INDEPENDENT SETS IN HYPERGRAPHS

Transcription

1 INDEPENDENT SETS IN HYPERGRAPHS Abstract. Many iportant theores and conjectures in cobinatorics, such as the theore of Szeerédi on arithetic progressions and the Erdős-Stone Theore in extreal graph theory, can be phrased as stateents about failies of independent sets in certain unifor hypergraphs. In recent years, an iportant trend in the area has been to extend such classical results to the so-called sparse rando setting. This line of research has recently culinated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving probles of this type. Although these two papers solved very siilar sets of longstanding open probles, the ethods used are very different fro one another and have different strengths and weaknesses. In this paper, we provide a third, copletely different approach to proving extreal and structural results in sparse rando sets that also yields their natural counting counterparts. We give a structural characterization of the independent sets in a large class of unifor hypergraphs by showing that every independent set is alost contained in one of a sall nuber of relatively sparse sets. We then derive any interesting results as fairly straightforward consequences of this abstract theore. In particular, we prove the wellknown conjecture of Kohayakawa, Luczak, and Rödl, a probabilistic ebedding lea for sparse graphs. We also give alternative proofs of any of the results of Conlon and Gowers and Schacht, such as sparse rando versions of Szeerédi s theore, the Erdős-Stone Theore, and the Erdős-Sionovits Stability Theore, and obtain their natural counting versions, which in soe cases are considerably stronger. For exaple, we show that for each positive β and integer k, there are at ost ( βn sets of size that contain no k-ter arithetic progression, provided that Cn 1 1/(k 1, where C is a constant depending only on β and k. We also obtain new results, such as a sparse version of the Erdős-Frankl-Rödl Theore on the nuber of H-free graphs and, as a consequence of the K LR conjecture, we extend a result of Rödl and Ruciński on Rasey properties in sparse rando graphs to the general, non-syetric setting. 1. Introduction A great any of the central questions in cobinatorics fall into the following general fraework: Given a finite set V and a collection H P(V of forbidden structures, what can be said about sets I V that do not contain any eber of H? For exaple, the celebrated theore of Szeerédi [62] states that if V = {1,..., n} and H is the collection of k-ter arithetic progressions in {1,..., n}, then every set I that contains no eber of H satisfies I = o(n. The archetypal proble studied in extreal graph theory, dating back Date: May 16, Research supported in part by: (JB NSF CAREER Grant DMS , UIUC Capus Research Board Grant 11067, and OTKA Grant K76099; (RM CNPq bolsa de Produtividade e Pesquisa; (WS ERC Advanced Grant DMMCA and a Trinity College JRF. 1

2 2 to the work of Turán [64] and Erdős and Stone [20], is the proble of characterizing such sets I when V is the edge set of the coplete graph on n vertices and H is the collection of copies of soe fixed graph H in K n. In this setting, a great deal is known, not only about the axiu size of I that contains no eber of H, but also what the largest such sets look like, how any such sets there are, and what the structure of a typical such set is. A collection H P(V as above is usually referred to as a hypergraph on the vertex set V and any set I V that contains no eleent (edge of H is called an independent set. Therefore, one ight say that a large part of extreal cobinatorics is concerned with studying independent sets in various specific hypergraphs. We ight add here that in any natural settings, such as the two entioned above, the hypergraphs considered are unifor, that is, all edges of H have the sae size. Although it ight at first see soewhat artificial to study concrete questions in such an abstract setting, the past few years have proved that taking such a general approach can be highly beneficial. The recently-proved general transference theores of Conlon and Gowers [14] and Schacht [58] (see also [23], which iply, aong other things, sparse rando analogues of the classical theores of Szeerédi and of Erdős and Stone, were stated in the language of hypergraphs. Roughly speaking, these transference theores say the following: Let H be a hypergraph whose edges are sufficiently uniforly distributed. Then the independence nuber of H is well-behaved with respect to taking subhypergraphs induced by (sufficiently dense rando subsets of the vertex set. More precisely, given p [0, 1] and a finite set V, we shall write V p to denote the p-rando subset of V, that is, the rando subset of V in which each eleent of V is included with probability p, independently of all other eleents. We write α(h and v(h to denote the size of the largest independent set and the nuber of vertices in a hypergraph H, respectively. The results of Conlon and Gowers [14] and Schacht [58] iply, in particular, that if the distribution of the edges of soe unifor hypergraph H is sufficiently balanced, then with probability tending to 1 as v(h, α ( H[V (H p ] pα(h + o ( pv(h, provided that p is sufficiently large. In this work, we give an approxiate structural characterization of the faily of all independent sets in unifor hypergraphs whose edge distribution satisfies a certain natural boundedness condition. More precisely, we shall prove that the faily I(H of independent sets of such a hypergraph H exhibits a certain clustering phenoenon. Our ain result (Theore 2.2, below states that I(H adits a partition into relatively few classes with the following property: all ebers of each class are essentially contained in a single alost independent subset of V (H (i.e., one which contains only a tiny proportion of all the edges of H. This soewhat abstract stateent has surprisingly any deep and interesting consequences, soe of which we list in the reainder of this section. We reark that Theore 2.2 was partly inspired by the work of Kleitan and Winston [38], who iplicitly considered a stateent of this type in the setting of graphs (2-unifor hypergraphs and subsequently used it to bound the nuber of n-vertex graphs without a 4-cycle. We also note that a result siilar to Theore 2.2 was independently proved by Saxton and Thoason [57], who also use it to derive any of the stateents that we present in Sections

