Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence

Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence Andrzej Dudek adudek@emory.edu Andrzej Ruciński rucinski@amu.edu.pl June 21, 2008 Joanna Polcyn joaska@amu.edu.pl Abstract We study the extremal parameter Nn, m, H which is the largest number of copies of a hypergraph H that can be formed of at most n vertices and m edges. Generalizing previous work of Alon [1], Friedgut and Kahn [6] and Janson, Oleszkiewicz and the third author [10], we obtain an asymptotic formula for Nn, m, H which is strongly related to the solution α q H of a linear programming problem, called here the fractional q-independence number of H. We observe that α q H is a piecewise linear function of q and determine it explicitly for some ranges of q and some classes of H. As an application, we derive exponential bounds on the upper tail of the distribution of the number of copies of H in a random hypergraph. 1 Introduction The subject of this paper touches upon three areas of discrete mathematics: extremal combinatorics, linear programming and probability. A hypergraph is a pair V, E, where V is a finite set of vertices, while E 2 V is a family of subsets of V, called edges. For two hypergraphs F and H, let NF, H be the number of isomorphic copies of H in F, that is, NF, H = {H F : H = H}, where = represents the isomorphism equivalence relation between hypergraphs. Furthermore, let Nn, m, H = max{nf, H : v F n, e F m}, where v F and e F represent the number of vertices and edges of F, respectively. In other words, Nn, m, H is the largest number of copies of H that can be 1

constructed using at most n vertices and m edges. Note that Nn, m, H = 0 if either n < v H or m < e H. Therefore, we will be always assuming that n v H and m e H. Throughout the paper we use notation a n = Θb n to mean that cb n < a n < Cb n for some C > c > 0. Originally, there was an interest in a related problem with no restriction on the number of vertices. Let Nm, H = max{nf, H : e F m}. Alon [1] proved that in the case of graphs, that is, when each edge of H has size 2, Nm, H = Θ m αh, 1 where αh is the fractional independence number of H see below for the definition. Typically, in the literature the independence number is denoted by α and its fractional counterpart by α. Here we drop the asterisk for the ease of notation. Later, Friedgut and Kahn [6] extended Alon s result 1 to all hypergraphs H using Shearer s entropy lemma [3]. In the random context, studied in [10], it was necessary to restrict also the number of vertices. Following the ideas from [6], an asymptotic formula for Nn, m, H was obtained in [10] in the case when H is a graph. The proof in [10] extends mutatis mutandis to hypergraphs yielding an asymptotic formula for Nn, m, H, for every hypergraph H. In order to state this result, we need to introduce a parameter which is central for the entire paper. For a real number q 0 and a hypergraph H, the fractional q-independence number of H, denoted by α q H, is the optimum value of the following linear program LP = LP q, H. LP Maximize subject to: v V H x v v e x v q, for every e EH x v [0, 1], for every v V H Note that for graphs, αh = α 1 H is the ordinary fractional independence number. The name fractional independence number comes from the fact that if we restrict ourselves to x v {0, 1}, then for any optimal solution the set of vertices v with x v = 1 is a maximum size independent set in H. There are several ways one for each integer q to generalize the notion of independence number to hypergraphs. Obviously, in the fractional relaxation there is no need to restrict q to integers. Now we are ready to state our result. Theorem 1.1 For every hypergraph H, where q = log n m. Nn, m, H = Θn αqh, 2 We will see in Section 3 that 1 is a special case of Theorem 1.1. In [10], Theorem 1.1, combined with an explicit formula for α q H, was just a tool in proving almost tight bounds on the upper tail of the distribution of the 2

