ARITHMETIC PROGRESSIONS OF LENGTH THREE IN SUBSETS OF A RANDOM SET

YOSHIHARU KOHAYAKAWA†, TOMASZ ŁUCZAK‡ and VOJTĚCH RÖDL*

Abstract. For integers 1 ≤ M ≤ n, let R(n, M) denote the uniform probability space which consists of all the M-element subsets of [n] = {0, 1, …, n − 1}. It is shown that for every α > 0 there exists a constant C such that if M = M(n) ≥ C\sqrt{n} then, with probability tending to 1 as n → ∞, the random set R ∈ R(n, M) has the property that any subset of R with at least α|R| elements contains a 3-term arithmetic progression. In particular, this result implies that for every α > 0 there exist ‘sparse’ sets S ⊆ [n] with the property that every subset of S with at least α|S| elements contains an arithmetic progression of length three.

1991 Mathematics subject classifications: 11B25, 05D99, 11B05, 11K99

Key words: Szemerédi’s theorem, arithmetic progressions, combinatorial number theory, regularity lemma, random sets of integers

§0. Introduction

In 1936 Erdős and Turán [ET 36] asked whether for every natural number k and every positive constant α, every subset A of [n] = {0, 1, …, n − 1} with at least αn elements contains a k-term arithmetic progression provided n is sufficiently large with respect to α and k. This conjecture was resolved by Roth [Ro 53] for k = 3, whereas for general k it was settled in the affirmative by the outstanding theorem of Szemerédi [Sz 75]. A few years after Szemerédi’s paper was published, an entirely different proof of this result, based on ergodic theory, was given by Furstenberg [Fu 77]. Since then, the main open problem concerning the original question of Erdős and Turán has been to find better lower bounds for the size of A that guarantee the existence of arithmetic progressions of length k in A. Unfortunately, not much has been accomplished for k ≥ 4. The explicit estimates that follow from Szemerédi’s original proof are very poor and Furstenberg’s approach does not provide such bounds at all. The case when k = 3 is much better understood. Roth’s original argument implies that it is enough to assume that |A| ≥ n/ log log n and the best lower bound to date has been given independently by Heath-Brown and Szemerédi (see [H-B 87]), who showed that for some absolute constant c > 0 every subset of [n] with

† Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508– 900 São Paulo, SP, Brazil. Research partially supported by FAPESP (Proc. 93/0603–1) and by CNPq (Proc. 300334/93–1 and ProTeM-CC-II Project ProComb).

‡ Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322. On leave from Mathematical Institute of the Polish Academy of Sciences, and Adam Mickiewicz University, Poznań, Poland.

* Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322. Research partially supported by NSF grant DMS–9401559.

at least n/(log n)^{c} elements contains an arithmetic progression of length three, provided n is sufficiently large.

In this paper we approach a related problem. Namely, here we are interested in the existence of a ‘small’ and ‘sparse’ set R ⊆ [n] with the property that every subset A ⊆ R that contains a fixed positive fraction of the elements of R contains also a 3-term arithmetic progression. The measure of sparseness here should be so that it reflects the fact that R is locally poor in 3-term arithmetic progressions. Clearly, a natural candidate for such a set R is an M-element set R_{M} uniformly selected from all the M-element subsets of [n], where 1 ≤ M = M(n) ≤ n is to be chosen suitably. Our main result here confirms this appealing, intuitive idea.

For integers 1 ≤ M ≤ n, let R(n, M) be the probability space of all the M-element subsets of [n] equipped with the uniform measure. In the sequel, given 0 < α ≤ 1 and a set R ⊆ [n], let us write R →_{α} 3 if any A ⊆ R with |A| ≥ α|R| contains a 3-term arithmetic progression. Our main result may then be stated as follows.

Theorem 1. For every constant 0 < α ≤ 1, there exists a constant C = C(α) such that if C\sqrt{n} ≤ M = M(n) ≤ n then the probability that R_{M} ∈ R(n, M) satisfies R_{M} →_{α} 3 tends to 1 as n → ∞.

From Theorem 1, it is easy to deduce the following analogous result for the random sets R_{p} ⊆ [n] (0 ≤ p = p(n) ≤ 1) whose elements are chosen from [n] independently with probability p. Thus, if we write R(n, p) for the probability space of such R_{p}, then for a given set R ⊆ [n] the probability that R_{p} = R is p^{|R|}(1 − p)^{n−|R|}.

Theorem 2. For every constant 0 < α ≤ 1, there exists a constant C = C(α) such that if C/\sqrt{n} ≤ p = p(n) ≤ 1 then the probability that R_{p} ∈ R(n, M) satisfies R_{p} →_{α} 3 tends to 1 as n → ∞.

Note that Theorems 1 and 2 are, in a way, close to being best possible: if M = M(n) = ⌊ε\sqrt{n}⌋ for some fixed ε > 0 then the number of 3-term arithmetic progressions in R_{M} ∈ R(n, M) is, with large probability, smaller than 2ε^{2}|R_{M}|, and hence all of them may be destroyed by deleting at most 2ε^{2}|R_{M}| elements from R_{M}; in other words, with large probability the relation R_{M} →_{α} 3 does not hold for α = 1 − 2ε^{2}. Clearly, a similar phenomenon happens for R_{p} with p = p(n) = ε/\sqrt{n}.

Our results above immediately imply the existence of ‘sparse’ sets S = S_{α} such that S →_{α} 3 for any fixed 0 < α ≤ 1. The following result makes this assertion precise.

Corollary 3. Suppose that s = s(n) = o(n^{1/8}) and g = g(n) = o(log n) as n → ∞. Then, for every fixed α > 0, there are constants C and N such that for every n ≥ N there exists S ⊆ [n] satisfying S →_{α} 3 for which the following three conditions hold.

(i) For every k ≥ 0 and ℓ ≥ 1 the set {k, k + ℓ, …, k + sℓ} contains at most three elements of S, and therefore, in particular, S contains no 4-term arithmetic progression.

(ii) Every set \{k, k + ℓ, …, k + mℓ\} with k ≥ 0, ℓ ≥ 1, and m ≥ \sqrt{n} log n contains

at most Cm/\sqrt{n} elements of S. (iii) If F = F(S) is the 3-uniform hypergraph on the vertex set S whose hyperedges

are the 3-term arithmetic progressions contained in S, then F has no cycle of length smaller than g.

In words, conditions (i) and (ii) above say that the set S intersects any arithmetic progression in a small number of elements. In particular, S contains no 4-term arithmetic progressions. Condition (iii) is more combinatorial in nature, and says that the 3-term arithmetic progressions contained in S locally form a tree-like structure, which makes the property S →_{α} 3 somewhat surprising.

Let us remark that the following extension of Szemerédi’s theorem related to Corollary 3 was proved in [Rö 90], thereby settling a problem raised by Spencer [Sp 75]. Let k, g ≥ 3 be fixed integers and 0 < α ≤ 1 be a fixed real. Theorem 4.3 in [Rö 90] asserts that then, for any large enough n, there exists a k-uniform hypergraph F on [n], all of whose hyperedges are k-term arithmetic progressions, such that F contains no cycle of length smaller than g but each subset A ⊆ [n] with |A| ≥ αn contains a hyperedge of F. For other problems and results in this direction, see Graham and Nešetřil [GN 86], Nešetřil and Rödl [NR 87] and Prömel and Voigt [PV 88]. Note that Corollary 3 strengthens the above result of [Rö 90] in the case in which k = 3.

The proof of Theorem 1 is unfortunately quite long. In the next section we describe our general approach, stressing the main ideas involved and ignoring several quite technical parts. We hope that the outline of our method presented there will be of some use in following the actual proof. The organisation of the paper is also discussed in the next section.

§1. Outline of the Method of Proof

The main lemma in the proof of Theorem 1 is Lemma 19. In essence, what this lemma says is quite simple. Assume C\sqrt{n} ≤ M′ = M′(n) ≤ n for some large C > 0. Disregarding some technicalities, Lemma 19 states the following: if we condition on our set R_{M′} ∈ R(n, M′) satisfying a certain ‘sparseness’ condition, the probability that R_{M′} fails to contain an arithmetic progression of length three is at most exp{−cM′}, where we may make c arbitrarily large by picking C appropriately large.

Theorem 1 is shown to follow from Lemma 19 in two steps. Suppose C is a large constant with respect to a given fixed α > 0 and M = M(n) ≤ α^{−1}M′. We aim at showing that R_{M} →_{α} 3 with probability approaching 1. Our first step consists of a quick calculation based on Lemma 19 to deduce that a typical random set R_{M} ∈ R(n, M) will not contain a ‘sparse’ M′-element subset R′ that is free of 3-term arithmetic progressions.

Our second step is then to show that our ‘sparseness’ condition is weak enough for every M′-element subset R′ of a typical R_{M} ∈ R(n, M) to be sparse. Hence Theorem 1 follows.

Thus, all our efforts go into proving Lemma 19. An important tool in the proof will be a version of Szemerédi’s regularity lemma [Sz 78]. As is well known, this is an important graph theoretical component of Szemerédi’s proof of his theorem on arithmetic progressions. It turns out that it is most convenient to phrase our arguments below in terms of graphs as well. Following an idea of Ruzsa and Szemerédi [RSz 78] (see also Erdős, Frankl and Rödl [EFR 86] or Graham and Rödl [GR 87]), for every subset R′ of [n] we construct a graph G(n, R′) that, roughly speaking, has the property that it contains a triangle (more precisely, a ‘spontaneous’ triangle) if and only if R′ contains a 3-term arithmetic progression (more precisely, an ‘arithmetic triple’). Lemma 19 is in fact stated in terms of sparse graphs and spontaneous triangles, and it asserts that sparse graphs free of such triangles are extremely rare.