3 INDEPENDENT SETS IN HYPERGRAPHS The nuber of sets with no k-ter arithetic progression. The celebrated theore of Szeerédi [62] says that for every k N, the largest subset of {1,..., n} that contains no k-ter arithetic progression (AP has o(n eleents. It iediately follows that there are only 2 o(n subsets of {1,..., n} with no k-ter AP. Our first result can be viewed as a sparse analogue of this stateent. Theore 1.1. For every positive β and every k N, there exist constants C and n 0 such that the following holds. For every n N with n n 0, if Cn 1 1/(k 1, then there are at ost ( βn -subsets of {1,..., n} that contain no k-ter AP. We shall deduce Theore 1.1 fro our ain theore, Theore 2.2, and a robust version of Szeerédi s theore, see Section 4. The sparse rando analogue of Szeerédi s theore, proved by Schacht [58] and independently by Conlon and Gowers [14], follows as an easy corollary of Theore 1.1. Following [14], we shall say that a set A N is (δ, k-szeerédi if every subset B A with at least δ A eleents contains a k-ter AP. For the sake of brevity, let [n] = {1,..., n} and recall that [n] p denotes the p-rando subset of [n]. Corollary 1.2. For every δ (0, 1 and every k N, there exists a constant C such that the following holds. If p n Cn 1/(k 1 for all sufficiently large n, then li P( [n] pn is (δ, k-szeerédi = 1. n We reark that Theore 1.1 and Corollary 1.2 are both sharp up to the value of the constant C, see the discussion in Section 4, where both of these stateents are proved. Our ain result has a variety of other applications in additive cobinatorics, see for exaple [1, 2] where, jointly with Alon, we used a uch sipler version of it to count sufree sets of fixed size in various Abelian groups and the set [n]. In Section 4, we shall ention two other applications: generalizations of Theore 1.1 to higher diensions and to k-ter APs whose coon difference is of the for d r. In each case, the rando version (which was proved in [14, 58] follows as an easy corollary Turán s proble in rando graphs. The faous theore of Erdős and Stone [20] states that the axiu nuber of edges in an H-free graph on n vertices, the Turán nuber for H, denoted ex(n, H, satisfies ex(n, H = ( 1 1 χ(h 1 + o(1 ( n 2, (1 where χ(h is the chroatic nuber of H. The analogue of this theore for the Erdős- Rényi rando graph G(n, p was first studied by Babai, Sionovits, and Spencer [4], who proved that asyptotically alost surely (a.a.s. for short, i.e., with probability tending to 1 as n, the largest triangle-free subgraph of G(n, 1/2 is bipartite, and by Frankl and Rödl [22], who proved that if p n 1/2+ε then a.a.s. the largest triangle-free subgraph of G(n, p has pn 2 /8 + o(pn 2 edges. The systeatic study of the Turán proble in G(n, p

4 4 was initiated by Haxell, Kohayakawa, and Luczak [34, 35] and by Kohayakawa, Luczak, and Rödl [42], who posed the following proble. For a fixed graph H, deterine necessary and sufficient conditions on a sequence p [0, 1] N of probabilities such that, a.a.s., ex ( G(n, p n, H = ( 1 1 χ(h 1 + o(1 ( n 2 p n, (2 where ex(g, H denotes the axiu nuber of edges in an H-free subgraph of G. By considering a rando (χ(h 1-partition of the vertex set of G(n, p, it is straightforward to show that the inequality ex ( G(n, p, H ( o(1 (n χ(h 1 2 p holds for every p [0, 1]. On the other hand, if the nuber of copies of soe subgraph H H in G(n, p is uch saller than the nuber of edges in G(n, p, then the converse inequality cannot hold, since one can ake any graph H-free by reoving fro it one edge fro each copy of H. This observation otivates the notion of 2-density of H, denoted by 2 (H, which is defined by { } e(h 1 2 (H = ax v(h 2 : H H with v(h 3. (3 It now follows easily that for every graph H with axiu degree at least 2 and every δ ( 0, 1/(χ(H 1, there exists a positive constant c such that if p n cn 1/2(H, then a.a.s. ex ( G(n, p n, H ( ( 1 n > 1 χ(h 1 + δ p n. 2 It was conjectured by Haxell, Kohayakawa, and Luczak [34] and Kohayakawa, Luczak, and Rödl [42] that the above siple arguent, reoving an arbitrary edge fro each copy of H in G(n, p, is the ain obstacle that prevents (2 fro holding asyptotically alost surely. The conjecture, often referred to as Turán s theore for rando graphs, has attracted considerable attention in the past fifteen years. Nuerous partial results and special cases had been established by various researchers [24, 28, 31, 34, 35, 42, 44, 61] before the conjecture was finally proved by Conlon and Gowers [14] (under the assuption that H is strictly 2- balanced 1 and by Schacht [58]. Theore 1.3. For every graph H with (H 2 and every positive δ, there exists a positive constant C such that if p n Cn 1/2(H, then a.a.s. ex ( G(n, p n, H ( ( 1 n 1 χ(h 1 + δ p n. 2 Our ethods give yet another proof of Theore 1.3. In fact, we shall deduce fro our ain result, Theore 2.2, a version of the general transference theore of Schacht [58, Theore 3.3], which easily iplies Theore 1.3 for such graphs H. Our version of Schacht s transference theore, Theore 5.2, is stated and proved in Section 5. We then, in Section 7, use it to derive a natural generalization of Theore 1.3 to t-unifor hypergraphs, Theore 7.2, which was also first proved in [14] and [58]. 1 A graph H is 2-balanced if the axiu in (3 is achieved with H = H, that is, if 2 (H = e(h 1 v(h 2. It is strictly 2-balanced if 2 (H > 2 (H for every proper subgraph H H.

5 INDEPENDENT SETS IN HYPERGRAPHS 5 Reark 1.4. In the original version of this paper, we only proved the results concerning H-free graphs under the additional assuption that H is 2-balanced. However, a siple odification of our ethod (peritting ultiple edges in our hypergraphs, as in [57] allowed us to reove this condition. We would like to thank David Saxton for pointing this out. Our ethods also yield the following sparse rando analogue of the faous stability theore of Erdős and Sionovits [16, 60], originally proved by Conlon and Gowers [14] in the case when H is strictly 2-balanced and then extended to arbitrary H by Saotij [56], who adapted the arguent of Schacht [58] for this purpose. Theore 1.5. For every graph H with (H 2 and every positive δ, there exist positive constants C and ε such that if p n Cn 1/ 2(H, then a.a.s. the following holds. Every H-free subgraph of G(n, p n with at least ( 1 1 χ(h 1 ε ( n 2 p n edges ay be ade (χ(h 1-partite by reoving fro it at ost δn 2 p n edges. As with Theore 1.3, we shall in fact deduce Theore 1.5 fro a ore general stateent, Theore 6.2, which is a version of the general transference theore for stability results proved in [56]. Theore 6.2 is stated and proved in Section 6; in Section 7, we use it to derive Theore The typical structure of H-free graphs. Let H be an arbitrary non-epty graph. For an integer n, denote by f n (H the nuber of labelled H-free graphs on the vertex set [n]. Since every subgraph of an H-free graph is also H-free, it follows that f n (H 2 ex(n,h. Erdős, Frankl, and Rödl [17] proved that this crude lower bound is in a sense tight, naely that f n (H = 2 ex(n,h+o(n2. (4 Our next result can be viewed as a sparse version of (4. Such a stateent was already considered by Luczak [46], who derived it fro the so-called K LR conjecture, which we discuss in the next subsection. For integers n and with 0 ( n 2, let fn, (H be the nuber of labelled H-free graphs on the vertex set [n] that have exactly edges. The following theore refines (4 to n-vertex graphs with edges. Theore 1.6. For every graph H and every positive δ, there exists a positive constant C such that the following holds. For every n N, if Cn 2 1/2(H, then ( ( ex(n, H ex(n, H + δn 2 f n, (H. In fact, we shall deduce fro our ain result, Theore 2.2, a counting version of the general transference theore of Schacht [58, Theore 3.3], which easily iplies Theore 1.6. This counting version of Schacht s theore (which refines and, in soe respects, strengthens the ain results of [14, 58] is stated and proved in Section 5. We then use it to derive Theore 1.6 in Section 8. We reark that (4 was refined in a different sense by Balogh, Bollobás, and Sionovits [5], who showed that f n (H = 2 ex(n,h+o(n2 c(h, where c(h is