number X H of the copies of a graph H in a random graph Gn, p cf. Theorem 4.1 in Section 4. In the hypergraph case, due to its generality, it is not so easy anymore to apply formula 2 in the random hypergraph context. The problem is that the parameter α q H is not always expressible as an explicit function of q. The goal of this paper is to initiate a study of the parameter α q H and, as application, obtain more explicit formulae for Nn, m, H, and, consequently, some estimates of the upper tails of subhypergraph counts in random hypergraphs. More hypergraph notation and terminology We set k H = max e EH e. If e = k for each e H, we call H k-uniform. For each v V H, let d H v be the degree of v in H, namely the number of edges in H that contain v. We set H = max v V H d H v. If d H v = d for each vertex v V H, then we say that H is d-regular. A hypergraph is called regular if it is d-regular for some d. For a hypergraph H, we will use notation v H = vh = V H and e H = eh = EH. 2 The q-independence number of hypergraphs In this section we give several facts about the parameter α q H which will be useful in computing explicit formulae for Nn, m, H. The most important feature of α q H is that it is a piecewise linear function of q. Indeed, if α q H = bh+chq for some range of q, then, by Theorem 1.1, Nn, m, H = Θn bh m ch in that range, which is a relatively simple formula. Let i H be the number of isolated vertices in H. Proposition 2.1 For every hypergraph H, i α q H is a nondecreasing function of q, i H α q H v H, α 0 H = i H, and α q H = v H for all q k H, ii α q H is a concave function of q, iii α q H is a piecewise linear, continuous function of q, iv if H is a spanning subhypergraph of H then α q H α q H, v if H consists of connected components H 1,..., H s then α q H = α q H 1 + + α q H s. Proof Parts i and v follow directly from the definition of α q H. For part ii, observe that the average of the optimal solutions of LP q 1, H and LP q 2, H is a feasible solution of LP 1q 2 1 + q 2, H for all q 1 < q 2. To verify part iii, we recall Theorem 10.2 from [4] which, in our context, implies that there exist linear functions f 1,..., f l such that α q H = min{f i q : 1 i l} for all q 0 this, of course, also implies ii. For iv, notice that every feasible solution of LP q, H is a feasible solution to LP q, H. 3

Note that property v implies, in particular, that α q H = α q H 0 + i H, where H 0 is H stripped from its isolated vertices. For simplicity, we will be assuming that i H = 0. For further references we reformulate parts ii and iii of Proposition 2.1 in a more detailed version. Corollary 2.2 There exists a unique finite sequence of real numbers 0 = q 0 < q 1 < < q l = k H, and a sequence of linear functions of q such that i 0 = b 1 < < b l and c 1 > c l 0, ii for each i = 1,..., l iii for all i = 1,, l and all q > 0, f H i q = b i + c i q 3 α q H = f H i q for q [q i 1, q i ], α q H f H i q. 4 We will now determine α q H at both ends of the range 0 q k H. Fact 2.3 i For all hypergraphs H, and all 0 q 1 α q H = α 1 Hq, that is, q 1 1, b 1 = 0, and c 1 = α 1 H; ii For all k-uniform hypergraphs H and all k 1 q k α q H = kα k 1 H k 1v H + v H α k 1 H q, that is, q l 1 k 1, b l = kα k 1 H k 1v H and c l = v H α k 1 H. Proof i Let x v, v V H, be an optimal solution of LP q, H. Set y v = x v /q and note that, because no vertex is isolated, 0 x v q, and so, 0 y v 1. Moreover, v e y v 1 for each e H. Thus, y v is a feasible solution of LP 1, H with v V H y v = α q H/q. This implies that α q /q α 1 H. The reverse inequality follows from the concavity of α q. ii Let x v, v V H, be an optimal solution of LP q, H. Since q k 1, we have x v q k 1 for all v V H, because otherwise we could increase an x v to q k 1 without violating any constraints. Therefore, we may write x v = q k 1 + k qξ v, where 0 ξ v 1. Note that for each e H, v e ξ v k 1, since v e x v q. 4

Thus, ξ v is a feasible solution of LP k 1, H, and x v s are optimal if and only if the ξ v s are. Therefore α q H = x v = q k 1v H + k q v V H v V H = q k 1v H + k qα k 1 H. ξ v Our next task is to derive bounds on the difference α q H α q H and the quotient α q H/α q H. From now on we assume that H is k-uniform. Fact 2.4 For all 0 < q < q k i ii and q q α q H α q H α 1 Hq q, α q H α q H q q. Proof Let q i 1 q q i and q j 1 q q j for some, possibly equal, i and j. Then, by 4, c j q q α q H α q H c i q q, and, by Corollary 2.2i and Fact 2.3 both parts, c j c l = v H α k 1 H 1 while c i c 1 = α 1 H. To bound the quotient, note that, again by 4, α q H α q H b i + c i q b i + c i q q q. Next, we prove general bounds on α q H. First, adding in LP all inequalities together yields x v x v e H q, 5 and consequently, v V H e EH v e α q H e H q. 6 Moreover, the equality in 6 holds if every edge e H contains t e q vertices of degree 1. Indeed, then x v = { q t e d H v = 1, v e 0 d H v 2, is a feasible solution of LP q, H with the value of the objective function e H q. More bounds are given in the next result. 5