Unfortunately, the proof of Lemma 19 is quite complex, and we shall not attempt to give a non-technical outline of it here. Probably any such sketch would fail to be of much help. Nonetheless, we remark that the argument below is divided into several steps, which are, to a large extent, independent from one another, and perhaps of some interest in their own right.

The organization of our paper is as follows. In §2 we introduce the notions of regularity, uniformity and sparseness of graphs, and state a version of Szemerédi’s regularity lemma for suitably sparse graphs together with a few related results. We start §3 with an analogue of a theorem of Ruzsa and Szemerédi [RSz 78] for sparse graphs (cf. Theorem 8 and Lemma 9), and then give an important but rather technical lemma, Lemma 10, concerning the existence of certain structures we call ‘flowers’ in edge-coloured sparse graphs. It is in proving Lemma 10 that we shall make use of Szemerédi’s regularity lemma in the form given in §2.

One of the main probabilistic ingredients in the proof of Lemma 19 is given in §4. Roughly speaking, we show in Lemma 11 that a random induced subgraph of a bipartite uniform graph contains with very large probability a fair number of edges. In §5 we give a simple sufficient condition for a regular bipartite graph to be uniform. In §6 we relate our graph theoretical results of the previous sections to subsets of [n]: we define the ‘difference graph’ G_{R} = G_{R}(n) for any given subset R of [n] and, using a result from §5, show that if R is a random set of suitably large expected size, then its difference graph G_{R} is uniform with large probability. Finally, the statement and proof of our main lemma, Lemma 19, and the proof of Theorem 1 are given in §7, together with a sketch of the proof of Corollary 3.

§2. Uniform Graphs and Szemerédi’s Lemma

Let G be a graph with vertex set V = V (G) and edge set E(G). Write |G| = |V (G)| = n for the order of G, and e(G) = |E(G)| for its size |E(G)|. Furthermore, let U, W ⊆ V be a pair of disjoint, non-empty subsets of G, let E_{G}(U, W) denote the set of edges of G that have one end-vertex in U and the other in W, and set e(U, W) = e_{G}(U, W) = |E_{G}(U, W)|. The density d_{G}(U, W) of the pair (U, W) is defined by

d_{G}(U, W) = \frac{e_{G}(U, W)}{|U||W |}/\frac{e(G)}{n^{2}} .

For 0 < ε ≤ 1, we say that the pair (U, W) is (ε, G)-regular, or simply ε-regular, if for all U′ ⊆ U, W′ ⊆ W with |U′| ≥ ε|U| and |W′| ≥ ε|W | we have

|d_{G}(U, W) − d_{G}(U′, W′)| ≤ ε .

We say that an ℓ-partite graph G is (ℓ, ε)-uniform, or just ε-uniform, if all the \binom{ℓ}{2} pairs of distinct vertex classes of the ℓ-partition of G are ε-regular.

We say that a partition Π = (V^{i})_{0}^{k} of the vertex set V = V (G) of G is (ε, k)-equitable if |V^{0}| ≤ εn and |V^{1}| = … = |V^{k}|. We refer to V^{0} as the exceptional class of Π. We say that the (ε, k)-equitable partition Π is a subpartition of a partition Π′ = (W^{j})_{0}^{s} of V if each V^{i} with 1 ≤ i ≤ k, that is every non-exceptional class of Π, is contained in some member of the partition Π′. For Π to be a subpartition of Π′ in the case in which Π′ is an equitable partition as well, we require every non-exceptional class of Π to be contained in some non-exceptional class of Π′. We say that an (ε, k)-equitable partition Π = (V^{i})_{0}^{k} is (ε, k; G)-regular, or simply (ε, G)-regular, if at most ε\binom{k}{2} pairs (V^{i}, V^{j}) with 1 ≤ i < j ≤ k are not (ε, G)-regular.

Finally, for a given b > 2 and 0 < η ≤ 1, we say that G is (b, η)-sparse if, for every disjoint pair of sets U, W ⊆ V such that |U|, |W | ≥ ηn, we have d_{G}(U, W) ≤ b. Thus, roughly speaking, a graph is (b, η)-sparse if all of its large induced subgraphs are not much denser that the graph itself. We can now state our extension of Szemerédi’s lemma for (b, η)-sparse graphs.

Lemma 4. For any given ε > 0, b > 2, k_{0} ≥ 1 and s ≥ 1, there are constants η = η(ε, b, k_{0}, s) > 0 and K_{0} = K_{0}(ε, b, k_{0}, s) ≥ k_{0} that depend only on ε, b, k_{0}, and s for which the following holds. For every (b, η)-sparse graph G and every partition (W^{j})_{0}^{s} of the vertex set of G, there exists an (ε, G)-regular (ε, k)-equitable subpartition of (W^{j})_{0}^{s} with k_{0} ≤ k ≤ K_{0}.

The proof of Lemma 4 goes along the same lines as the proof of Szemerédi’s original result [Sz 78], and hence we omit it here. As a matter of fact, in order to prove Lemma 4, one may just rewrite the argument from [Sz 78] putting the ‘scaled’ density d_{G}(U, W) (or, more precisely, d_{G}(U, W)/b) in place of Szemerédi’s original density d(U, W) = e_{G}(U, W)/|U||W |.

From Lemma 4 it immediately follows that one can uniformly partition any fixed number of graphs at the same time. Let G_{1}, …, G_{m} be a sequence of graphs defined on the same vertex set V, and let G = ⋃_{i=1}^{m} G_{i} be their union, i.e., the graph on V with the edge set ⋃_{i=1}^{m} E(G_{i}). We say that G_{1}, …, G_{m} is (b, η)-sparse if G is (b, η)-sparse. Furthermore, an (ε, k)-equitable partition Π = (V^{i})_{0}^{k} is (ε, k; G_{1}, …, G_{m})-regular if it is (ε, k; G_{i})-regular for all 1 ≤ i ≤ m.

One can easily deduce from Lemma 4 that every (b, η)-sparse sequence of graphs G_{1}, …, G_{m} admits an (ε, k; G_{1}, …, G_{m})-regular partition, provided η is small enough with respect to ε, b and 1/k. Roughly speaking, we first choose a rapidly decreasing sequence ε = ε_{m} ≥ ε_{m−1} ≥ ⋯ ≥ ε_{1} > 0 of constants, and then invoke Lemma 4 in turn to define a sequence Π_{1}, …, Π_{m} of finer and finer partitions of V. To be precise, Π_{i} is required to be a subpartition of Π_{i−1} for all 1 < i ≤ m, and Π_{i} (1 ≤ i ≤ m) is required to be an (ε_{i}, G_{i})-regular partition of V with a ‘small’ number of classes (as small as Lemma 4 can guarantee). Choosing the ε_{i} (1 ≤ i ≤ m) carefully enough, the final partition Π_{m} of our sequence Π_{1}, …, Π_{m} will be the (ε, k; G_{1}, …, G_{m})-regular partition we seek.

It will be important later that, for any 1 < i ≤ m, the partition Π_{i} above may be determined solely from Π_{i−1} and G_{i}. In other words, the graphs G_{i+1}, …, G_{m} play no rôle in the definition of Π_{i}.

We now make the above informal discussion precise. Thus, let k_{0} and m ≥ 3 be natural numbers, let \underline{ε} be a sequence of numbers ε_{1}, …, ε_{m} such that 0 < ε_{1} ≤ ⋯ ≤ ε_{m} < 1 and let G_{1}, …, G_{m} be graphs with the same vertex set V, where |V | ≥ k_{0}. In the definition below, we shall assume that the set of all equitable partitions of V have been given a fixed ordering, say ≺. The (\underline{ε}, k_{0})-canonical sequence of partitions Π̃_{1}, …, Π̃_{m} for G_{1}, …, G_{m} is defined recursively in the following way:

(i) Among all the (ε_{1}, k; G_{1})-regular partitions which minimize k ≥ k_{0}, let Π̃_{1} be the first one according to ≺.

(ii) Assume that 2 ≤ i ≤ m and that the partition Π̃_{i−1} has already been defined. Then we let Π̃_{i} be the ≺-first (ε_{i}, k; G_{1}, …, G_{i})-regular subpartition of Π̃_{i−1} which minimizes k.

Note that any partition of the vertices of G into singletons is (ε, k; G)-regular for k = |V | and every ε > 0. Thus, for each sequence G_{1}, …, G_{m} as above, an (ε, k_{0})-canonical sequence of partitions does exist and is of course unique by definition.

Now we can use Lemma 4 to deduce that for (b, η)-sparse sequences of graphs the sizes of the partitions in the associated canonical partition can be uniformly bounded from above.

Lemma 5. For every ε > 0, b > 2, k_{0} ≥ 1 and m ≥ 1, there are constants η = η(ε, b, k_{0}, m) > 0 and K_{0} = K_{0}(ε, b, k_{0}, m) ≥ k_{0} and a sequence \underline{ε} = \underline{ε}(m, b, ε, k_{0}) = (ε_{1}, …, ε_{m}) with 0 < ε_{1} ≤ ⋯ ≤ ε_{m} = ε such that η, K_{0} and \underline{ε} depend only on ε, b, k_{0} and m and the following holds. For every (b, η)-sparse sequence of graphs G_{1}, …, G_{m}, the (\underline{ε}, k_{0})-canonical sequence of partitions Π̃_{1}, …, Π̃_{m} associated to G_{1}, …, G_{m} only contains partitions of sizes bounded by K_{0} + 1. In fact, we have

k_{0} + 1 ≤ |Π̃_{1}| ≤ ⋯ ≤ |Π̃_{m}| ≤ K_{0} + 1.

Note that \underline{ε} in the lemma above depends only on m, b, ε, and k_{0}. In fact, throughout the paper we shall assume that, for any given m, b, ε, and k_{0} as in Lemma 5, we have a fixed vector \underline{ε} = \underline{ε}(m, b, ε, k_{0}) as in that lemma associated with this choice of m, b, ε, and k_{0}.