6 6 soe positive constant, and also gave a very precise structural description of alost all H- free graphs. We would also like to point out that our proof of Theore 1.6 does not use Szeerédi s regularity lea, unlike the proof given in [46] or the proofs of Erdős, Frankl, and Rödl [17] and Balogh, Bollobás, and Sionovits [5]. The result of Erdős, Frankl, and Rödl has, in soe cases, a structural counterpart that significantly strengthens (4. For exaple, Erdős, Kleitan, and Rothschild [18] proved that alost all triangle-free graphs are bipartite, that is, that with probability tending to 1 as n, a graph selected uniforly at rando fro the faily of all triangle-free graphs on the vertex set [n] is bipartite or, in other words (since clearly every bipartite graph is triangle-free, f n (K 3 is asyptotic to the nuber of bipartite graphs on the vertex set [n]. Extending this result, Osthus, Pröel, and Taraz [49] proved that if Cn 3/2 log n for soe C > 3/4, then alost all n-vertex triangle-free graphs with edges are bipartite. The corresponding result for K r+1 -free graphs was proved recently in [7]. Our next result, which is a strengthening of Theore 1.6, is an approxiate version of this stateent for an arbitrary graph H. Such a stateent was also considered by Luczak [46], who derived it fro the K LR conjecture. Following [46], given a positive real δ and an integer k, let us say that a graph G is (δ, k-partite if G can be ade k-partite by reoving fro it at ost δe(g edges. Theore 1.7. For every graph H with χ(h 3, and every positive δ, there exists a positive constant C such that the following holds. If Cn 2 1/ 2(H, then alost all H-free graphs with n vertices and edges are ( δ, χ(h 1 -partite. As with Theore 1.6, we shall in fact deduce Theore 1.7 fro a counting version of the general transference theore for stability results proved in [56]. Our version of it, Theore 6.3, is stated and proved in Section 6. In Section 8, we use it to derive Theore 1.7. Once again, our proof does not use the regularity lea, unlike that in [46]. Finally, we would like to ention that, as observed by Luczak [46], Theore 1.7 has the following elegant corollary. Corollary 1.8. For every graph H with χ(h 3 and every positive ε, there exist positive constants C and n 0 such that the following holds. For every n N with n n 0 and every N with Cn 2 1/2(H n 2 /C, ( χ(h 2 χ(h 1 ε P ( G n, H where G n, is a uniforly selected rando n-vertex graph with edges. ( χ(h 2 χ(h 1 + ε, (5 Note that (5 does not hold if is too large; for exaple, if > n 2 /4 then P ( G n, K 3 = 0. We reark that a great deal ore is known about the structure of a typical H-free graph (drawn uniforly at rando fro the set of all n-vertex H-free graphs, see [6] and the references therein for ore details The K LR conjecture. The celebrated Szeerédi regularity lea [63], which is considered to be one of the ost iportant and powerful tools in extreal graph theory, says

7 INDEPENDENT SETS IN HYPERGRAPHS 7 that the vertex set of every graph ay be divided into a bounded nuber of parts of approxiately the sae size in such a way that ost of the bipartite subgraphs induced between pairs of parts of the partition satisfy a certain pseudo-randoness condition tered ε-regularity. The strength of the regularity lea lies in the fact that it ay be cobined with the so-called ebedding lea to show that a graph contains particular subgraphs. The cobination of the regularity and ebedding leas allows one to prove any wellknown theores in extreal graph theory, such as the theore of Erdős and Stone [20] and the stability theore of Erdős and Sionovits [16, 60], both entioned in Section 1.2. For sparse graphs, that is, n-vertex graphs with o(n 2 edges, the original version of the regularity lea is vacuous since if the vertex set of a sparse graph is partitioned into a bounded nuber of parts, then all induced bipartite subgraphs thus obtained are trivially ε-regular, provided that n is sufficiently large. However, it was independently observed by Kohayakawa [39] and Rödl (unpublished that the notion of ε-regularity ay be extended in a eaningful way to graphs with density tending to zero. Moreover, with this ore general notion of regularity, they were also able to prove an associated regularity lea which applies to a large class of sparse graphs, including (a.a.s. the rando graph G(n, p. Given a p [0, 1] and a positive ε, we say that a bipartite graph between sets V 1 and V 2 is (ε, p-regular if for every W 1 V 1 and W 2 V 2 with W 1 ε V 1 and W 2 ε V 2, the density d(w 1, W 2 of edges between W 1 and W 2 satisfies d(w1, W 2 d(v 1, V 2 εp. A partition of the vertex set of a graph into r parts V 1,..., V r is said to be (ε, p-regular if V i V j 1 for all i and j and for all but at ost εr 2 pairs (V i, V j, the graph induced between V i and V j is (ε, p-regular. The class of graphs to which the Kohayakawa- Rödl regularity lea applies are the so-called upper-unifor graphs. Given positive η and K, we say that an n-vertex graph G is (η, p, K-upper-unifor if for all W V (G with W ηn, the density of edges within W satisfies d(w Kp. This condition is satisfied by any natural classes of graphs, including (a.a.s. all subgraphs of rando graphs of density p. The sparse regularity lea of Kohayakawa [39] and Rödl says the following. The sparse Szeerédi regularity lea. For all positive ε, K, and r 0, there exist a positive constant η and an integer R such that for every p [0, 1], the following holds. Every (ε, p, K-upper-unifor graph with at least r 0 vertices adits an (ε, p-regular partition of its vertex set into r parts, for soe r {r 0,..., R}. We reark that a version of this theore avoiding the need for the upper-unifority assuption was recently proved by Scott [59]. The aforeentioned ebedding lea roughly says that if we start with an arbitrary graph H, replace its vertices by large independent sets and its edges by ε-regular bipartite graphs with density uch larger than ε, then this blown-up graph will contain a copy of H. To ake it ore precise, let H be a graph on the vertex set {1,..., v(h}, let ε and p be as above, and let n and be integers satisfying 0 n 2. Let us denote by G(H, n,, p, ε the collection of all graphs G constructed in the following way. The vertex set of G is a disjoint union V 1... V v(h of sets of size n, one for each vertex of H. For each edge {i, j} of H, we add to G an (ε, p-regular bipartite graph with edges between the sets V i and