Fact 2.5 For all 0 q k v H k q α qh kα k 1 H k 1v H +v H α k 1 H q v H k qe H H. 7 Proof The first inequality follows from Fact 2.4ii and Proposition 2.1i α k = v H. The second inequality is a consequence of Fact 2.3ii combined with 4. The last inequality is equivalent to α k 1 H v H e H H. 8 To prove 8, observe that any optimal solution of LP k 1, H satisfies 1 v e 1 x v, for every e EH. Thus, e H 1 x v H 1 x v = H v H α k 1. e EH v e v V Note that in view of Fact 2.3ii, the sharper upper bound in 7 is achieved for k 1 q k. Also, for regular, nonempty k-uniform hypergraphs all bounds in 7 coincide. Indeed, if H is d-regular, d > 0, then ke H = dv H, and v H k qe H H = v H k qv H k = v H k q. In fact, the lower bound in 7 is attained for all q by a broader class of hypergraphs, including those with perfect matchings. Fact 2.6 If a k-uniform hypergraph H contains a spanning subhypergraph H such that each of the connected components of H is regular but not an isolated vertex, then, for all 0 q k, α q H = v H k q. Proof By Proposition 2.1iv, we have α q H α q H, so, in view of the lower bound in 7, all we need is that α q H = v H k q note that v H = v H. As we have just pointed out, this is true for each connected component of H, and thus, by Proposition 2.1v, it is true for H as well. We conclude this section with finding the values of α q H for two particular classes of k-uniform hypergraphs. A loose cycle L k t, where k 1 divides t, is a hypergraph with vertices v 0,..., v t 1 and edges {v 1, v 2,..., v k }, {v k, v k+1,..., v 2k 1 },..., {v t k+2, v t k+3,..., v 0, v 1 }. Fact 2.7 Let L k t be a loose k-uniform cycle of length t. Then α q L k t = { t q k 1 q k 2, k 2t 2k 1 2k 1 k 2 q k. 6

Proof We may assume that k 3 for k = 2 the loose cycles become to 2-regular graphs and are covered by Fact 2.6. The case when 0 q k 2 follows from 6 because each edge contains precisely k 2 q vertices of degree 1. Suppose that k 2 q k. Then, 7 implies α q L k k qt t t 2k 1 = k 2t 2k 1 + t 2k 1 q. The converse inequality is derived by taking the feasible solution { 1 d H v = 1, x v = q k 2 d 2 H v = 2. Fact 2.8 Let K k t 1,..., t k be the k-uniform, k-partite complete hypergraph with k-partition V = V 1 V k, where V i = t i and t 1 t k. Further, let j = 1,..., k and j 1 < q j. Then α q K k t 1,..., t k = t 1 + + t j 1 j 1t j + t j q. Proof Suppose that 0 j 1 < q j k. Let d i = d H v for every v V i. Then, d i = e H ti and d 1 d 2 d k. By 5, we have d 1 x v + d 2 x v + + d j 1 x v + d j x v e H q, v V 1 v V 2 v V j 1 v V j V k or, equivalently, d j x v e H q + d j d 1 x v + d j d 2 x v + + d j d j 1 v V 1 v V 2 v V Moreover, since x v 1 and d i t i = e H, we have d j x v e H q + d j d 1 t 1 + d j d 2 t 2 + + d j d j 1 t j 1 v V which yields, = e H q + d j t 1 + t 2 + + t j 1 j 1e H, α q K k t 1,..., t k t 1 + + t j 1 j 1t j + t j q. v V j 1 x v. 7

