4, ay o1 term would suffice, so oe could i fact use Theorem 17 istead of 14. We are ow ready to prove the followig 14 Propositio 18. Let c > 0. The there exists b > 0 such that for ay A P Xwith A c /,theiduced subgraph H[A] ca be made bipartite by removig at most edges. log log b A log Proof. Fix c > 0; let A P X with A =m c /. First, we show that for ay fixed l N, thereexists t N such that the homomorphism desity of C l+1 t s i H[A] is at most o1. The argumet is a stregtheig of that used by Alo ad Fral to prove Lemma 4. i [1]. Let t N to be chose later. Choose l + 1t members of A uiformly at radom with replacemet, A j i 1 i l+1, 1 j t. The homomorphism desity of C l+1 t i H[A] is precisely the probability that the uios

19 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A U i = t j=1 A j i satisfy U i U i+1 = for each i where the additio is modulo l + 1. We claim that if this occurs, the U i < 1 η for some i, provided η < 1/4l +. Suppose for a cotradictio that U i U i+1 = for each i, ad U i 1 η for each i. The U i+ \ U i U i+1 U i η for each i [l 1]. SiceU l+1 \U 1 lj=1 U j+1 \U j 1,wehave U l+1 \U 1 l j=1 U j+1 \ U j 1 lη. It follows that U 1 U l+1 1/ l + 1η > 0ifη < 1/4l +, a cotradictio. We ow show that the probability of this evet is very small. Fix i []. Observe that { Pr U i 1/ } t { j η = Pr A } S i provided t /η. Hece, Pr Therefore, l+1 = S X: S 1/ η S 1/ η j=1 S t /m S 1/ η 1/ η c / = ηt 1 c t c t, j=1 t { j Pr A } S i t { U i 1/ } l+1 η { Pr U i 1/ } η l + 1 c t. h Cl+1 t H[A] l + 1 c t. Choose η = 1 8l ad t = /η = 16l. By Lemma 7, h Cl+1 H[A] l + 1 c t 1/t l+1 = l + 1 1/16ll+1 /16ll+1 c 1/16ll = O /16ll+1. Observe that the umber of s + 1-subsets of A cotaiig a odd cycle of H is at most s m l+1 m l + 1 h Cl+1 H[A]. s l l=1 Hece, the probability that a uiform radom s + 1-subset of A cotais a odd cycle of H is at most s m l+1 mm 1 m l s + 1s s l + 1 h Cl+1 H[A] l=1 ss + 1!O /16ss+1

20 338 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A provided s O m. This ca be made < 1/ by choosig s = a log / log log, for some suitable a > 0 depedig oly o c. By 14, it follows that H[A] is log log b / log - close to beig bipartite, for some suitable b > 0 depedig oly o c, provig the propositio. Before provig Theorem 4 for odd, we eed some more defiitios. Let X be a fiite set. If A P X, ad i X, wedefie A i ={x A: i / x}, A + i = { x \{i}: x A, i x } ; these are respectively called the lower ad upper i-sectios of A. If Y ad Z are disjoit subsets of X, we write H[Y, Z] for the bipartite subgraph of the Keser graph H cosistig of all edges betwee Y ad Z.IfB is a bipartite subgraph of H with vertex-sets Y ad Z, ad F P X, we say that B -geerates F if for every set x F,thereexisty Y ad z Z such that y z =, yz EB, ad y z = x, i.e.everysetif correspods to a edge of B. Proof of Theorem 4 for odd. Suppose that = l is odd, X is a -elemet set, ad G P X is a -geerator for X with G =m F, =3 l. Observe that e H[G] l+1 G 1 l+1 3 l + 1, ad therefore H[G] has edge-desity at least l+1 3 l + 1 G l+1 3 l l 3 l 3 > 4 9. Here, the last iequality rearrages to the statemet l > 0. By Propositio 18 applied to G, weca remove at most log log b G < log log b 9 l log log edges from H[G] to produce a bipartite graph B. LetY, Z be the vertex-classes of B; we may assume that Y Z = G. Defieɛ > 0by {y z: y Y, z Z, y z = } = 1 ɛ l+1 ; the clearly, we have eb 1 ɛ l Note that ɛ 9 log log b + 3 l+1 log log = b O = o1. log log Let α = Y / l, β = Z / l. By assumptio, α + β 3 l 1 < 3. Sice Y Z eb ɛ l,wehaveαβ ɛ. This implies that 1 ɛ < α,β < + ɛ. 16