Let us conclude this section with two simple observations. The definition of ε-regularity deals with the distribution of edges between ‘large’ sets. Nonetheless, it turns out that each ε-uniform ℓ-partite graph G contains a large 3ε-uniform ℓ-partite subgraph Ḡ such that each vertex of Ḡ has a fairly large degree. In fact, more is true as shows the following lemma.

Fact 6. Suppose ℓ ≥ 2 and 0 < ε < 1/5ℓ. Let G be an ε-uniform (ℓ+1)-partite graph with (ℓ + 1)-partition V (G) = V_{0} ∪ ⋯ ∪ V_{ℓ}. Then there exist subsets V̄_{i} ⊆ V_{i} (1 ≤ i ≤ ℓ) such that, for every 1 ≤ i ≤ ℓ, we have |V̄_{i}| ≥ (1 − ℓε)|V_{i}| and for every vertex v ∈ V̄_{i} we have

|N_{G}(v) ∩ V̄_{j}| ≥ (1 − 2ℓε)e_{G}(V_{i}, V_{j})/|V_{j}|

for every j ≠ i (1 ≤ j ≤ ℓ). In particular, the graph Ḡ induced in G by ⋃_{i=1}^{ℓ} V̄_{i} is 3ε-uniform.

Proof. Since proofs of very similar statements may be found in [HKŁ ⋆⋆], here we only mention the simple idea behind the argument. In order to find the V̄_{i} (0 ≤ i ≤ ℓ) we successively delete from V_{i} the vertices that violate the conditions we seek. Then one can easily show that, because of the ε-uniformity of G, this process finishes with the required sets V̄_{i} (0 ≤ i ≤ ℓ).

Finally, since every graph G on n vertices contains a bipartite subgraph H whose vertex classes are of cardinality ⌊n/2⌋ and ⌈n/2⌉ and such that e(H) ≥ e(G)/2, the following fact is an immediate consequence of the definition of a (b, η)-sparse graph.

Fact 7. If G is (b, η)-sparse then every subgraph H of G with |V (H)| ≥ 2η|V (G)| + 2 vertices contains at most b|V (H)|^{2}e(G)/|V (G)|^{2} edges.

§3. The Ruzsa–Szemerédi Theorem for Sparse Graphs

In this section we state and prove an analogue of a result of Ruzsa and Szemerédi [RSz 78] (see also Erdős, Frankl and Rödl [EFR 86] or Graham and Rödl [GR 87]) which remains valid for (b, η)-sparse graphs. Let us say that a graph is 3-decomposable if it is the union of edge-disjoint triangles. Then the Ruzsa–Szemerédi theorem that we shall need (and generalize below) may be stated as follows.

Theorem 8. For every constant c > 0 there exists a constant δ̂(c) > 0 such that every 3-decomposable graph G with at least cn^{2} edges contains at least δ̂(c)n^{3} triangles.

We would like to apply a similar result for graphs which are not so dense. Unfortunately, in this case the above theorem is no longer valid: there are 3-decomposable graphs G with n vertices and at least n^{2} exp(−3\sqrt{log n}) edges which contain only e(G)/3 triangles (see, e.g., Theorem 6.6 in [GR 87]). We are thus forced to take another approach. Very roughly speaking, our method will consist in proving a probabilistic version of Theorem 8, Lemma 19, asserting that, in some sense, ‘counterexamples’ as above are rare. However, we need to work for a while before we may state and prove Lemma 19.

We start with a result saying that if a 3-decomposable graph G admits an ε-regular partition then, although G may contain only a few triangles, it must contain many ‘dense’ triples of partition classes. To emphasize the difference between triangles on the one hand and triples of partition classes on the other, we shall refer to the latter as triads.

Thus, let G be a (b, η)-sparse graph on n vertices and let Π = (V^{i})_{0}^{k} be an (ε, k)- equitable (ε, G)-regular partition of the vertex set of G. We say that a pair (V^{r}, V^{s}) (1 ≤ r < s ≤ k) is thick if it is (ε, G)-regular and

e_{G}(V^{r}, V^{s}) ≥ |V^{r}||V^{s}|e(G)/50n^{2} .

We say that a triad (V^{r}, V^{s}, V^{t}) (1 ≤ r < s < t ≤ k) is thick if all three pairs (V^{r}, V^{s}), (V^{r}, V^{t}) and (V^{s}, V^{t}) are thick.

Lemma 9. For every b > 2 there exist constants δ̄ = δ̄(b) > 0 and k̄_{0} = k̄_{0}(b) ≥ 1 that depend only on b such that the following holds. For every (b, η)-sparse 3-decomposable graph G, if Π is an (ε, k)-equitable (ε, G)-regular partition of G such that 200εb ≤ 1, k ≥ k̄_{0}, and 0 < η ≤ min{ε, 1/2k}, then Π contains at least δ̄k^{3} thick triads.

Proof. Let k̄_{0} = k̄_{0}(b) = 40000b and δ̄ = δ̄(b) = δ̂(0.3/b) > 0, where δ̂ is as given by Theorem 8. We shall show that these values will do for our lemma. Suppose 200εb ≤ 1 and let Π be an (ε, k)-equitable (ε, G)-regular partition of a (b, η)-sparse 3-decomposable graph G, where k ≥ k̄_{0} and 0 < η ≤ min{ε, 1/2k}. Then the following assertions hold:

(i) the number of edges of G that have both ends in one set of Π is, by Fact 7, less than

ε^{2}n^{2}be(G)/n^{2} + k(n/k)^{2}be(G)/n^{2} = (ε^{2}b + b/k)e(G) ≤ e(G)/100 ,

(ii) the number of edges of G incident to the vertices in V^{0} is less than

εn^{2}be(G)/n^{2} ≤ εbe(G) ≤ e(G)/200 ,

(iii) the number of edges between pairs (V^{r}, V^{s}) (1 ≤ r < s ≤ k) that are not ε-regular is less than

ε\binom{k}{2} (\frac{n}{k})^{2} \frac{be(G)}{n^{2}} ≤ εbe(G) ≤ e(G)/200 ,

(iv) the number of edges between pairs (V^{r}, V^{s}) (1 ≤ r < s ≤ k) that are not thick

is less than

\binom{k}{2} (\frac{n}{k})^{2} \frac{e(G)}{50n^{2}} ≤ e(G)/100 .

Hence, at least 0.97e(G) edges of G belong to thick pairs (V^{r}, V^{s}). Let these edges form the edge set of the spanning subgraph G^{0} of G. Let F be a family of edge-disjoint triangles of G such that G = ⋃ F. Clearly, just considering F, we see that G^{0} contains at least e(G)/3 − 0.03e(G) ≥ 0.3e(G) triangles. In particular, the partition Π has at least one thick triad, say (V^{r(1)}, V^{s(1)}, V^{t(1)}). Let us delete from G^{0} all edges between the sets V^{r(1)}, V^{s(1)} and V^{t(1)}, and let G^{1} be the graph obtained in this way. Since the number of edges we delete is smaller than 3b(n/k)^{2}e(G)/n^{2} ≤ 3be(G)/k^{2}, we destroy at most 3be(G)/k^{2} < 0.3e(G) triangles from F. Thus, the graph G^{1} contains a triangle and hence the partition Π, viewed as a partition of G^{1}, contains at least one thick triad. We repeat the procedure above and obtain a sequence G = G^{0} ⊃ ⋯ ⊃ G^{ℓ} of spanning subgraphs of G with G^{ℓ} such that Π, viewed as a partition of G^{ℓ}, contains no thick triad. Since in every step we decrease the number of triangles in F by at most 3be(G)/k^{2}, we have ℓ ≥ 0.1k^{2}/b. Hence, the graph G(Π), whose vertices are the sets V^{1}, …, V^{k} and V^{r}, V^{s} (1 ≤ r < s ≤ k) are joined by an edge if and only if the pair (V^{r}, V^{s}) is thick, contains at least 0.1k^{2}/b edge-disjoint triangles. Thus, by Theorem 8 and our choice of δ̄ = δ̄(b) = δ̂(0.3/b), the graph G(Π) contains at least δ̄k^{3} triangles. Consequently, there exist at least δ̄k^{3} thick triads in Π and Lemma 9 follows.

We now turn to the main lemma of this section, Lemma 10. As we have already mentioned in §1, this is a rather technical result, and before we may state it we need to introduce a few definitions, including the definition of a ‘flower’ in an edge-coloured graph. Let us say that a sequence G_{1}, …, G_{m} of graphs on the same vertex set is balanced if

(i) E(G_{i}) ∩ E(G_{j}) = ∅ for all 1 ≤ i < j ≤ m, (ii) e(G_{i}) = e(G_{j}) for all 1 ≤ i < j ≤ m, (iii) for all 1 ≤ i ≤ m, all the vertices of G_{i} have the same degree.