α k α k 1 α q f 1 q f l q α 1 1 k 1 k q Figure 1: The placement of q, α q points, for 1 q k 1. The converse inequality follows by taking the feasible solution 1 v V 1 V j 1, x v = q j + 1 v V j, 0 otherwise. In summary, we have found explicit expressions for α q H as a linear function of q for both, 0 q 1 and k 1 q k. By the concavity, in the remaining range 1 q k 1, the points q, α q H must lie in the triangle T H located above the straight line segment joining the points 1, α 1 H and k 1, α k 1 H, and, by Corollary 2.2iii, below the lines f 1 q and f l q see Figure 1. This triangle may have an empty interior, as is the case of hypergraphs satisfying the assumptions of Fact 2.6 where l = 1. Otherwise, it can be still traversed by the graph of α q H along the upper boundary, lower boundary or through the interior. Indeed, let H 3,2 be the 3-uniform hypergraph in Figure 2a. Then { 3q q < 3, 2 α q H 3,2 = 5 q + 2 q 3, 3 2 and therefore, l = 2, q 0 = 0, q 1 = 3 2 and q 2 = 3, and α q H 3,2 traverses the upper boundary of the triangle T H3,2. As we have seen in Fact 2.8, the 3-partite complete hypergraph K = K 3 3, 2, 1 has l = 3, q i = i, i = 0, 1, 2, 3 and thus α q K traverses the bottom of T K. Finally, for the disjoint union H 3,2 K we have l = 4, q 0 = 0, q 1 = 1, q 2 = 3/2, q 3 = 2 and q 4 = 3, which means that the graph of α q H 3,2 K traverses the interior of T H3,2 K. It seems to be interesting to determine, for a fixed k, the largest value of l = lh, the number of straight line segments of the function α q H. Let τk = max lh, H 8

a b Figure 2: Hypergraphs H 3,2 a and H 4,2 b. where the maximum is taken over all k-uniform hypergraphs H. We know only that τ2 = 2 cf. Fact 2.3 for k = 2. However, for k 3 it might be the case that τk =. In fact we believe that this is indeed true. Conjecture 2.9 For k 3 we have τk =. The example of H 3,2 K 3 3, 2, 1 shows that τ3 4. Moreover, it suggests the right approach to the problem. Namely, in order to obtain a hypergraph H with large lh, it is sufficient to construct a family of hypergraphs H 1, H 2,... with distinct break points of the function α q H i, i = 1, 2,..., and then take their vertex-disjoint union i H i. Recently, Matoušek [11] found two further such hypergraphs yielding τ3 6. The construction of H 3,2 can be easily generalized for arbitrary k 3 as follows. For a given k 3 and 0 j k let H k,j be the k-uniform hypergraph with the edge set {e 1,..., e k+1 } satisfying, for some S V H k,j of size k j, the following conditions: i e i1 e i2 = S, for any 1 i 1 < i 2 k, ii e k+1 S =, iii e i e k+1 = 1, for any 1 i k. In Figure 2 there are two examples of graphs H k,j. It is easy to check that { kq q < j 1 + j 1 k 1 α q H k,j =, k 1j 1 + 2q q otherwise. k Let K = K k k,..., 1 be the k-partite complete hypergraph with parts of size k,..., 1. Then one can check that l k 1 j=2 H k,j K = 2k 2, implying τk 2k 2. 3 Subhypergraph counts in extremal hypergraphs As we mentioned in the introduction, the proof of Theorem 1.1 goes along the lines of the proof of its special case for graphs given in [10], and therefore we omit it here. However, to give the reader some indication how the parameter α q H becomes involved in the formula for Nn, m, H, we present the proof of the lower bound, that is, we show that Nn, m, H cn αqh for some c > 0. 9

Proof of Theorem 1.1 the lower bound Let n, m and H be given, and let q = log n m. Let x v, v V, be an optimal solution of LP q, H, and y v = x v log n. We construct a graph F by blowing up each vertex v V H to a set of size n v = c H expy v, where c H = v 1 H e 1 H. In other words, V F = v V H V v, where V v s are pairwise disjoint sets, V v = n v, and each k-tuple of sets V v1, V v2,..., V vk spans the complete k-uniform, k-partite hypergraph if {v 1, v 2,..., v k } EH, and it spans the empty hypergraph otherwise. Since y v log n and v e y v log m, for every v V H and e EH, we have v F n and e F m. Moreover, NF, H n v c v H H v V H exp log n v V H x v = c v H H n αqh. Note that if H 0 denotes H stripped from its isolated vertices and i H = V H \ V H 0 then Nn, m, H = Θ Nn, m, H 0 n i H. Therefore, we will be assuming that i H = 0, that is, H 0 = H. Note also that it follows from Theorem 1.1 that the order of magnitude of Nn, m, H is not affected by a linear change in n, that is N Cn, m, H = ΘNn, m, H. With the help of Fact 2.4i one can show that the same is true with respect to m. This observation will be utilized in the next section. Corollary 3.1 For all k-uniform hypergraphs H and for every C > 1, Nn, Cm, H = ΘNn, m, H. Proof Given C > 1, let q = logcm/ log n = q + log C/ log n. Then, q > q, and consequently, Fact 2.4 yields n α q H n αqh n q q = Cn αqh and n α q H n αqh n α 1Hq q = C α 1H n αqh. Our knowledge about α q H can be used to derive explicit formulae for Nn, m, H and Nm, H. First note that, by Theorem 1.1, for all hypergraphs H and all n q i 1 m n q i, Nn, m, H = Θn b i m c i, 9 where q i, b i and c i were defined in 3. In particular, if m n, then, using Fact 2.3, Nn, m, H = Θn α 1Hq = Θm α 1H and thus Nm, H = Nk H m, m, H = Θm α 1H, 10 10