21 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A To see this, simply observe that to maximize αβ subject to the coditios α 1 ɛ ad α + β 3, it is best to tae α = 1 ɛ ad β = + ɛ, givigαβ = ɛ 4ɛ < ɛ, a cotradictio. It follows that we must have α > 1 ɛ, soβ< + ɛ; 16 follows by symmetry. From ow o, we thi of X as the set []={1,,...,}. Let W 1 = { i []: Y + } Y /3, W = { i []: Z + i First, we prove the followig Claim 1. W 1 W =[]. i } Z /3. Proof. Suppose for a cotradictio that W 1 W []. Without loss of geerality, we may assume that / W 1 W.Let θ = Y + / Y, φ = Z + / Z ; the we have θ,φ 1/3.Observethattheumbere of edges betwee Y ad Z which geerate a set cotaiig satisfies 1 ɛ l e θα1 φβ + φβ1 θα l = θ + φ θφαβ l. 17 Here, the left-had iequality comes from the fact that B -geerates all but at most ɛ l+1 subsets of [], ad therefore B -geerates at least 1 ɛ l sets cotaiig. Notice that the fuctio f θ, φ = θ + φ θφ, 0 θ,φ 1/3 is a strictly icreasig fuctio of both θ ad φ for 0 θ,φ 1/3, ad therefore attais its maximum of 4/9 atθ = φ = 1/3. Therefore, 1 ɛ 4 9 αβ; sice α + β 3, we have 3/ 3 ɛ/ α,β 3/ + 3 ɛ/. Moreover, by the AM/GM iequality, αβ 9/4, so 1 ɛ 9 f θ, φ, 4 18 ad therefore 1/3 8ɛ/3 θ,φ 1/3. Thus Y, Z =3/ o1 l ad θ,φ = 1/3 o1. Therefore, we have Y + = l 1 1 o1, Z + = l 1 1 o1, Y = l 1 + o1, Z = l 1 + o1. Observe that G = Y Z must -geerate all but at most ol of the sets i P{1,,..., 1}= P{1,,...,l}, ad therefore, by Propositio 9 for = ad eve, there exists a equipartitio

22 340 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A S 1 S of {1,,...,l} such that Y cotais at least 1 o1l members of P S 1, ad Z cotais at least 1 o1 l members of P S.Defie U ={y Y : y S = }, V ={z Z: z S 1 = }. Sice U =1 o1l ad V =1 o1l,wemusthave Y \ U =ol, ad Z \ V = o l. Our aim is ow to show that Y + \ U + =ol, ad Z + \ V + =ol. Clearly, we have U P S 1, ad V P S, so U l ad V l. Moreover, each set x Y + \ U + cotais a elemet of S, ad therefore x {} is disjoit from at most l 1 sets i V P S. Similarly, each set x Z + \ V + cotais a elemet of S 1, ad therefore x {} is disjoit from at most l 1 sets i U P S 1. It follows that e U + V + Y + \ U + l 1 + V + U + Z + \ V + + Y \ U Z + + Z \ V Y + U + l + Y + \ U + l 1 + V + l + Z + \ V + l 1 l 1 + o l. O the other had, by 17, we have e 1 o1 l. Sice Y + =l 1 1 o1, ad Z + = l 1 1 o1, wemusthave Y + \ U + =ol, ad Z + \ V + =ol,asrequired. We may coclude that Y \ U =o l ad Z \ V =o l. Hece, there are at most o l sets i Y Z = G that itersect both S 1 ad S. O the other had, sice Y + =1 o1l 1 ad Z + =1 o1l 1, there are at least 1 + o1 l 1 sets s 1 S 1 such that s 1 {} / Y, ad there are at least 1 + o1 l 1 sets s S such that s {} / Z. Taig all pairs s 1, s gives at least 1 + o1 l sets of the form {} s 1 s s1 S 1, s 1 {} / Y, s S, s {} / Z. 19 Each of these requires a set itersectig both S 1 ad S to express it as a disjoit uio of two sets from G. Sicethereareo l members of G itersectig both S 1 ad S, G geerates at most G +1 o l = o l sets of the form 19, a cotradictio. This proves the claim. We ow prove the followig Claim. W 1 W =. Proof. Suppose for a cotradictio that W 1 W. Without loss of geerality, we may assume that W 1 W. As before, let θ = Y + / Y, φ = Z + / Z ; this time, we have θ,φ 1/3. Observe that ɛ l eb 1 θφαβ l. 0 Here, the left-had iequality is 15, ad the right-had iequality comes from the fact that there are o edges betwee pairs of sets y, z Y Z such that y z. Sice1 θφ 8/9, we have ɛ 8 9 αβ. Sice α + β 3, it follows that 3 1 ɛ α,β ɛ.