8 8 V j. These are the only edges of G. With this notation in hand, we can state the ebedding lea. Given any graph G as above, we define canonical copies of H to be all copies of H in G in which (the iage of each vertex i V (H lies in the set V i V (G. The ebedding lea. For every graph H and every positive d, there exist a positive ε and an integer n 0 such that for every n and with n n 0 and dn 2, every G G(H, n,, 1, ε contains a canonical copy of H. One ight hope that a siilar stateent holds when one replaces 1 by an arbitrary p and the assuption dn 2 by pdn 2, even if p is a decreasing function of n. However, for an arbitrary function p, this is too uch to hope for. Indeed, consider the rando blow-up of H, that is, the rando graph G obtained fro H by replacing each vertex of H by an independent set of size n and each edge of H by a rando bipartite graph with pn 2 edges. With high probability, the nuber of canonical copies of H in G will be about p e(h n v(h and hence if p e(h n v(h pn 2, then one can reove all copies of H fro G by deleting a tiny proportion of all edges. Since in the above arguent one ay replace H with an arbitrary subgraph H H, it follows easily 2 that if p n 1/ 2(H, then there are graphs in G(H, n, pn 2, p, ε that do not contain any canonical copies of H. As in the case of Turán s theore for rando graphs, see Section 1.2, one ight still hope that if p Cn 1/ 2(H for soe large constant C, then the natural sparse analogue of the ebedding lea discussed above holds. However, it was observed by Luczak (see [32, 43] that, soewhat surprisingly, for any graph H which contains a cycle and any function p satisfying p = o(1, there are graphs in G(H, n, pn 2, p, ε with no canonical copy of H. Nevertheless, it still seeed likely that such atypical graphs coprise so tiny a proportion of G(H, n,, p, ε that they do not appear in G(n, p asyptotically alost surely. This was foralized in the following conjecture of Kohayakawa, Luczak, and Rödl [42], usually referred to as the K LR conjecture. Given a graph H, integers and n, a p [0, 1], and a positive ε, let G (H, n,, p, ε denote the collection of graphs in G(H, n,, p, ε that contain no canonical copy of H. We will prove the conjecture in Section 9. Theore 1.9 (The K LR conjecture. For every graph H and every positive β, there exist positive constants C, n 0, and ε such that the following holds. For every n N with n n 0 and N with Cn 2 1/2(H, G (H, n,, /n 2, ε ( n β 2 e(h. The K LR conjecture has been one of the central open questions in extreal graph theory and has attracted substantial attention fro any researchers over the past fifteen years. It has been verified in several special cases. It is easy to see that it holds for all graphs H which do not contain a cycle. The cases H = K 3, K 4, and K 5 were resolved in [41], [30], and [31], respectively. The case H = C l has also been resolved, but here the history is soewhat ore coplex. A proof under soe extra technical assuptions was given in [40]. Those extra assuptions were later reoved in [29] and, independently, in [12]. We reark here that in parallel to this work, Conlon, Gowers, Saotij, and Schacht [15] have proved a sparse 2 Note that we also replace p with soe p = (1 + o(1p, and that the reoval of o(pn 2 edges does not affect the ε-regularity conditions.

9 INDEPENDENT SETS IN HYPERGRAPHS 9 analogue of the counting lea for subgraphs of the rando graph G(n, p, which ay be viewed as a version of the K LR conjecture that is stronger in soe aspects and weaker in other aspects. It is well-known that Theore 1.9 easily iplies Turán s theore for rando graphs, Theore 1.3, and also its stability version, Theore 1.5. In fact, this was the original otivation behind the K LR conjecture, see [42]. Moreover, it was proved by Luczak [46] that Theore 1.9 iplies Theores 1.6 and 1.7. The work of Conlon and Gowers [14] and Schacht [58] (see also [56], as well as this work, have shown that one does not need to appeal to the sparse regularity lea and to the K LR conjecture in order to prove such extreal stateents in rando graphs. Nevertheless, there are still any beautiful corollaries of the conjecture that cannot (yet be proved by other eans. For discussion and derivation of soe of the, we refer the reader to [15]. Here, we present only one corollary of the K LR conjecture, the threshold for asyetric Rasey properties of rando graphs, which does not follow fro the version of the conjecture proved in [15]. The deduction of this result fro the K LR conjecture is essentially due to Kohayakawa and Kreuter [40] Rasey properties of rando graphs. Let H be a fixed graph and let r be a positive integer. For an arbitrary graph G, we write G (H r if every r-coloring of the edges of G contains a onochroatic copy of H. It follows fro the classical result of Rasey [51] that K n (H r, provided that n is sufficiently large. Rasey properties of rando graphs were first investigated by Frankl and Rödl [22] and since then uch effort has been devoted to their study. Most notably, Rödl and Ruciński [52, 53] established the following general threshold result. Theore For every graph H that is not a forest, and every positive integer r, there exist positive constants c and C such that { li P( 1 if p n Cn 1/2(H, G(n, p n (H r = n 0 if p n cn 1/2(H. In the above discussion, a copy of the sae graph H is forbidden in each of the r color classes. A natural generalization of Theore 1.10 would deterine thresholds for so-called asyetric Rasey properties. For any graphs G, H 1,..., H r, we write G (H 1,..., H r if for every coloring of the edges of G with colors 1,..., r, there exists, for soe i [r], a copy of H i all of whose edges have color i. In the context of asyetric Rasey properties of rando graphs, the following generalization of the 2-density 2 ( was introduced in [40]. For two graphs H 1 and H 2, define 3 { 2 (H 1, H 2 = ax e(h 1 v(h / 2 (H 2 : H 1 H 1 with v(h 1 3 }. (6 Kohayakawa and Kreuter [40] forulated the following conjecture and proved it in the case when all H i are cycles. 3 To otivate this definition, set p = n 1/2(H1,H2 and observe that the edges of G(n, p which are contained in a copy of each subgraph H 1 H 1 have density roughly n 1/2(H2.