which is the result from [6]. If H is k-uniform, that is, k H = k, then at the other end of the range, that is, for n k 1 m n k, we have again by Fact 2.3 Nn, m, H = Θ n kα k 1H k 1v H m v H α k 1 H. In the special case of graphs we thus recover a result from [10]. Theorem 3.2 [10] For every graph H { Θm αh if m n, Nn, m, H = Θn 2αH v H m vh αh if n m n 2. Turning to specific classes of graphs, by Fact 2.6 we know that if a k-uniform hypergraph H contains a spanning subhypergraph H such that each of the connected components of H is regular but not an isolated vertex, then for m n k, Nn, m, H = Θ m v H/k. 11 In particular, the above formula remains true for all k-uniform hypergraphs H containing perfect matchings, as well as for all regular k-uniform hypergraphs H, including complete hypergraphs. Moreover, as consequences of Facts 2.7 and 2.8, we know that Nn, m, L k t = and, for n j 1 < m n j, Θ Θ m t k 1 n tk 2 t 2k 1 m 2k 1 m n k 2, n k 2 m n k, Nn, m, K k t 1,..., t k = Θ n t 1+ +t j 1 j 1t j m t j. 12 4 Subhypergraph counts in random hypergraphs Let G k n, p, k 2, denote the random k-uniform hypergraph with n labeled vertices and such that each of the n k possible edges exists with probability p, independently of all other edges. Let X G be the number of isomorphic copies of G contained in G k n, p and µ G = EX G. In this section we reserve the letter H to denote subhypergraphs of G. In the case of graphs the distribution of X G has been studied extensively since the pioneering paper by Erdős and Rényi [5]. A general threshold for {X G > 0} was established by Bollobás [2] at p = n 1/m G, where e H m G = max. 13 H G v H Next, it was shown that the lower tail of the distribution of X G decays exponentially in the expectation of the least expected subgraph of G, see [8] and [7]. Namely, 11

let Ψ H = n v H p e H, which is roughly the expected number of copies of H in Gn, p. Then, for all ε 0, 1], with c ε > 0 depending on H and ε, P X G 1 εµ G exp c ε min Ψ H. H G, e H >0 This is best possible, provided p stays away from 1, as by the FKG inequality, log PX G = 0 log PX H = 0 = OΨ H for every H G see [9] for details. Finally, in [10], almost as tight bounds were found for the upper tail of X G, that is, for P X G 1 + εµ G. All these results are easily extendable to k-uniform hypergraphs. Here we focus on a generalization of the latter one. Let G be a k-uniform hypergraph and for any integer n and 0 < p < 1, let Ψ G = n v G p e G. Let K r k denote the complete k-uniform hypergraph on r vertices. Finally, let M G = M G n, p = max { m n k : H G } Nn, m, H ΨH 14 for p n k and 1 otherwise. A reason for the special definition in the extreme case p < n k when Ψ k K = n k p < 1, is to prevent M G = 0. Since Nn, 1, H = 0 k unless e H 1, it is easily checked that 1 M G n k. Now we can formulate a result, which is a straightforward generalization of Theorem 1.2 in [10]. Theorem 4.1 For every k-uniform hypergraph G and for every ε > 0 there exist constants cε, G > 0 and Cε, G > 0 such that for all n v G and p 0, 1 PX G 1 + εµ G exp { cε, GM G n, p}, 15 and, provided 1 + εµ G NK n k, G, PX G 1 + εµ G exp { Cε, GM G n, p log1/p}. We omit the complete proof and present only one part which points to the connection between the upper tail of X G and the extremal parameter Nn, m, H. Sketch of proof of 15 Note that by Markov s inequality, for every positive integer m we have EX m G PX G 1 + εµ G. 16 1 + ε m µ m G In order to turn 16 into an exponential bound, it will be enough to show that EXG m 1 + ε m µ m 2 G, 17 for some large m. Let G 1, G 2,..., G k NK be all copies of G in the complete n,g hypergraph K n k. Moreover, let I Gi be an indicator random variable corresponding 12