23 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A Sice αβ 9/4, we have ɛ 9 1 θφ, 4 ad therefore 1/3 θ,φ 1/3 + 8ɛ/3. Hece, we have Y + = l 1 1 o1, Z + = l 1 1 o1, Y = l 1 + o1, Z = l 1 + o1, so exactly as i the proof of Claim 1, we obtai a cotradictio. Claims 1 ad together imply that W 1 W is a partitio of {1,,...,}={1,,...,l + 1}. We will ow show that at least a /3 o1-fractio of the sets i Y are subsets of W 1, ad similarly at least a /3 o1-fractio of the sets i Z are subsets of W.Let σ = Y \ PW 1, τ = Z \ PW. Y Z Let y Y \ P W 1, ad choose i y W ; sice at least Z /3 ofthesetsiz cotai i, y has at most Z /3 eighbours i Z.Hece, ɛ l eb σαβ + 1 σ αβ l = 1 σ /3αβ l 1 σ / l, 1 ad therefore σ 1/3 + 8ɛ/3, so Y PW 1 /3 8ɛ/3 Y. Similarly, τ 1/3 + 8ɛ/3, ad therefore Z PW /3 8ɛ/3 Z. If W 1 l 1, the Y PW 1 l 1,so l 1 Y /3 8ɛ/3 = ɛ <1 ɛl, l cotradictig 16. Hece, we must have W 1 l. Similarly, W l, so{ W 1, W } = {l,l + 1}. Without loss of geerality, we may assume that W 1 =l ad W =l + 1. We ow observe that Z 3/ 6ɛ l. 3 To see this, suppose that Z =3/ η l.sice Z + Y < 3 l,wehave Y 3/ + η l.recall that ay y Y \ P W 1 has at most Z /3 eighbours i Z. Thus, we have