It will be convenient to think of a balanced sequence as above as a graph G, namely G = ⋃_{i=1}^{m} G_{i}, whose edges have been coloured with m colours. Let G̃ = (G_{i})_{i=1}^{m} be a balanced sequence of graphs and G = ⋃_{i=1}^{m} G_{i}. Let n = |V (G)|. Suppose also that Π̃_{1}, …, Π̃_{m} is the (\underline{ε}, k_{0})-canonical sequence of partitions associated to G_{1}, …, G_{m} for some given k_{0} and some \underline{ε} = (ε_{1}, …, ε_{m}). Then, for any constant 0 < δ ≤ 1, a (δ, k_{0}, \underline{ε}; G̃)-flower, or, for short, a (δ, G̃)-flower, consists of three indices 1 ≤ w(1) < w(2) < w(3) ≤ m together with a vertex v of G and a family {(X^{(i)}, Y^{(i)}): 1 ≤ i ≤ g} of pairs of non-exceptional elements X^{(i)} and Y^{(i)} of Π̃_{w(1)} such that

(i) 2g|X^{(i)}| = 2g|Y^{(i)}| ≥ δn,
(ii) the 2g sets X^{(i)}, Y^{(i)} (1 ≤ i ≤ g) are all distinct,

(iii) all pairs (X^{(i)}, Y^{(i)}) are (ε_{w(i)}, G_{w(1)})-regular and

\begin{aligned}e_{G_{w(1)}}(X^{(i)}, Y^{(i)}) & ≥ δ|X^{(i)}||Y^{(i)}|e(G_{w(1)})/10^{6}n^{2} \\ & = δ|X^{(i)}||Y^{(i)}|e(G)/10^{6}mn^{2} , \end{aligned}

(iv) the vertex v is joined to each X^{(i)} (1 ≤ i ≤ g) by at least δ|X^{(i)}|e(G_{w(2)})/10^{6}n^{2} =

δ|X^{(i)}|e(G)/10^{6}mn^{2} edges of G_{w(2)},

(v) the vertex v is joined to each Y^{(i)} (1 ≤ i ≤ g) by at least δ|Y^{(i)}|e(G_{w(3)})/10^{6}n^{2} =

δ|Y^{(i)}|e(G)/10^{6}mn^{2} edges of G_{w(3)}.

We may now state and prove Lemma 10.

Lemma 10. Let b > 2 be given. Then there exist integers m = m(b) ≥ 3 and k_{0} = k_{0}(b), and a real number 0 < δ = δ(b) ≤ 1 that depend only on b such that, for any 0 < ε ≤ 1, there exists a constant 0 < η = η(b, ε) ≤ 1 for which the following holds. Let \underline{ε} = \underline{ε}(m, b, ε, k_{0}). If G̃ = (G_{i})_{i=1}^{m} is a balanced m-edge-colouring of a (b, η)-sparse 3-decomposable graph G = ⋃_{i=1}^{m} G_{i}, then the (\underline{ε}, k_{0})-canonical sequence of partitions Π̃_{1}, …, Π̃_{m} associated to G̃ = (G_{i})_{i=1}^{m} admits a (δ, k_{0}, \underline{ε}; G̃)-flower.

Remark. In the sequel, when considering (b, η)-sparse sequences of graphs G̃ as above, we shall often say that ‘a (δ, k_{0}, \underline{ε}; G̃)-flower exists’ or that ‘G̃ contains a (δ, G̃)-flower’. In such cases, we are tacitly assuming that these flowers are built from the (\underline{ε}, k_{0})-canonical sequence of partitions associated to our G̃, where \underline{ε} = \underline{ε}(m, b, ε, k_{0}) is the vector we have explicitly associated with m, b, ε and k_{0}.

Proof of Lemma 10. Let δ̄(b) > 0 be as given by Lemma 9. Set m = m(b) = ⌈3 × 10^{6}/δ̄(b)⌉, δ = δ(b) = δ̄(b)/1600m^{3} > 0 and k_{0} = k_{0}(b) = max{k̄_{0}(b), 800m^{3}/δ̄(b)} ≥ 1, where k̄_{0}(b) is given by Lemma 9. We shall show that the assertion holds with this choice of m and δ and k_{0}. Thus, let an arbitrary 0 < ε ≤ 1 be given. Clearly, we may assume that ε ≤ min{1/200b, δ̄(b)/800m^{3}}. Now let us invoke Lemma 5 to obtain η(ε, b, k_{0}, m) > 0 and K_{0} = K_{0}(ε, b, k_{0}, m) ≥ k_{0} as given by that lemma. We may and shall assume that η ≤ min{ε, 1/2K_{0}}. Our aim is to show that the assertion holds for η = η(b, ε) = η(ε, b, k_{0}, m) given above. Therefore, let G̃ = (G_{i})_{i=1}^{m} be a balanced m-edge-colouring of a (b, η)-sparse 3-decomposable graph G = ⋃_{i=1}^{m} G_{i}. We need to verify that, under these conditions, a (δ, k_{0}, \underline{ε}; G̃)-flower does indeed exist.

Let Π̃_{1}, …, Π̃_{m} be the (\underline{ε}, k_{0})-canonical sequence of partitions associated G_{1}, …, G_{m}. We first concentrate our attention on the graph G = ⋃_{i=1}^{m} G_{i} and the partition Π̃_{m}. We show that each of at least a half of the thick triads (V^{r}, V^{s}, V^{t}) in Π̃_{m}, whose existence is guaranteed by Lemma 9, has the property that we may assign to its ‘sides’, i.e., the pairs (V^{r}, V^{s}), (V^{r}, V^{t}) and (V^{s}, V^{t}), some three distinct colours so that, if a side is assigned colour 1 ≤ w ≤ m, then it contains a substantial number of edges from G_{w}.

Let us eliminate first the triads for which an assignment as we seek is not possible. Given 1 ≤ w ≤ m, we say that a thick triad (V^{r}, V^{s}, V^{t}) of the ε-regular partition Π̃_{m} of G is dominated by w if, putting u = |V^{r}| = |V^{s}| = |V^{t}|, at least two out of the three pairs of sets (V^{r}, V^{s}), (V^{r}, V^{t}) and (V^{s}, V^{t}) are joined by more than

u^{2}e(G)/150n^{2} ≥ (n/2k_{m})^{2}e(G)/150n^{2} = e(G)/600k_{m}^{2}

edges of G_{w}, where k_{m} = |Π̃_{m}| − 1 ≤ K_{0}. If a thick triad is not dominated by any w (1 ≤ w ≤ m), we say that it is balanced. Since every vertex of G_{w} has degree 2e(G_{w})/n = 2e(G)/mn, at most 2e(G)/mk_{m} edges of G_{w} are incident to a given set V^{r} of Π̃_{m}. Consequently, the number of thick triads dominated by w is less than

k_{m} [\frac{2e(G)/mk_{m}}{e(G)/600k_{m}^{2}} ]^{2} ≤ \frac{1.5 × 10^{6}k_{m}^{3}}{m^{2}} ≤ \frac{δ̄(b)k_{m}^{3}}{2m} ,

so the number of thick triads dominated by some w (1 ≤ w ≤ m) is less than δ̄(b)k_{m}^{3}/2. Thus, by Lemma 9, the graph G contains at least δ̄(b)k_{m}^{3}/2 balanced thick triads.

We call a pair (V^{r}, V^{s}) of partition classes of Π̃_{m} a w-rich pair, where 1 ≤ w ≤ m, if the number of edges of G_{w} between V^{r} and V^{s} is larger than |V^{r}||V^{s}|e(G_{w})/150n^{2} = |V^{r}||V^{s}|e(G)/150mn^{2}. We say that (V^{r}, V^{s}, V^{t}) is (w(1), w(2), w(3))-colourable, where the w(i) (i ∈ {1, 2, 3}) are three distinct colours from {1, …, m}, if and only if we may assign the colours w(1), w(2) and w(3) to the pairs (V^{r}, V^{s}), (V^{r}, V^{t}) and (V^{s}, V^{t}), in some order, in such a way that a colour w is assigned to a pair only if this pair is w-rich. Now, let (V^{r}, V^{s}, V^{t}) be a balanced thick triad. A moment’s thought reveals that any such triad is (w(1), w(2), w(3))-colourable for some 1 < w(1) < w(2) < w(3) ≤ m. Consequently, there is a choice of (w(1), w(2), w(3)) with 1 ≤ w(1) < w(2) < w(3) ≤ m such that Π̃_{m} contains at least δ̄(b)k_{m}^{3}/2\binom{m}{3} thick triads that are (w(1), w(2), w(3))-colourable.

Thus, there exist a non-exceptional partition class V̄ of Π̃_{m} and pairs (U_{m}^{i}, W_{m}^{i}) (1 ≤ i ≤ f = ⌈δ̄(b)k_{m}^{2}/2m^{3}⌉) of non-exceptional partition classes of Π̃_{m} such that, for all 1 ≤ i ≤ f, the triple (V̄ , U_{m}^{i}, W_{m}^{i}) satisfies the following conditions:

(i) the pair (U_{m}^{i}, W_{m}^{i}) is (ε, G_{w(1)})-regular, and
e_{G_{w(1)}}(U_{m}^{i}, W_{m}^{i}) ≥ |U_{m}^{i}||W_{m}^{i}|e(G_{w(1)})/150n^{2},
(ii) the pair (V̄ , U_{m}^{i}) is (ε, G_{w(2)})-regular, and
e_{G_{w(2)}}(V̄ , U_{m}^{i}) ≥ |V̄ ||U_{m}^{i}|e(G_{w(2)})/150n^{2},
(iii) the pair (V̄ , W_{m}^{i}) is (ε, G_{w(3)})-regular, and
e_{G_{w(3)}}(V̄ , W_{m}^{i}) ≥ |V̄ ||W_{m}^{i}|e(G_{w(3)})/150n^{2}.

Now, it follows from Fact 6 with ℓ = 3 that, for every given i (1 ≤ i ≤ f), at least a half of the vertices of V̄ is joined both by at least |U_{m}^{i}|e(G_{w(2)})/300n^{2} edges of G_{w(2)} to U_{m}^{i} and by at least |W_{m}^{i}|e(G_{w(3)})/300n^{2} edges of G_{w(3)} to W_{m}^{i}. Thus, by an elementary averaging argument, there is a vertex v in V̄ and a set Ξ = {(U_{m}^{i}, W_{m}^{i}): i ∈ Λ} of w(1)- rich pairs (U_{m}^{i}, W_{m}^{i}) of cardinality |Λ| ≥ f/2 ≥ δ̄(b)k_{m}^{2}/4m^{3} such that v is joined by at least |U_{m}^{i}|e(G_{w(2)})/300n^{2} edges of G_{w(2)} to U_{m}^{i} and by at least |W_{m}^{i}|e(G_{w(3)})/300n^{2} edges of G_{w(3)} to W_{m}^{i} for all (U_{m}^{i}, W_{m}^{i}) ∈ Ξ.