to G i. Then, EXG m = EI Gi1 I Gim i 1,...,i m = p egi1 Gim i 1,...,i m = i 1,...,i m 1 p egi1 Gim 1 = EX m 1 G where F = G i1 G im 1. Furthermore, p e G ef G i = i i m p eg egi1 Gim 1 Gim i m p eg ef Gim, 18 i:f G i = µ G + µ G + p eg + H G,e H >0 H G,e H >0 i:f G i p e G ef G i i:f G i =H p eg eh NF, Hn v G v H p e G e H. 19 The last inequality follows because there are at most NF, H copies of H in F, and every such copy can be extended to at most n v G v H copies of G. Since v F n and e F m 1e G, the definition of Nn, m, H and Corollary 3.1 imply, that NF, H Nn, m 1e G, H = ΘNn, m, H. 20 Consequently, 18, 19 and 20 yield that EXG m EX m 1 G µ G + ΘNn, m, Hn v G v H p e G e H = EX m 1 G µ G Hence, by induction on m, EX m G µ m G H G,e H >0 1 + H G,e H >0 1 + H G,e H >0 Nn, m, H Θ. Ψ H m 1 Nn, m, H Θ. Now, by the definition of M G and in view of the proof of Corollary 3.1, one can choose a constant c such that Nn, cmg, H Θ < ε 2, H G,e H >0 13 Ψ H Ψ H

and consequently, 17 holds with m = cm G. To apply Theorem 4.1, we need to estimate M G, and for this, in turn, it is crucial to have a fair estimate of the extremal parameter Nn, m, H for every graph H G. In the case of graphs, Theorem 3.2 allows to completely determine the asymptotic order of M G. Theorem 4.2 [10] For every graph G Θ1 if p n 1/m G, M G n, p = Θ min H G Ψ 1/αH H if n 1/m G p n 1/ G, Θn 2 p G if p n 1/ G. Unfortunately, in the case of k-uniform hypergraphs, k 3, we can do it only for those H and for those ranges of p for which the value of α q H has been explicitly determined as a function of q, and, consequently, the asymptotic range of Nn, m, H is known as a function of m and n. We will first illustrate the technique of estimating M G by two examples regular k-uniform hypergraphs and loose k-cycles, before turning to the general case. Owing to Corollary 3.1, in determining the order of magnitude of M G it is sufficient to find an m for which Nn, m, H = OΨ H for all H G, and Nn, m, H 0 = ΘΨ H0 for some H 0 G. Note that if p < n 1/m G and H G satisfies m G = e H /v H cf. 13, then and thus, M G = Θ1. Ψ H = n v H p e H < n v H n 1 m G e H = n v H n v H eh e H = 1, Proposition 4.3 If G is a d-regular k-uniform hypergraph and p n 1/m G = n k/d, then M G n, p = Θ n k p d. Proof Set m = n k p d and recall that q = log n m. Theorem 1.1 together with the upper bound in 7 yields that, for every H G, Nn, m, H = Θ n αqh = O n v H k qe H / H = O n v H ke H / H n qe H/ H = O n v H ke H / H m e H/ H = O n v H ke H / H n ke H/ H p de H/ H = O n v H p e H = O Ψ H. On the other hand, with H 0 = G, by 11, Nn, m, H 0 = Θ m v H 0 /k = Θ n v H 0 p dv H0 /k = Θ n v H 0 p e H0 = Θ ΨH0. Note that for the loose k-cycle G = L k t we have m G = 1 k 1. 14