10 10 Conjecture Let H 1,..., H r be graphs with 1 < 2 (H r... 2 (H 1. Then there exist constants c and C such that li P( G(n, p n (H 1,..., H r { 1 if p n Cn 1/ 2(H 1,H 2, = n 0 if p n cn 1/ 2(H 1,H 2. More accurately, the above conjecture was stated in [40] only in the case r = 2, but the above generalization is quite natural. 4 There had been little progress on Conjecture 1.11 until quite recently, when the 0-stateent was proved by Marciniszyn, Skokan, Spöhel, and Steger [47] in the case where all of the H i are cliques, and the 1-stateent in the case r = 2 was established 5 by Kohayakawa, Schacht, and Spöhel [45] under very ild extra assuptions on H 1 and H 2. It was observed in [47, Theore 31] that, using Theore 1.9, the approach of Kohayakawa and Kreuter [40], which eploys the sparse regularity lea, can be adapted to yield a proof of the 1-stateent in Conjecture 1.11 for the following class of graphs. Theore Let H 1,..., H r be graphs with 1 < 2 (H r... 2 (H 1 and such that H 1 is strictly 2-balanced. Then there exists a constant C such that if p n Cn 1/ 2(H 1,H 2, then a.a.s. G(n, p n (H 1,..., H r. For the deduction of Theore 1.12 fro Theore 1.9, see [40] and [47, Section 4] Outline of the paper. The reainder of this paper is organized as follows. In Section 2, we state and discuss our ain result, Theore 2.2, which we then prove in Section 3. In Section 4, we discuss the applications of Theore 2.2 in the context of subsets of [n] with no k-ter arithetic progressions. In particular, we prove Theore 1.1 and use it to derive Corollary 1.2. In Section 5, we prove two versions of the general transference theore of Schacht [58, Theore 3.3] (obtained independently, in a slightly different for, by Conlon and Gowers [14] a rando version suited for extreal probles in sparse rando discrete structures and its counting counterpart that generalizes Theore 1.1. In Section 6, we prove rando and counting versions of the general stability result of Conlon and Gowers [14] in a for that is easily coparable with [56, Theore 3.4]. In Section 7, we discuss several applications of Theore 2.2 in the context of the Turán proble in sparse rando graphs. In particular, using the results of Sections 5 and 6 we give new proofs of the sparse rando analogues (stated above of the classical theores of Erdős and Stone, and Erdős and Sionovits, see Section 1.2. In Section 8, we discuss applications of Theore 2.2 to the proble of describing the typical structure of a sparse graph without a forbidden subgraph. In particular, we prove sparse analogues of classical theores of Erdős, Frankl, and Rödl and Erdős, Kleitan, and Rothschild, see Section 1.3. Finally, in Section 9, we use Theore 2.2 to prove the K LR conjecture for every graph H. 4 To see why the graphs H 3,..., H r do not appear in the threshold, replace each of H 2,..., H r by the disjoint union H = H 2 H r, and note that 2 (H = 2 (H 2, see [47]. 5 In their concluding rearks, the authors of [45] oreover clai that their ethod can be extended to the setting with ore than two colours, using ideas fro [53].

11 INDEPENDENT SETS IN HYPERGRAPHS The Main Theore In this section, we present the ain result of this paper, Theore 2.2, which gives a structural characterization of the collection of all independent sets in a large class of unifor hypergraphs. Let us stress here that all of the hypergraphs we consider are allowed to have ultiple edges; oreover, we shall always count edges with ultiplicities. We start with an iportant definition. Recall that a faily of sets F P(V is called increasing (or an upset if it is closed under taking supersets, that is, if for every A, B V, A F and A B iply that B F. Definition 2.1. Let H be a unifor hypergraph with vertex set V, let F be an increasing faily of subsets of V and let ε (0, 1]. We say that H is (F, ε-dense if for every A F. e(h[a] εe(h A oent of thought reveals that for an arbitrary hypergraph H and ε (0, 1], it is extreely siple to find failies F P(V (H for which H is (F, ε-dense. To this end, let F ε = { A V (H: e(h[a] εe(h } and note that F ε is increasing and H is (F ε, ε-dense. In fact, the failies F for which H is (F, ε-dense are precisely all increasing subfailies of F ε. In this work, we will be interested in upsets that adit a uch ore constructive description than that of F ε. Many such failies arise naturally in the study of extreal and structural probles in cobinatorics. For exaple, consider the k-unifor hypergraph H 1 on the vertex set [n] whose edges are all k-ter arithetic progressions in [n] and let F 1 be the collection of all subsets of [n] with at least δn eleents. Clearly, F 1 is an upset and it follows fro the faous theore of Szeerédi [62] that H 1 is (F 1, ε-dense for soe positive ε depending only on δ and k, see Section 4. Siilarly, consider the 3-unifor hypergraph H 2 on the vertex set E(K n whose edges are edge sets of all copies of K 3 in the coplete graph K n and let F 2 be the faily of all n-vertex graphs (subgraphs of K n with at least (1/2 ε ( n 2 edges such that every 2-coloring of its vertices yields at least δn 2 onochroatic edges. Again, F 2 is increasing and it follows fro the stability theore of Erdős and Sionovits [16, 60] and the triangle reoval lea of Ruzsa and Szeerédi [55] that H 2 is (F 2, ε-dense, provided that ε is sufficiently sall as a function of δ. Our ain result roughly says the following. If H is a unifor hypergraph that is (F, ε- dense for soe faily F and whose edge distribution satisfies certain natural boundedness conditions, then the collection I(H of all independent sets in H adits a partition into relatively few classes such that all independent sets in one class are essentially contained in a single set A F. Before we state the result, we first need to quantify the above boundedness conditions for the edge distribution of a hypergraph. Given a hypergraph H, for each T V (H, we define 6 deg H (T = {e H: T e}, 6 We ephasize that if H has ultiple edges, then {e H: T e} should be thought of as a ulti-set. In other words, deg H (T is the nuber of edges of H, counted with ultiplicities, which contain T.