We shall show that the existence of such a vertex v and such a set Ξ implies the existence of a (δ, G̃)-flower. The three colours associated to the flower we shall exhibit are w(1), w(2), w(3). Our only problem is that so far we have only considered the partition Π̃_{m}, whereas the definition of the flower deals with subsets of the partition Π̃_{w(1)}. Thus, in the rest of the proof we shall try to relate the properties of Π̃_{m} with those of Π̃_{w(1)}. In order to do this, we shall consider the graph Ĝ(Π̃_{m}, Ξ), whose vertex set is the set of non-exceptional partition classes of Π̃_{m}, with two such vertices U_{m}, W_{m} being connected by an edge in Ĝ(Π̃_{m}, Ξ) if and only if (U_{m}, W_{m}) ∈ Ξ.

Note first that Ĝ(Π̃_{m}, Ξ) has N = k_{m} vertices and at least cN^{2} edges, where c = δ̄(b)/8m^{3}. Furthermore, since Π̃_{m} is a subpartition of Π̃_{w(1)}, each non-exceptional partition class of Π̃_{m} is contained in some non-exceptional partition class of Π̃_{w(1)}. Thus, the partition Π̃_{w(1)} of G naturally induces a partition Π̂_{w(1)} of the vertex set of Ĝ(Π̃_{m}, Ξ). Unfortunately, because of the exceptional class V_{m}^{0} of Π̃_{m}, not all partition classes of Π̂_{w(1)} need have the same size. However, since |V_{w(1)}^{0}|, |V_{m}^{0}| ≤ εn we immediately see that all of them contain at most 2k_{m}/k_{w(1)} elements, and a little argument shows that at least (1 − 3ε)k_{w(1)} of them have at least k_{m}/2k_{w(1)} elements.

Now suppose ℓ ≥ 100/c and ε ≤ c/100. Then simple calculations show that the following holds: for any graph Ĝ with N vertices and at least cN^{2} edges, and every partition Π̂ = (V^{i})_{i=1}^{ℓ} of the vertex set of Ĝ into ℓ classes such that all the partition classes V^{i} have cardinality at most 2N/ℓ and not fewer than (1 − 3ε)ℓ of them have cardinality greater than or equal to N/2ℓ, there exist at least cℓ^{2}/100 pairs {V̂ , V̂′} of distinct partition classes of Π̂ such that |V̂ |, |V̂′| ≥ N/2ℓ and V̂, V̂′ are joined by at least c|V̂ ||V̂′|/10 edges. Hence, since every graph on at most k vertices and at least ck^{2} edges contains a matching of size at least ck/2, the above partition Π̂ of Ĝ must contain at least g = ⌈cℓ/200⌉ disjoint pairs (X̂^{i}, Ŷ^{i}) (1 ≤ i ≤ g) such that |X̂^{i}|, |Ŷ^{i}| ≥ N/2ℓ and such that the number of edges between X̂^{i} and Ŷ^{i} is at least c|X̂^{i}||Ŷ^{i}|/10 for all 1 ≤ i ≤ g. Finally, we may again use the fact that dense graphs contain large matchings to deduce that there is a matching of size at least cN/40ℓ between X̂^{i} and Ŷ^{i} for all 1 ≤ i ≤ g.

We apply this observation to Ĝ = Ĝ(Π̃_{m}, Ξ) and the partition Π̂ = Π̂_{w(1)}. Thus let us check the required hypotheses for our observation to apply. First note that the condition ℓ ≥ 100/c becomes in our context k_{w(1)} ≥ 800m^{3}/δ̄(b), whilst ε ≤ c/100 corresponds to ε ≤ δ̄(b)/800m^{2}. The number of disjoint pairs (X̂^{i}, Ŷ^{i}) (1 ≤ i ≤ g) guaranteed by the observation is g = ⌈ck_{w(1)}/200⌉ = ⌈δ̄(b)k_{w(1)}/1600m^{3}⌉. Moreover, X̂^{i} is joined to Ŷ^{i} by at least

\frac{c|X̂^{i}||Ŷ^{i}|}{10} ≥ \frac{c}{10} (\frac{k_{m}}{2k_{w(1)}} )^{2} ≥ \frac{1}{10} · \frac{δ̄(b)}{8m^{3}} · \frac{k_{m}^{2}}{4k_{w(1)}^{2}} = \frac{δ̄(b)k_{m}^{2}}{320k_{w(1)}^{2}m^{3}}

edges of Ĝ(Π̃_{m}, Ξ), and hence the corresponding pair (X_{w(1)}^{i}, Y_{w(1)}^{i}) of partition classes of Π̃_{w(1)} is joined by at least

\frac{δ̄(b)k_{m}^{2}}{320k_{w(1)}^{2}m^{3}} (\frac{n}{2k_{m}} )^{2} \frac{e(G_{w(1)})}{150n^{2}} ≥ \frac{δ̄(b)|U_{w(1)}^{i}||W_{w(1)}^{i}|e(G)}{2 × 10^{5}m^{4}n^{2}}

edges of G_{w(1)}.

Furthermore, we know that, for every 1 ≤ i ≤ g, the graph Ĝ(Π̃_{m}, Ξ) contains a matching M̂_{i} between X̂^{i} and Ŷ^{i} of size at least

\frac{c}{40} · \frac{k_{m}}{k_{w(1)}} = \frac{δ̄(b)}{320m^{3}} · \frac{k_{m}}{k_{w(1)}} .

Recall that every edge of Ĝ(Π̃_{m}, Ξ) corresponds to a pair (U_{m}, W_{m}) from Ξ. We may and shall assume that at least (δ̄(b)/640m^{3})k_{m}/k_{w(1)} of the edges (U_{m}, W_{m}) from M̂_{i} are such that U_{m} ∈ X̂^{i} and W_{m} ∈ Ŷ^{i} (i.e., U_{m} ⊆ X_{w(1)}^{i} and W_{m} ⊆ Y_{w(1)}^{i}). Recall that, for every (U_{m}, W_{m}) ∈ Ξ, the vertex v is joined to U_{m} by at least

\frac{|U_{m}|e(G_{w(2)})}{300n^{2}} ≥ \frac{n}{2k_{m}} · \frac{e(G)}{300mn^{2}} = \frac{e(G)}{600k_{m}mn}

edges of G_{w(2)}, and similarly it is joined to W_{m} by at least e(G)/600k_{m}mn edges of G_{w(3)}. Therefore the vertex v is joined to X_{w(1)}^{i} by at least

\frac{δ̄(b)}{640m^{3}} · \frac{k_{m}}{k_{w(1)}} · \frac{e(G)}{600k_{m}mn} ≥ |X_{w(1)}^{i}|\frac{δ̄(b)e(G)}{4 × 10^{5}m^{4}n^{2}}

edges of G_{w(2)}. Clearly, the same argument applied to Y_{w(1)}^{i} shows that v is joined to Y_{w(1)}^{i} by at least |Y_{w(1)}^{i}|δ̄(b)e(G)/4 × 10^{5}m^{4}n^{2} edges of G_{w(3)}. Since δ = δ̄(b)/1600m^{3}, we conclude that there are at least g = ⌈δk_{w(1)}⌉ pairwise disjoint pairs (X_{w(1)}^{i}, Y_{w(1)}^{i}) of non-exceptional partition classes of Π̃_{w(1)} such that every such pair is (ε, G_{w(1)})-regular, between every such pair there are at least δ̄(b)|U_{w(1)}^{i}||W_{w(1)}^{i}|e(G)/10^{6}m^{4}n^{2} edges of G_{w(1)}, and one set of every such pair is joined to v by at least

|X_{w(1)}^{i}|\frac{δ̄(b)e(G)}{4 × 10^{5}m^{4}n^{2}} ≥ \frac{δ̄(b)|X_{w(1)}^{i}|e(G)}{10^{6}m^{4}n^{2}} = \frac{δ̄(b)|Y_{w(1)}^{i}|e(G)}{10^{6}m^{4}}

edges of G_{w(2)}, while the other is joined to v by at least δ̄(b)|Y_{w(1)}^{i}|e(G)/10^{6}m^{4}n^{2} edges of G_{w(3)}. Finally, if u is the common cardinality of the sets X_{w(i)}^{i}, Y_{w(1)}^{i} (1 ≤ i ≤ g), then 2gu ≥ δn. Therefore 1 ≤ w(1) < w(2) < w(3) ≤ m, together with v ∈ V (G), and {(X_{w(1)}^{i}, Y_{w(1)}^{i}): 1 ≤ i ≤ g} form a (δ, G̃)-flower, and Lemma 10 follows.

§4. Random Subgraphs of Uniform Bipartite Graphs

In this section we give a result that, although a little technical, may be of independent interest. Namely, we prove that, under quite weak hypotheses, with very large probability a random induced subgraph H′ of a bipartite uniform graph H contains a fair number edges. In fact, we show that, in selecting the random subgraph H′ in question, we may allow an ‘adversary’ to ‘mark’, during the selection process, a few vertices of H as forbidden vertices for H′ so as to minimize our chances of getting many edges in our random subgraph; even with this rule the probability that we fail to get a few edges in H′ is essentially superexponentially small in the number of edges of the original bipartite graph H.