Proposition 4.4 For the loose k-cycle G = L k t we have Θ1 if p < n 1 k, M G n, p = Θ n k 1 p if n 1 k p < n 1, Θn k p 2 if p n 1. Proof Let p 1/n and set m = n k 1 p. Then m n k 2 and q = log n m k 2. Note that for each H G, by 6, α q H = e H q. Moreover, m G = 1, that is, k 1 e H /v H m G = 1. Hence, k 1 Nn, m, H = Θ m e H = Θ n k 1e H p e H = O n v H p e H = OΨ H, while Nn, m, G = Θ m t k 1 = Θ n t p t k 1 = ΘΨ G. For p 1/n, set m = n k p 2 n k 2. Then, n k q = p 2, and so, by Theorem 1.1 and the rightmost upper bound in 7, Nn, m, H = Θ n αqh = O n v H k qe H / H = O n v H p 2e H/ H = O ΨH, where the last inequality follows from the fact that H 2. On the other hand, with H 0 = G, by 12, Nn, m, H 0 = Θ n k 2t t 2k 1 m 2k 1 = ΘΨ H0. For a general k-uniform hypergraph G, by Corollary 2.2 applied simultaneously to all subhypergraphs H G, there exists a finite sequence q 0 < q 1 < < q r, such that for each i = 1,..., r, each subhypergraph H G, and for all q i 1 q q i we have α q H = fi H q = b i H + c i Hq, for some b 1 H b r H and c 1 H c r H. Note that r H G lh, q r 1 k 1 and q r = k. Set also f0 H q = 0 for convenience. For i = 0, 1,..., r 1, define e H s i G = s i = max H G v H fi H q i and p i = n 1/s i, i = 0,..., r 1, and p r = 1. Theorem 4.5 For all k-uniform hypergraphs G M G n, p = Θ min H G n b ih/c i H Ψ 1/c ih H for p i 1 p p i, i = 1,..., r. 15

Proof Fix G and i, and let m = min H G n b ih/c i H Ψ 1/c ih H. Further, let H 0 and H 1 be subhypergraphs of G achieving, respectively, the maximum in s i G and the minimum in m. We will first show that n q i 1 m n q i. Indeed, and m n b ih 0 /c i H 0 Ψ 1/c ih 0 H 0 m = n b ih 1 /c i H 1 Ψ 1/c ih 1 H 1 b n i H 0 +v H0 e H 0 /c s i H 0 i = n q i b n i H 1 +v H1 e H 1 /c s i H 1 i 1 n q i 1, because, by continuity, f H i q i 1 = f H i 1q i 1. Hence, by 9 and the definition of m, for all H G, Nn, m, H = Θ n b ih m c ih = OΨ H, while Nn, m, H 1 = ΘΨ H1, what was to be proved. e Observe that s 0 = m G and s 1 = max H H G. Moreover, since q v H α 1 H r 1 k 1, 8 yields that s r 1 G. Thus, in particular, Θ1 if p n 1/m G, M G n, p = Θ min H G Ψ 1/α 1H H if n 1/m G p n 1/s 1, Θn k p G if p n 1/ G. It would be nice to find similar explicit formulae for M G n, p in the range n 1/s 1 p n 1/ G. This boils down, however, to finding explicit formulae for Nn, m, G in the corresponding range n m n k 1, which, in turn, depends on the more explicit determination of α q H for all 1 < q < k 1. 5 Acknowledgments The authors would like to thank the referee for his or her careful reading. References [1] N. Alon, On the number of subgraphs of prescribed type of graphs with a given number of edges, Israel J. Math. 38 1981, no. 1 2, 116 130. [2] B. Bollobás, Threshold functions for small subgraphs, Math. Proc. Cambridge Philos. Soc. 90 1981, no. 2, 197 206. [3] F. Chung, R. Graham, P. Frankl & J. Shearer, Some intersection theorems for ordered sets and graphs, J. Combin. Theory Ser. A 43 1986, no. 1, 23 37. [4] V. Chvátal, Linear Programming, W. H. Freeman and Company, New York, 1983. 16

[5] P. Erdős & A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 1960, 17 61. [6] E. Friedgut & J. Kahn, On the number of copies of one hypergraph in another, Israel J. Math. 105 1998, 251-256. [7] S. Janson, Poisson approximation for large deviations, Random Structures Algorithms 1 1990, 221 230. [8] S. Janson, T. Luczak & A. Ruciński, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, Random Graphs 87 Proceedings, Poznań 1987, eds. M. Karoński, J. Jaworski & A. Ruciński, Wiley, Chichester, 1990, pp. 73 87. [9] S. Janson, T. Luczak & A. Ruciński, Random Graphs, John Wiley and Sons, New York, 2000. [10] S. Janson, K. Oleszkiewicz & A. Ruciński, Upper tails for subgraph counts in random graphs, Israel J. Math. 142 2004, 61-92. [11] J. Matoušek, private communication, 2008. 17