24 34 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A ɛ l eb Y P W 1 Z + Y \ P W 1 3 Z 3 l η 1 l + + η l 3 3 η l = 13 η 3 η l. Therefore η 6ɛ, i.e. Z 3/ 6ɛ l, as claimed. Sice Z + Y < 3 l,wehave Y 3/ + 6ɛ l. We ow prove the followig 4 Claim 3. a PW 1 \ Y ɛ l ; b Z \ P W ɛ + ɛ l. Proof. We prove this by costructig aother bipartite subgraph B of H with the same umber of vertices as B, ad comparig eb with eb. First,let D = mi { PW \ Z, Z \ P W }, add D ew members of PW \ Z to Z, ad delete D members of Z \ P W, producig a ew set Z ad a ew bipartite graph B 1 = H[Y, Z ].Sice Z = Z + ɛ l,wehave Z \ P W ɛ l+1, i.e. Z is almost cotaied withi P W. Notice that every member z Z \ P W had at most Y /3 eighbours i Y, ad every ew member of Z has at least Y PW 1 /3 8ɛ/3 Y eighbours i Y, usig. Hece, eb eb 1 8ɛ 3 Y D 8ɛ 3 Y 16ɛ 9 Z l = 4ɛ l, ad therefore eb 1 eb ɛ l+1 1 3ɛ l+1. Secod, let C = mi { P W 1 \ Y, Y \ P W 1 }, add C ew members of PW 1 \ Y to Y, ad delete C members of Y \ P W 1, producig a ew set Y ad a ew bipartite graph B = H[Y, Z ].Sice Y 1 ɛ l,wehave Y P W 1 1 ɛ l. Sice every deleted member of Y cotaied a elemet of W, it had at most 1 + ɛ l eighbours i Z.Ideed,suchmemberofY itersects l sets i P W, so has at most l eighbours i Z P W ; there are Z \ P W ɛ l+1 other sets i Z. O the other had, every ew member of Y is joied to all of Z P W, which has size at least Z P W 3/ 8ɛ l. It follows that eb eb 1 + C 1 10ɛ l 1 3ɛ l+1 + C 1 10ɛ l. 5 We ow show that eb 1 + ɛ l+1. If Y l, the write Y =1 + φ l where φ 0; Y cotais all of P W 1, ad φ l extra sets. We have Z φ l, ad therefore by 3, φ 1/+6ɛ < 1. Note that every extra set i Y \PW 1 has at most l eighbours i P W, ad therefore at most 1 + ɛ l eighbours i Z.Hece, eb l φ l + φ l 1 + ɛ l = 1 + φɛ l ɛ l+1.

25 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A If, o the other had, Y l,thesice Y + Z 3 l,wehaveeb Y Z l+1. Hece, we always have eb 1 + ɛ l+1. 6 Combiig 5 ad 6, we see that C 8ɛ 1/ 10ɛ l 0ɛ l, provided ɛ 1/100. This implies a. Ideed, if P W 1 \ Y C 0ɛ l, the we are doe. Otherwise, by the defiitio of C, wehave Y \ P W 1 0ɛ l. Recall that by 16, Y 1 ɛ l, ad therefore Y P W 1 = Y Y \ P W 1 1 ɛ l 0ɛ l = 1 ɛ l. Hece, PW 1 \ Y ɛ l, 7 provig a. Sice eb 1 ɛ l+1, eb 1 + ɛ l+1, ad eb eb 1,wehave eb 1 eb eb eb 1 + ɛ l+1 1 ɛ l+1 = ɛ l+ 8 We ow use this to show that D = mi { PW \ Z, Z \ P W } ɛ l. Suppose for a cotradictio that D ɛ l ; the it is easy to see that there must exist z Z \PW with at least Y /3 8 ɛ l eighbours i Y. Ideed, suppose that every z Z \ P W has less tha Y /3 8 ɛ l eighbours i Y. Recall that every ew member of Z has at least /3 8ɛ Y eighbours i Y.Hece, eb 1 eb>8d ɛ ɛ Y 8 ɛ l ɛ ɛ1 ɛ l ɛ l+1 sice ɛ < 1/16, cotradictig 8. Hece, we may choose z Z \ P W with at least Y /3 8 ɛ l eighbours i Y. Without loss of geerality, we may assume that z W 1 ; the oe of these eighbours ca cotai. Hece,Y cotais at most Y /3 + 8 ɛ l sets cotaiig. But by 7, Y cotais at least 1 44ɛ l 1 of the subsets of W 1 that cotai, ad therefore Y 3/ o1 l. By 3, it follows that Y =3/ o1 l ad Z =3/+o1 l, so Y cotais 1 o1 l 1 sets cotaiig. Hece, by 18, so does Z. As i the proof of Claim 1, we obtai a cotradictio. This implies that as desired. D = mi { PW \ Z, Z \ P W } ɛ l,