Let H = H(u, ρ, ε) be an ε-uniform bipartite graph with bipartition V (H) = V_{1} ∪ V_{2}, where |V_{1}| = |V_{2}| = u ≥ 1, and with edge-density e(H)u^{−2} = ρ. Let d_{1}, d_{2} ≤ u be two given positive integers. Now select a random induced subgraph of H in the following manner. First, an adversary chooses a set S_{1} ⊂ V_{1} with |S_{1}| ≤ u/ log log u. Then we randomly pick a set D_{1} ⊂ V_{1} \ S_{1} with |D_{1}| = d_{1}, with all the d_{1}-subsets of V_{1} \ S_{1} equiprobable. Next, under the full knowledge of the sets S_{1} and D_{1}, our adversary picks a set S_{2} ⊂ V_{2} with |S_{2}| ≤ u/ log log u, and we randomly pick a set D_{2} ⊂ V_{2} \ S_{2} with |D_{2}| = d_{2}, with all the d_{2}-subsets of V_{2} \ S_{2} equiprobable. Let us call the outcome of the above procedure a random (d_{1}, d_{2}; S_{1}, S_{2})-subgraph of H, or simply a (d_{1}, d_{2})-subgraph of H.

Lemma 11. For every constant 0 < β ≤ 1, there exist a constant 0 < ε = ε(β) ≤ 1 and a natural number u_{0} such that, for any real d ≥ 2(u/ε)^{1/2} and any given graph H = H(u, ρ, ε) as above with u ≥ u_{0} and ρ ≥ d/u, the following assertion holds. If d_{1}, d_{2} ≥ d, regardless of the choices for S_{1} and S_{2} of our adversary, the probability that a random (d_{1}, d_{2}; S_{1}, S_{2})-subgraph of H fails to contain at least d/2 edges is at most β^{d}.

Proof. Given 0 < β ≤ 1, we choose ε = β^{2}/16 > 0 and show that this choice of ε > 0 will do. In the sequel, we assume that u is large enough for our inequalities to hold. Let our adversary choose the set S_{1} ⊂ V_{1}. Recall that |S_{1}| ≤ u/ log log u. We show first that the set U of those vertices of V_{2} that are adjacent to the vertices in our random set D_{1} ⊂ V_{1}\S_{1} is larger than (1 − ε)n with probability at least 1 − (4ε)^{d/2}. In order to do so we generate the vertices of D_{1} one by one and prove that, typically, in each step we enlarge the set U by a fair number of vertices.

Let us randomly choose a vertex v^{1} among all the vertices of V_{1}\S_{1} to be the first vertex of D_{1}. Denote by W^{1} the set of the vertices of V_{1} that have fewer than dε/2 < (1 − ε)d neighbours in V_{2}. Then, by the ε-regularity of (V_{1}, V_{2}), we have that |W^{1}| ≤ εu. If v^{1} belongs to W^{1}, let us say that it is a bad vertex, whereas if v^{1} ∉ W^{1} let us say that it is a good vertex. Finally, let U^{1} ⊆ V_{2} be the set of the neighbours of v^{1} in V_{2}.

Similarly, let us suppose that for some 2 ≤ i ≤ d_{1} the vertices v^{1}, …, v^{i−1} have already been put into D_{1}. We randomly pick a vertex v^{i} from V_{1} \ ({v^{1}, …, v^{i−1}} ∪ S_{1}) to be the ith vertex of D_{1} and denote by U^{≤i−1} the set of neighbours of v^{1}, …, v^{i−1}. Then, if |U^{≤i−1}| ≤ (1 − ε)u, we let W^{i} be the set of all the vertices in V_{1} that have fewer than dε/2 ≤ (1 − ε)εd neighbours in V_{2} \ U^{≤i−1}. Note that, by the ε-regularity of the pair (V_{1}, V_{2}), we have |W^{i}| ≤ εu. As before, we call v^{i} bad if it belongs to W^{i} and good otherwise. Moreover, in the case in which |U^{≤i−1}| > (1 − ε)u, we always say that v^{i} is good. This process is continued until all the d_{1} elements of D_{1} have been chosen.

Now, suppose that our process has terminated with a set D_{1} with U = U^{≤d_{1}} of cardinality |U| < (1−ε)u. Since each good vertex v^{i} increases the size of the neighbourhood of D_{1} in V_{2} by at least dε/2, the number of good elements in D_{1} must be less than 2u/dε ≤ d_{1}/2. Hence, at least a half of all the elements of D_{1} must be bad, but the probability of this happening is smaller than 2^{d_{1}}ε^{d_{1}/2} ≤ (4ε)^{d/2}. Thus, |U| ≥ (1−ε)n with probability at least 1 − (4ε)^{d/2}.

Assume now that our process has terminated with a set D_{1} with U = U^{≤d_{1}} of cardinality |U| ≥ (1 − ε)u. We now let our adversary pick his set S_{2} ⊂ V_{2}. Then the probability that at least a half of the d_{2} ≥ d vertices of D_{2} should lie outside U is at most 2^{d_{2}}(εn/(n − |S_{2}|))^{d_{2}/2}, which, for sufficiently large u, is less than 2^{d_{2}}(2ε)^{d_{2}/2} ≤ (8ε)^{d/2}.

Thus, the probability that our random (d_{1}, d_{2})-subgraph of H contains fewer than d/2 edges is bounded from above by (4ε)^{d/2} + (8ε)^{d/2} ≤ (16ε)^{d/2} = β^{d}, as required.

§5. A Sufficient Condition for Uniformity

Let G be an ℓ-partite graph with ℓ-partition V (G) = V_{1} ∪ ⋯ ∪ V_{ℓ} (ℓ ≥ 2). Recall that G is ε-uniform if all pairs (V_{i}, V_{j}) (1 ≤ i < j ≤ ℓ) are ε-regular. Moreover, observe that in order to check the ε-regularity for any such pair (V_{i}, V_{j}), in principle one must examine the density of many pairs (A, B) with A ⊆ V_{i}, B ⊆ V_{j}. However, it turns out that the ε-regularity of (V_{i}, V_{j}) is implied by a rather simple condition imposed upon the intersection of the neighbourhoods of pairs of vertices. This idea has been exploited in many places; see, e.g., Alon, Duke, Lefmann, Rödl and Yuster [ADLRY 94], Frankl, Rödl and Wilson [FRW 88] and Thomason [Th 87a] (see also [Th 87b]). The following fact is a slight refinement of earlier results in [Th 87a] and [FWR 88].

Lemma 12. Let G be a d-regular bipartite graph with bipartition V (G) = V_{1} ∪ V_{2}, where |V_{1}| = |V_{2}| = n and d = pn (0 < p ≤ 1). Assume that for a subset B of V_{2} with b vertices and some ε > 0 we have

∑_{x,x′∈B} |N(x) ∩ N(x′)| ≤ (1 + ε)\binom{b}{2}np^{2} ,  (1)

where the sum is taken over all unordered pairs x, x′ ∈ B with x ≠ x′. Then, for every subset A of V_{1} with a vertices, we have

(e(A, B) − abp)^{2} ≤ εa(n − a)b^{2}p^{2} + abnp .

Proof. Suppose V_{1} = {x_{1}, …, x_{n}} and let d_{i} (1 ≤ i ≤ n) be the number of neighbours the vertex x_{i} has in B. For simplicity of notation, assume that A = {x_{1}, …, x_{a}}. Then, since G is d-regular, we have

∑_{i=1}^{n} d_{i} = e(V_{1}, B) = db = e(A, B) + e(V_{1} \ A, B) .

Furthermore, using (1), counting directed paths of length two with both ends in B leads to

(1 + ε)b(b − 1)np^{2} ≥ ∑_{i=1}^{n} d_{i}(d_{i} − 1) = ∑_{i=1}^{a} d_{i}(d_{i} − 1) + ∑_{i=a+1}^{n} d_{i}(d_{i} − 1) .

Since ∑_{i=1}^{a} d_{i} = e(A, B) and ∑_{i=a+1}^{n} d_{i} = e(V_{1} \ A, B) = db − e(A, B), by the Cauchy– Schwarz inequality we have

\begin{aligned}(1 + ε)b(b − 1)np^{2} ≥ \frac{1}{a} & e(A, B)(e(A, B) − a) \\ & + \frac{1}{n − a}(db − e(A, B))(db − e(A, B) − n + a) , \end{aligned}

which, after elementary calculations, may be reduced to

abpn(n − a)(1 − p) + εab(b − 1)p^{2}n(n − a) ≥ n(e(A, B) − abp)^{2} ,

from which the assertion easily follows.

As an immediate consequence of the above result we obtain a simple sufficient condition for the η-uniformity of a regular bipartite graph G.

Lemma 13. Let G be a d-regular bipartite graph with bipartition V (G) = V_{1} ∪ V_{2}, where |V_{1}| = |V_{2}| = n, d = pn, and p = ω/n with ω = ω(n) → ∞ as n → ∞. Suppose that for some constant 0 < ε < 1/2 independent of n we have

∑_{x,x′} (|N(x) ∩ N(x′)| − (1 + ε)np^{2}) ≤ 4ε^{3}n^{3}p^{2} ,  (2)

where the sum is taken over all unordered pairs x, x′ ∈ V_{2} with x ≠ x′ and |N(x)∩N(x′)| ≥ (1 + ε)np^{2}. Then G is 8ε^{1/3}-uniform provided n is large enough.

Proof. Take any A ⊆ V_{1}, B ⊆ V_{2} of sizes a = |A| ≥ 2ε^{1/3}n and b = |B| ≥ 2ε^{1/3}n. Then, writing ∑′_{x,x′∈B} for the sum over all unordered pairs of distinct vertices x, x′ ∈ B, and ∑′′_{x,x′∈V_{2}} for the sum over all unordered pairs of distinct vertices x, x′ ∈ V_{2} with |N(x) ∩ N(x′)| ≥ (1 + ε)np^{2}, we have from (2) that

∑′_{x,x′∈B}|N(x) ∩ N(x′)| ≤ (1 + ε)\binom{b}{2}np^{2} + ∑′′_{x,x′∈V_{2}} (|N(x) ∩ N(x′)| − (1 + ε)np^{2}))

≤ (1 + ε)\binom{b}{2}np^{2} + 4ε^{3}n^{3}p^{2} ≤ (1 + ε + 2ε^{2})\binom{b}{2}np^{2} ≤ (1 + 2ε)\binom{b}{2}np^{2} .