12 12 and let l (H = ax { deg H (T : T V (H and T = l }. Recall that I(H denotes the faily of all independent sets in H. The following theore is our ain result. Theore 2.2. For every k N and all positive c and ε, there exists a positive constant C such that the following holds. Let H be a k-unifor hypergraph and let F P(V (H be an increasing faily of sets such that A εv(h for all A F. Suppose that H is (F, ε-dense and p (0, 1 is such that, for every l [k], l 1 e(h l (H c p v(h. Then there exists a faily S ( V Cp v(h and functions f : S F and g : I(H S such that for every I I(H, g(i I and I \ g(i f(g(i. Roughly speaking, if H satisfies certain technical conditions, then each independent set I in H can be labelled with a sall subset g(i in such a way that all sets labelled with soe S S are essentially contained in a single set f(s that contains very few edges of H. We reark that the constant C in the theore has only a polynoial dependence on ε. Unfortunately, however, in ost of our applications ε will have a tower-type dependence on soe other paraeter. Theore 2.2 will be proved in Section 3. We end this section with a short inforal discussion of its consequences. As we have already entioned, Theore 2.2 cobined with soe classical extreal results on discrete structures has strikingly strong iplications. Let us briefly explain why this is so. Many classical extreal probles ask for an estiate on the nuber of independent sets (of a certain size in soe auxiliary unifor hypergraph. If applicable, Theore 2.2 iplies that all such independent sets are alost contained in one of very few sets that are alost independent, that is, contain a sall nuber of copies of soe forbidden substructure. If we know a good characterization of sets that are alost independent in the above sense, which is often the case, we can easily obtain an upper bound on the nuber of independent sets. For exaple, consider the proble of counting subsets of [n] with no k-ter AP and recall the definition of H 1 and F 1 fro the beginning of this section. Theore 2.2, applied to this pair, iplies that every subset of [n] with no k-ter AP is essentially contained in one of at ost ( n O(n sets of size at ost δn each, where 1 1/(k 1 δ is an arbitrarily sall positive constant. This easily iplies that if n 1 1/(k 1, then there are at ost ( 2δn sets of size with no k-ter AP. For ore details, we refer the reader to Section Proof of the ain theore In this section, we shall prove Theore 2.2. The ain ingredient in the proof is the following proposition, which (roughly says that Theore 2.2 holds in the special case when

13 INDEPENDENT SETS IN HYPERGRAPHS 13 F is the faily of all subsets of V (H with at least (1 δv(h eleents. Theore 2.2 follows by applying Proposition 3.1 a constant nuber of ties. Proposition 3.1. For every integer k and positive c, there exists a positive δ such that the following holds. Let p (0, 1 and suppose that H is a k-unifor hypergraph such that, for every l [k], l 1 e(h l (H c p v(h. Then there exist a faily S ( V (H (k 1p v(h and functions f0 : S P(V (H and g 0 : I(H S such that for every I I(H, g 0 (I I f 0 (g 0 (I g 0 (I and f0 (g 0 (I (1 δv(h. Moreover, if for soe I, I I(H, g 0 (I I and g 0 (I I, then g 0 (I = g 0 (I. The final line of Proposition 3.1 states that the labelling function g 0 exhibits a certain consistency. This property of g 0, which ay look soewhat puzzling, will be crucial in the proof of Theore 2.2. In order to prove Proposition 3.1, given an independent set I I(H, we shall construct a sequence (B k 1,..., B q of subsets of I with B k 1,..., B q pv(h, for soe q [k 1], and use it to define a sequence (H k 1,..., H r, where r {q, q + 1}, of hypergraphs such that the following holds for each i {r,..., k 1}: (a H i is an i-unifor hypergraph on the vertex set V (H, (b I is an independent set in H i, (c 1 (H i O ( e(h i /v(h i, and (d e(h i Ω(p k i e(h. We shall be able to do it in such a way that in the end, there will be a set A V (H of size at ost (1 δv(h such that the reaining eleents of I (i.e., the set I \ S, where S = B k 1 B q ust all lie inside A. If r = 1, then we will siply let A be the set of non-edges of the 1-unifor hypergraph H 1 ; in this case, the upper bound on A will follow fro (c and (d. If r > 1, then we will obtain an appropriate A while trying (and failing to construct the hypergraph H r 1 using the hypergraph H r and the set B r 1. Crucially, this set A will depend solely on S, that is, if for soe pair I, I I(H our procedure generates (S, A and (S, A, respectively, and if S = S, then also A = A. This will allow us to set g 0 (I = S and f 0 (S = A The Algorith Method. For the reainder of this section, let us fix k, c, p, and H as in the stateent of Proposition 3.1. Without loss of generality, we ay assue that c 1. Let I be an independent set in H. We shall describe a procedure of choosing the sets B i I and constructing the hypergraphs H i as above. This procedure, which we shall ter the Scythe Algorith, lies at the heart of the proof of Proposition 3.1. The general strategy used in the Scythe Algorith, that of selecting a sall set S of high-degree vertices and using it to define a set A such that S I A S, dates back to the work of Kleitan and Winston [38], who used it to bound the nuber of independent sets in graphs satisfying the following local density condition: all sufficiently large vertex

14 14 sets induce subgraphs with any edges. Recently, Balogh and Saotij [10, 11] refined the ideas of Kleitan and Winston and obtained a bound on the nuber of independent sets in unifor hypergraphs satisfying a siilar local density condition. Even ore recently, Alon, Balogh, Morris and Saotij [1] used siilar ideas to bound the nuber of independent sets in alost linear 3-unifor hypergraphs satisfying a ore general density condition tered (α, B-stability, see Definition 6.1. Here, we cobine, generalize, and refine all of the above approaches and ake the work in the general setting of (F, ε-dense unifor hypergraphs. At each step of the Scythe Algorith, we shall order the vertices of a certain subhypergraph of H with respect to their degrees in that subhypergraph. For the sake of brevity and clarity of the presentation, let us ake the following definition. Definition 3.2 (Max-degree order. Given a hypergraph G, we define the ax-degree order on V (G as follows: (1 Fix an arbitrary total ordering of V (G. (2 For each j {1,..., v(g}, let u j be the axiu-degree vertex in the hypergraph G [ V (G \ {u 1,..., u j 1 } ] ; ties are broken by giving preference to vertices which coe earlier in the order chosen in (1. (3 The ax-degree order on V (G is (u 1,..., u v(g. Finally, we write W (u to denote the initial segent of the ax-degree order on V (G that ends with u, i.e., for every j, we let W (u j = {u 1,..., u j }. We reark here that the only property of the ax-degree order that will be iportant for us is that for every j {1,..., v(g}, the degree of the vertex u j in the hypergraph G[V (G \ W (u j 1 ] is at least as large as the average degree of this hypergraph. We next define the nubers i l, where 1 l i k, which will play a crucial role in the description and the analysis of the algorith. Definition 3.3. For every l [k], let k l = l(h and for all i [k 1] and l [i], let i l = ax { 2 i+1 l+1, p } i+1 l. (7 We use the nubers i l to define the following failies of sets with high degree. Definition 3.4. Given an i [k], an i-unifor hypergraph G and an l [i], let { ( } V (G Ml(G i = T : deg l G (T i l. 2 Let b = pv(h and for each i [k], let c i = (ck2 k+1 i k. Properties. The key properties that we would like the constructed hypergraph H i to possess are: (P1 H i is i-unifor and V (H i = V (H, (P2 I is an independent set in H i, (P3 l (H i i l for each l [i], (P4 e(h i c i p k i e(h.