26 344 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A This implies b. Ideed, if Z \ P W ɛ l, the we are doe. Otherwise, by the defiitio of D, PW \ Z ɛ l, ad therefore Z P W ɛ l. Sice Z + ɛ l,wehave Z \ P W = Z Z P W + ɛ l ɛ l = ɛ + ɛ l, provig b. We coclude by provig the followig Claim 4. PW \ Z 4 ɛ l. Proof. Let F = PW \ Z be the collectio of sets i P W which are missig from Z, ad let E 1 = Y \ P W 1 be the set of extra members of Y. Sice G is a -geerator for X, we ca express all F l sets of the form w 1 f w 1 W 1, f F as a disjoit uio of two sets i G. All but at most ɛ l+1 of these uios correspod to edges of B. Sice Z \ P W ɛ + ɛ l, there are at most ɛ + ɛ l Y edges of B meetig sets i Z \ P W. Call these edges of B bad, ad the rest of the edges of B good. Fix f F ; we ca express all l sets of the form w 1 f w 1 W 1 as a disjoit uio of two sets i G. Ifw 1 f is represeted by a good edge, the we may write w 1 f = y 1 w where y 1 E 1 with y 1 W 1 = w 1, ad w W,soforeverysuchw 1, there is a differet y 1 E 1. By 4, Y 3/ + 6ɛ l, ad by 7, Y P W 1 1 ɛ l,so E 1 = Y PW 1 Y 3/ + 6ɛ l 1 ɛ l = 1/ + 8ɛ l. Thus, for ay f F,atmost1/ + 8ɛ l uios of the form w 1 f correspod to good edges of B. All the other uios are geerated by bad edges of B or are ot geerated by B at all, so 1/ 8ɛ l F ɛ + ɛ l Y +ɛ l+1. Sice Y 3/ + 6ɛ l ad ɛ is small, F 4 ɛ l,asrequired. We ow ow that Y cotais all but at most o l of P W 1, ad Z cotais all but at most o l of P W.Sice Y + Z < 3 l, we may coclude that Y =1 o1 l ad Z = o1 l. It follows from Propositio 16 that provided is sufficietly large, we must have G = PW 1 PW \{ }, completig the proof of Theorem 4.

27 D. Ellis, B. Sudaov / Joural of Combiatorial Theory, Series A Coclusio We have bee uable to prove Cojecture 1 for 3adall sufficietly large. Recall that if G is a -geerator for a -elemet set X, the G /. I view of Propositio 18, it is atural to as whether for ay fixed, all iduced subgraphs of the Keser graph H with Ω / vertices ca be made -partite by removig at most o / edges. This is false for = 3, however, as the followig example shows. Let be a multiple of 6, ad tae a equipartitio of [] ito 6 sets T 1,...,T 6 of size /6. Let A = T i T j ; {i, j} [6] the A =15 /3, ad H[A] cotais a /3 -blow-up of the Keser graph K 6,, which has chromatic umber 4. It is easy to see that H[A] requires the removal of at least /3 edges to mae it tripartite. Hece, a differet argumet to that i Sectio 3 will be required. We believe Cojecture 1 to be true for all ad, but it would seem that differet techiques will be required to prove this. Refereces [1] N. Alo, P. Fral, The maximum umber of disjoit pairs i a family of subsets, Graphs Combi [] N. Alo, M. Krivelevich, Testig -colorability, SIAM J. Discrete Math [3] B. Bollobás, P. Erdős, M. Simoovits, E. Szemerédi, Extremal graphs without large forbidde subgraphs, A. Discrete Math [4] F.K.R. Chug, P. Fral, R.L. Graham, J.B. Shearer, Some itersectio theorems for ordered sets ad graphs, J. Combi. Theory Ser. A [5] D. Ellis, Note o geeratig all subsets of a fiite set with disjoit uios, Electro. J. Combi , Note 16. [6] P. Erdős, O the umber of complete subgraphs cotaied i certai graphs, Publ. Math. Ist. Hug. Acad. Sci. Ser. A [7] P. Erdős, M. Simoovits, Supersaturated graphs ad hypergraphs, Combiatorica [8] Y. Frei, B. Lévêque, A. Sebő, Geeratig all sets with bouded uios, Combi. Probab. Comput [9] Z. Füredi, G.O.H. Katoa, -Bases of quadruples, Combi. Probab. Comput [10] O. Goldreich, S. Goldwasser, D. Ro, Property testig ad its coectio to learig ad approximatio, Proc. 37th Aual IEEE FOCS