Furthermore, since ω = pn → ∞ as n → ∞, for large enough n we have that abpn ≤ εanb^{2}p^{2} and hence, by Lemma 12, we have

|e(A, B) − abp| ≤ \sqrt{4εanb^{2}p^{2}} ≤ 2ε^{1/3}abp.

Now, to complete the proof, it is enough to observe that, according to our definition of density, d_{G}(A, B) = 4e(A, B)/abp while d_{G}(V_{1}, V_{2}) = 4. Hence

|d_{G}(A, B) − d_{G}(V_{1}, V_{2})| = \frac{4|e(A, B) − abp|}{abp} ≤ 8ε^{1/3} ,

as required.

§6. Difference-Graphs

Let A be a subset of [n] = {0, 1, …, n − 1}. The associated difference-graph G_{A} = G_{A}(n) is the bipartite graph with bipartition V (G_{A}) = V ∪ W, where both V and W are copies of [n], and for v ∈ V and w ∈ W the pair {v, w} is an edge of G_{A} if and only if w ≡ v + a (mod n) for some a ∈ A. It is immediate that each vertex of G_{A} has degree |A|, and that the number of common neighbours of two distinct vertices v and v′ that belong to the same class of the bipartition depends only on the value of v − v′. In fact, this value is the same as the number t_{A}(v − v′) of ordered pairs (a, a′) ∈ A × A such that a′ − a ≡ v − v′ (mod n). Therefore, the structure of G_{A} is closely related to the behaviour of the numbers t_{A}(j) (1 ≤ j < n). Our next result deals with the distribution of the t_{R}(j) for a random set R ∈ R(n, p). In the sequel, we write ⊕ and ⊖ for addition and subtraction modulo n, respectively.

Lemma 14. For every fixed 0 < ε ≤ 1 and 0 < η ≤ 1 there exists a constant C = C(ε, η) for which the following holds. For every p = p(n) ≥ C/\sqrt{n}, the probability that R ∈ R(n, p) satisfies

∑ {t_{R}(j) − (1 + ε)np^{2}} ≤ ηn^{2}p^{2},  (3)

where the sum is taken over all 1 ≤ j < n such that t_{R}(j) ≥ (1 + ε)np^{2}, tends to 1 as n → ∞.

Proof. For i ∈ [n], let I_{i} be the characteristic function of the event {i ∈ R}. Thus I_{i} is a 0–1 random variable with I_{i} = 1 if and only if i ∈ R. For a given 1 ≤ j < n, let us divide the set of all n pairs (i, i ⊕ j) (i ∈ [n]) into three classes B^{1}(j), B^{2}(j) and B^{3}(j) in such a way that ⌊n/3⌋ ≤ |B^{ℓ}(j)| ≤ ⌈n/3⌉ for all ℓ ∈ {1, 2, 3}, and such that, for any i ∈ [n], no B^{ℓ}(j) contains both pairs (i⊖j, i) and (i, i⊕j). The fact that such a partition is always possible follows from the simple observation that every 2-regular graph admits a proper 3edge-colouring in which the sizes of any two colour classes differ by at most one. In fact, for later convenience, we may and shall further require of the B^{ℓ}(j) that, for all ℓ ∈ {1, 2, 3}, if j_{1} + j_{2} = n, then the set of unordered pairs that naturally correspond to the elements of B^{ℓ}(j_{1}) should be the same as the corresponding set for B^{ℓ}(j_{2}). Now, for all 1 ≤ j < n and ℓ ∈ {1, 2, 3}, define the random variables X(j, ℓ) and X̂(j, ℓ) by setting

X(j, ℓ) = ∑_{(i,i⊕j)∈B^{ℓ}(j)} I_{i}I_{i⊕j},

and

X̂(j, ℓ) = \{ \begin{matrix}0 & if X(j, ℓ) < (1 + ε)|B^{ℓ}(j)|p^{2} \\ X(j, ℓ) & otherwise \end{matrix}

or, shortly, X̂(j, ℓ) = 1I(j, ℓ)X(j, ℓ), where 1I(j, ℓ) is the characteristic function of the event {X(j, ℓ) ≥ (1+ε)|B^{ℓ}(j)|p^{2}}. Then, clearly, we have t_{R}(j) = X(j, 1)+X(j, 2)+X(j, 3) for all 1 ≤ j < n. Notice now that the left-hand side of (3) is at most

∑_{ℓ=1}^{3} ∑_{j=1}^{n−1} (X̂(j, ℓ) − 1I(j, ℓ)(1 + ε)|B^{ℓ}(j)|p^{2}) .

Hence it is enough to show that, with probability tending to 1 as n → ∞, we have

Z = ∑_{ℓ=1}^{3} ∑_{j=1}^{n−1} X̂(j, ℓ) ≤ ηn^{2}p^{2}.  (4)

Let us estimate first the expectation E X̂(j, ℓ) of X̂(j, ℓ). Note that X(j, ℓ) is a sum of independent 0–1 random variables, and thus it has binomial distribution Bi(m, p^{2}), where m = |B^{ℓ}(j)|. (It was for achieving the above independence that the classes B^{1}(j), B^{2}(j) and B^{3}(j) were introduced). Therefore, setting r_{0} = ⌈(1 + ε)p^{2}m⌉, we get

E(X̂(j, ℓ)) = ∑_{r=r_{0}}^{m} r\binom{m}{r}p^{2r}(1 − p^{2})^{m−r} = mp^{2} ∑_{r=r_{0}}^{m} \binom{m − 1}{r − 1}p^{2r−2}(1 − p^{2})^{m−r}

= mp^{2} ∑_{r=r_{0}−1}^{m−1} \binom{m − 1}{r}p^{2r}(1 − p^{2})^{m−r−1} = mp^{2}b_{≥}(r_{0} − 1; m − 1, p^{2}), (5)

where b_{≥}(r_{0} − 1; m − 1, p^{2}) is the probability that a binomial random variable with parameters m − 1 and p^{2} takes values greater than or equal to r_{0} − 1. The behaviour of the function b_{≥} is, of course, well studied, and it is known that for every choice of m′, p′ and k′ such that k′ ≤ m′p′ we have

b_{≥}(m′p′ + k′; m′, p′) ≤ exp ( − \frac{(k′)^{2}}{3m′p′} ) .

(See, for instance, §6 in McDiarmid [McD 89]). Thus, in our case,

b_{≥}(r_{0} − 1; m − 1, p^{2}) ≤ exp ( − \frac{(εp^{2}m − 1)^{2}}{3(m − 1)p^{2}} ) ≤ exp \{ − \frac{1}{10}ε^{2}np^{2}\} ,

where here and below we assume that n is large enough for all our inequalities to hold. Hence E X̂(j, ℓ) ≤ |B^{ℓ}(j)|p^{2} exp(−ε^{2}np^{2}/10) and, consequently,

E(Z) = E ( ∑_{ℓ=1}^{3} ∑_{j=1}^{n−1} X̂(j, ℓ)) ≤ n^{2}p^{2} exp \{ − \frac{1}{10}ε^{2}np^{2}\} .

Thus, if p = p(n) is such that np^{2} → ∞ as n → ∞, then the right-hand side of the above inequality is o(n^{2}p^{2}). Therefore, by Markov’s inequality, relation (4) holds with probability 1 − o(1) as n → ∞, and hence our lemma follows in this case.

In order to complete the proof, it is enough to consider the case when p = p(n) > C/\sqrt{n} for some large C > 0 but, say, np^{2} < log log n. Thus we henceforth assume that p = p(n) satisfies these conditions. In the remaining, rather technical part of the proof, we shall compute the variance of Z = ∑_{ℓ=1}^{3} ∑_{j=1}^{n−1} X̂(j, ℓ) for such a p and then show that this random variable is concentrated around its expectation through a direct application of Chebyshev’s inequality.

First, let us note that with m = |B^{ℓ}(j)| as before we have that the variance Var(X̂(j, ℓ)) of X̂(j, ℓ) is no greater than

∑_{r=r_{0}}^{m} r^{2}\binom{m}{r}p^{2r}(1 − p^{2})^{m−r} = ∑_{r=r_{0}}^{m} r(r − 1)\binom{m}{r}p^{2r}(1 − p^{2})^{m−r} + E X̂(j, ℓ)

= m(m − 1)p^{4} ∑_{r=r_{0}−2}^{m−2} \binom{m − 2}{r}p^{2r−2}(1 − p^{2})^{m−r−2} + E X̂(j, ℓ)

= m(m − 1)p^{4}b_{≥}(r_{0} − 2; m − 2, p^{2}) + mp^{2}b_{≥}(r_{0} − 1; m − 1, p^{2})

≤ n^{2}p^{4} exp \{ − \frac{1}{10}ε^{2}np^{2}\} = O(1).