15 INDEPENDENT SETS IN HYPERGRAPHS 15 Set H k = H and note that (P1 (P4 are vacuously satisfied for i = k. The ain step of the Scythe Algorith will be a procedure that, given H i+1 and I satisfying (P1 (P4, outputs a set B i I of cardinality at ost b, a set A i V (H with the property that I \B i A i, and a hypergraph H i satisfying (P1 (P3. Moreover, if the constructed H i does not satisfy (P4, then we have A i (1 c i v(h. Crucially, these A i and H i depend solely on B i and H i+1, that is, if on two inputs (H i+1, I and (H i+1, I, the procedure outputs the sae set B i, it also outputs the sae A i and H i. The Scythe Algorith. Given an (i + 1-unifor hypergraph H i+1 and an independent set I I(H i+1, set A (0 i+1 = H i+1 and let H (0 i be the epty hypergraph on the vertex set V (H. For j = 0,..., b 1, do the following: (1 If I V ( A (j i+1 =, then set Hi = H (0 i, A i =, and B i = {u 0,..., u j 1 } and STOP. (2 Let u j be the first vertex of I in the ax-degree order on V ( A (j i+1. (3 Let H (j+1 i be the hypergraph on the vertex set V (H defined by: { ( } H (j+1 i = H (j V (H i D : D {u j } A (j i+1. i (4 Let A (j+1 i+1 be the hypergraph on the vertex set V ( A (j i+1 \ W (uj defined by: 7 { i A (j+1 i+1 = D A (j i+1 : D W (u ( j = and T D for every T Ml i (j+1 } H i. Finally, set H i = H (b i, A i = V ( A (b i+1, and Bi = {u 0,..., u b 1 }. We shall now establish various properties of the Scythe Algorith. We begin by aking soe basic (but key observations. Lea 3.5. The following hold for every i [k 1]: (a H i is i-unifor and V (H i = V (H. (b If I I(H i+1, then I I(H i. (c B i I A i B i. (d The hypergraph H i and the set A i depend only on H i+1 and the set B i. Proof. Property (a is trivial. To see (b, siply observe that each edge of H i is of the for D \ {u} for soe D H i+1 and u I. Thus, if I contains an edge of H i, it ust also contain an edge of H i+1. To see (c, observe that for each j, u j is the first vertex of I in the ax-degree order on V ( A (j i+1 and hence W (uj I = {u j }. It follows that B i I and that I \ A i = B i. Note in particular that if A i =, then I V ( A (j i+1 = for soe j {0,..., b}, which iplies that B i = I. Finally, to prove (d, observe that all steps of the Scythe Algorith are deterinistic and that every eleent of I that we need to observe in order to define A i and H i is placed in B i. More precisely, note that while choosing the vertex u j, we only need to know the first vertex of I in the ax-degree order on V ( A (j i+1 ; the reaining vertices reain unobserved. Since we have W (u j B i = W (u j I = {u j }, 7 We ephasize that W (u j is defined relative to the ax-degree order on V (A (j i+1. l=1

16 16 this inforation can be recovered fro B i. Thus, at each step, the hypergraph H (j+1 i can be and B i, and the hypergraph A (j+1 i+1 can be recovered fro A (j i+1, H(j+1 i recovered fro H (j i and B i. Hence, a trivial inductive arguent proves that, if the algorith does not stop in step (1, for each j {0,..., b}, the hypergraphs H (j i and A (j i+1 are deterined by H i+1 and the set B i, as required. Finally, the algorith stops in step (1 if and only if B i < b. If this happens, then H i and A i are epty. We next show that the Scythe Algorith exhibits a certain consistency while generating its output. This property will be iportant in the proof of Proposition 3.1. Lea 3.6. Suppose that on inputs (H i+1, I and (H i+1, I, the Scythe Algorith outputs (A i, B i, H i and (A i, B i, H i, respectively. If B i I and B i I, then (A i, B i, H i = (A i, B i, H i. Proof. By Lea 3.5, it suffices to show that B i = B i. Let us first consider the (degenerate case when in{ B i, B i } < b. Without loss of generality, we ay assue that B i < b. This eans that, while running on (H i+1, I, the Scythe Algorith stopped in step (1. By Lea 3.5, it follows that B i = I and hence B i B i, which eans that B i < b and therefore B i = I. Hence, B i = B i, as claied. On the other hand, if B i = B i = b and B i B i, then there ust exist soe j such that u j u j. Let j be the sallest such index. Note that by the iniality of j, we have A (j i+1 = ( A (j i+1 = A. Since uj u j, one of these vertices coes earlier in the ax-degree order on V (A; without loss of generality, we ay suppose that it is u j. Since B i I, it follows that u j I and hence the Algorith, while running on the input (H i+1, I, would not pick u j in step j, a contradiction. This shows that in fact B i = B i, as required. The next lea shows that if H i+1 satisfies (P3, then so does H i. The lea follows easily fro the definitions of i l and M i l (G. Lea 3.7. If l+1 (H i+1 i+1 l+1 for soe l [i], then l(h i i l. Proof. The crucial observation is that if deg (j H (T i l i 2 (j and j [b], then all edges containing T are reoved fro A for soe T ( V (H l no ore such edges are added to H i. It follows that deg Hi (T = deg (j H i we extend H (j 1 i and hence It follows that deg H (j i to H (j i i+1 and hence (T. Moreover, when, then we only add to it sets D such that D {u j } A (j 1 i+1 H i+1 (T deg (j 1 H (T deg Hi+1 (T {u j } T +1 (H i+1. i l (H i i l 2 + l+1(h i+1 i l 2 + i+1 where the last inequality follows fro (7. l+1 i l

17 INDEPENDENT SETS IN HYPERGRAPHS 17 Next, let us establish an easy bound on the nubers i 1. Lea 3.8. i 1 c2 k p k i e(h v(h for every i {1,..., k}. Proof. To prove the lea, siply note that, by the definition of i l, for every i [k] and every l [i], i l = 2 d p k i d d+l (H for soe d {0,..., k i}. (8 One easily proves (8 by induction on k i. Intuitively, d in (8 is the nuber of ties that the first ter in the axiu in (7 is larger than the second ter when following the recursive definition of i l back to k d+l. Since l (H c p l 1 e(h v(h i 1 as required., as in the stateent of Proposition 3.1, it follows fro (8 that { 2 d p k i d cp d e(h } c 2 k k i e(h p v(h v(h, { ax 2 d p k i d d+1 (H } ax 0dk i 0dk i Finally, we show that if H i+1 satisfies (P3 and (P4, then either H i+1 also satisfies (P4 or we have A i (1 c i v(h. Recall that c i = (ck2 k+1 i k. Lea 3.9. Let i [k 1] and suppose that e(h i+1 c i+1 p k (i+1 e(h and that l (H i+1 i+1 l for every l [i + 1]. Then either p e(h i c 2 k+1 k e(h i+1 c i p k i e(h (9 or A i (1 c i v(h. Proof. If the Scythe Algorith stops in step (1, then A i = 0 and there is nothing to prove. Hence, we ay assue that steps (2 (4 are executed b ties. Note that, for each j {0,..., b 1}, we have e ( H (j+1 i ( (j e H i = dega (j (u j. (10 i+1 By the definition of the ax-deg order, the right-hand side of (10 is at least the average degree of the hypergraph Ã(j i+1, the subhypergraph of A(j i+1 induced by the set ( V ( A i+1 (j \ W (u j {u j }. Therefore, by the definition of A (j+1, we have e ( H (j+1 i Hence, if (i + 1e ( A (j+1 i+1 e(h i i+1 ( (j (i + 1e ( Ã (j e H i i+1 v ( Ã (j (i + 1e( A (j+1 i+1 v ( H. i+1 ( e Hi+1 for every j {0,..., b 1}, then b 1 j=0 (i + 1e ( A (j+1 i+1 v ( H b e(h i+1 v(h = p e(h i+1, since b = p v(h, as required. Thus, we ay assue that for soe j, e ( A (b ( (j+1 e ( H i+1 i+1 e A i+1 < i + 1. (11

18 18 Intuitively, (11 eans that while running the Scythe Algorith on H i+1 and I, any edges are reoved fro A i+1 (that is, H i+1 in step (4. This ay happen for one of the following two reasons: either any of the initial segents W (u j are long or one of the failies M i l (H i of sets with high degree in H i is large. Clai. Either or for soe l [i], b 1 W (u j 1 4 i+1 e(h i+1 1 j=0 ( M i l Hi 1 2(i + 1 i+1 e(h i+1. l Proof of clai. Recall that A (0 i+1 = H i+1 and observe that for every j {0,..., b 1}, e ( A (j ( (j+1 i ( i+1 e A i+1 W (uj 1 (H i+1 + Ml i (j+1 ( H i \ M i (j l H i l (H i+1. (12 Inequality (12 follows since in step (4 of the Scythe Algorith, we reove fro A (j i+1 ( only the edges that contain either a vertex of W (u j or a eber of Ml i (j+1 H i for soe l [i]. Thus, since l (H i+1 i+1 l for every l [i], suing (12 over all j, we get e ( b 1 (b H i+1 e(a i+1 i W (u j i+1 ( 1 + Ml i (b H i i+1 l. j=0 Since we assued that e(a (b i+1 < e( H i+1 /(i + 1, see (11, and Hi = H (b, it follows that if then as claied. j=0 l=1 b 1 W (uj i+1 1 < e(h i+1 4 ( M i l Hi i+1 l l=1 i 2(i + 1 e(h i+1, 1 2(i + 1 e(h i+1 for soe l [i], Finally, let us deal with ( the two cases iplied by the clai. In the reainder of the proof, we will show that if Ml i Hi is large for soe l [i], then e(hi is large and if b 1 j=0 W (u j is large, then A i is sall. Case 1: ( M i l Hi 1 e(h 2(i+1 i+1 i+1 for soe l [i]. l Since deg Hi (T i l /2 for every T M l( i Hi, it follows by the handshaking lea that ( 1 i M i e(h i = deg l Hi (T l (H i i l 2 ( i. (13 T ( V (H l l i

19 Recalling that i l as required. e(h i INDEPENDENT SETS IN HYPERGRAPHS 19 p i+1 l, see (7, we have e(h i+1 4(i + 1 ( i l i l i+1 l p 2 i+2 (i + 1 e(h i+1 p 2 k+1 k e(h i+1, Case 2: b 1 j=0 W (u j 1 e(h 4 i+1 i+1. 1 We clai that in this case, A i (1 c i v(h. Indeed, we have v(h A i = v ( A (0 ( (b b 1 i+1 v A i+1 = W (u j e(h i+1 4 i+1. 1 Recall that i+1 1 c2 k p k i 1 e(h v(h v(h A i pi+1 k c2 k+2 j=0 by Lea 3.8. Thus, since e(h i+1 c i+1 p k (i+1 e(h and c i+1 /(c2 k+2 c i. v(h e(h e(h i+1 c i v(h, 3.2. The proof of Proposition 3.1 and Theore 2.2. Proof of Proposition 3.1. Let k be an integer and let c be a positive constant. Furtherore, let p (0, 1 and let H be a k-unifor hypergraph that satisfy the assuptions of Proposition 3.1. Let δ = (ck2 k+1 k and b = pv(h. We will use the Scythe Algorith, described in Section 3.1, to construct a faily S and functions f 0 and g 0 as in the stateent of Proposition 3.1. We obtain the by running the following algorith (with H k = H on every independent set I I(H. We shall define f 0 soewhat iplicitly by defining a function f0 : I(H P(V (H that is constant on the set g0 1 (S for every S S. Constructing g 0 and f 0. Given an I I(H, set i = k 1 and repeat the following: (1 Apply the Scythe Algorith to H i+1 and I. Suppose that it outputs H i, A i and B i. (2 If A i (1 δv(h, then set q = i, r = i + 1 and STOP. (3 If i > 1, then set i = i 1. Otherwise, set q = r = 1 and STOP. Let I be an independent set and let us execute the above procedure (with H k = H on I. We clai that for every i {r,..., k}, the hypergraph H i satisfies properties (P1 (P4 defined in Section 3.1. This follows by induction on k i. The base of the induction, the case i = k, follows vacuously fro the definitions of c k and k l for l [k]. The inductive step follows fro Leas 3.5, 3.7, and 3.9. To see this, note that since A i > (1 δv(h (1 c i v(h for all i {r,..., k 1}, then (9 in Lea 3.9 always holds. Now, let us define g 0 (I and f0 (I. Suppose first that r > 1 and note that in this case, the algorith stopped in step (2, which eans that A q (1 δv(h; we set On the other hand, if r = 1, then we set g 0 (I = B k 1... B q and f 0 (I = A q. g 0 (I = B k 1... B 1 and f 0 (I = { v V (H 1 : {v} H 1 }.