Note that for j_{1}, j_{2} ∈ {1, …, n − 1}, if j_{1} + j_{2} = n, then X̂(j_{1}, ℓ) = X̂(j_{2}, ℓ) for all ℓ ∈ {1, 2, 3}. Thus the covariance Cov(X̂(j_{1}, ℓ), X̂(j_{2}, ℓ)) between X̂(j_{1}, ℓ) and X̂(j_{2}, ℓ) coincides with Var(X(j_{1}, ℓ)) = O(1). Now let (j_{1}, ℓ_{1}) ≠ (j_{2}, ℓ_{2}) and suppose also that if j_{1} + j_{2} = n then ℓ_{1} ≠ ℓ_{2}. We shall estimate Cov(X̂(j_{1}, ℓ_{1}), X̂(j_{2}, ℓ_{2})). For simplicity, put U_{1} = X̂(j_{1}, ℓ_{1}) and U_{2} = X̂(j_{2}, ℓ_{2}). We have Cov(X̂(j_{1}, ℓ_{1}), X̂(j_{2}, ℓ_{2})) = Cov(U_{1}, U_{2}) = E(U_{1}U_{2}) − E(U_{1}) E(U_{2}). We shall first estimate from above the value of E(U_{1}U_{2}). Recall that we write R for a random element of our probability space R(n, p). We have

E(U_{1}U_{2}) = ∑_{R_{0}⊆[n]} U_{1}(R_{0})U_{2}(R_{0}) Prob(R = R_{0}).  (6)

Let us say that R ∈ R(n, p) is exceptional if either U_{1}(R) or U_{2}(R) is at least as large as u_{max} = ⌊log n⌋. Standard inequalities for the tail of the binomial distribution gives that the probability that R ∈ R(n, p) is exceptional is no greater than n^{−(1/2) log log n}. Since U_{1} and U_{2} are at most n, we have that the exceptional R_{0} contribute with n^{2−(1/2) log log n} to the sum in (6). In the sequel we concentrate our attention on non-exceptional R ∈ R(n, p).

Let us introduce some notation. Let G_{s} (s ∈ {1, 2}) be the directed graph with vertex set [n] and edge set B^{ℓ_{s}}(j_{s}). Thus each G_{s} simply consists of some isolated edges. Moreover, the directed graph G_{1} ∪ G_{2} contains no cycle of length 2. Let H_{s} = H_{s}(R) (s ∈ {1, 2}) be the random subgraph of G_{s} induced by the elements of R ∈ R(n, p), and put H = H(R) = H_{1}(R) ∪ H_{2}(R) ⊆ G_{1} ∪ G_{2}.

We now consider the structure of H. Let us say that R ∈ R(n, p) is typical if H = H(R) is a matching. The probability that R ∈ R(n, p) is not typical is at most np^{3} ≤ (log log n)^{3/2}/\sqrt{n}. Thus, the contribution of the atypical, non-exceptional R_{0} ⊆ [n] in (6) is at most u_{max}^{2}(log log n)^{3/2}n^{−1/2} = O((log n)^{2}(log log n)^{3/2}n^{−1/2}). From here onwards we only take into account R ∈ R(n, p) that are typical and non-exceptional. Thus H = H(R) will always consist of isolated edges.

To bound the contribution of the typical, non-exceptional R_{0} ⊆ [n] to the sum in (6), we shall need the following consequence of a large deviations inequality of Janson, Łuczak, and Ruciński [JŁR 90]. Let J be a graph with maximal degree at most 2, with m̄ edges, and n_{2} vertices of degree two. Let J^{p} be a random induced subgraph of J obtained by selecting its vertices randomly and independently each with probability p. Then the inequality of Janson, Łuczak and Ruciński gives that the probability that J^{p} has no edges is at most exp{−m̄p^{2} + 2n_{2}p^{3}}.

We are now ready to bound E′(U_{1}U_{2}) = ∑ U_{1}(R_{0})U_{2}(R_{0}) Prob(R = R_{0}), where the sum is taken over all typical and non-exceptional R_{0} ⊆ [n]. For s ∈ {1, 2}, let m_{s} = |B^{ℓ_{s}}(j_{s})| and u_{s} = ⌈(1 + ε)m_{s}p^{2}⌉. Then E′(U_{1}U_{2}) is at most

∑_{u_{1}=r_{1}}^{u_{max}} u_{1}\binom{m_{1}}{u_{1}} ∑_{u_{2}=r_{2}}^{u_{max}} u_{2}\binom{m_{2}}{u_{2}}p^{2u_{1}+2u_{2}} exp(−(m_{1} + m_{2} − 3u_{1} − 3u_{2})p^{2} + 2np^{3})

≤ ∑_{u_{1}=r_{1}}^{u_{max}} u_{1}\binom{m_{1}}{u_{1}}p^{2u_{1}}(1 − p^{2})^{m_{1}−u_{1}} ∑_{u_{2}=r_{2}}^{u_{max}} u_{2}\binom{m_{2}}{u_{2}}p^{2u_{2}}(1 − p^{2})^{m_{2}−u_{2}}

× exp(2u_{max}p^{2} + np^{4} + 2np^{3})

≤ (1 + O(n^{−1/2}(log log n)^{3/2}) E(U_{1}) E(U_{2}) .

Therefore

Cov(X̂(j_{1}, ℓ_{1}), X̂(j_{2}, ℓ_{2})) = Cov(U_{1}, U_{2}) = E(U_{1}U_{2}) − E(U_{1}) E(U_{2})
≤ n^{2−(1/2) log log n} + O(n^{−1/2}(log n)^{2}(log log n)^{3/2}) + O(n^{−1/2}(log log n)^{7/2})
= O(n^{−1/2}(log n)^{2}(log log n)^{3/2}).

Thus,

Var(Z) = Var ( ∑_{ℓ=1}^{3} ∑_{j=1}^{n−1} X̂(j, ℓ)) = ∑_{ℓ_{1}=1}^{3} ∑_{j_{1}=1}^{n−1} ∑_{ℓ_{2}=1}^{3} ∑_{j_{2}=1}^{n−1} Cov(X̂(j_{1}, ℓ_{1}), X̂(j_{2}, ℓ_{2}))
= 6nO(1) + 9n^{2}O(n^{−1/2}(log n)^{3}) = O(n^{3/2}(log n)^{3}).

We now note that (5) and the trivial fact that

b_{≥}(r_{0} − 1; m − 1, p^{2}) ≥ \binom{m − 1}{r_{0} − 1}p^{2(r_{0}−1)}(1 − p^{2})^{m−r_{0}} ≥ exp\{−3np^{2}\}

implies that

E(Z) = ∑_{ℓ=1}^{3} ∑_{j=1}^{n−1} E X̂(j, ℓ) ≥ n(n − 1)p^{2} exp{−3np^{2}} ≥ C^{2}n/2(log n)^{3}.

Thus Var(Z) = o((E(Z))^{2}) and, hence, from Chebyshev’s inequality, with probability tending to 1 as n → ∞, we have Z ≤ 2 E(Z) ≤ 2n^{2}p^{2} exp(−ε^{2}np^{2}/10). Thus (4) holds with probability tending to 1 as n → ∞ provided C^{2} ≥ 10ε^{−2} log(2/η).

Lemmas 13 and 14 immediately imply that the difference graph G_{R} = G_{R}(n) for a random subset R ∈ R(n, p) is η-uniform, provided the probability p is large enough.

Fact 15. For every 0 < η ≤ 1 there exists a constant C = C(η) such that, if p = p(n) ≥ C/\sqrt{n} and R ∈ R(n, p), then the bipartite graph G_{R} = G_{R}(n) is η-uniform with probability tending to 1 as n → ∞.

§7. Proof of Theorem 1

Let S be a subset of [n] = {0, …, n − 1}, where n is an odd natural number. Following Ruzsa and Szemerédi [RSz 78] (see also Erdős, Frankl and Rödl [EFR 86] and Graham and Rödl [GR 87]) we introduce a graph G(n, S) that reflects the arithmetic structure of S. Thus, G(n, S) is a tripartite graph whose vertex set consists of three copies V_{1}, V_{2}, V_{3} of the set [n] and {i, j} is an edge of G(n, S) if and only if one of the following three conditions holds:

(i) i ∈ V_{1}, j ∈ V_{2} and j = i ⊕ k for some k ∈ S,
(ii) i ∈ V_{2}, j ∈ V_{3} and j = i ⊕ k for some k ∈ S,
(iii) i ∈ V_{1}, j ∈ V_{3} and j = i ⊕ 2k for some k ∈ S,

where here and below ⊕ and ⊖ stand for addition and subtraction modulo n. Clearly, if k ∈ S then the vertices i ∈ V_{1}, i ⊕ k ∈ V_{2} and i ⊕ 2k ∈ V_{3} induce a triangle in G(n, S). A triangle of G(n, S) of this type is said to be trivial. We are interested in the non-trivial, or spontaneous, triangles of G(n, S), since they reflect the arithmetic structure of S in the sense made precise below.

Clearly, each G(n, S) contains precisely n|S| trivial triangles, and in fact G(n, S) is the edge-disjoint union of those triangles. More importantly, the number of spontaneous triangles in G(n, S) depends on the number of arithmetic triples in S, that is triples of distinct elements a, b, and c ∈ S such that a ⊖ c = c ⊖ b. Indeed, for any arithmetic triple Δ = (a, b, c) and every i ∈ [n], the graph G(n, S) contains the associated spontaneous triangle with vertices i ∈ V_{1}, i⊕a ∈ V_{2} and i⊕a⊕b ∈ V_{3}, and, conversely, any spontaneous triangle is associated to such a pair (Δ, i). Thus, in order to verify whether S contains an arithmetic triple it is enough to look for a spontaneous triangle in G(n, S).

Naturally, we shall be particularly interested in the structure of G(n, R) for a random subset R of [n]. It will be later crucial that, for a large random set R ⊆ [n], the graph G(n, R) is typically uniform, and hence sparse, as show our next two results. In the sequel, it will be convenient to extend the definition of η-uniformity to subsets of [n] in the obvious way. If S ⊆ [n] is such that G(n, S) is η-uniform for some 0 < η ≤ 1, then let us say that S itself is η-uniform. Moreover, given b > 2 and 0 < η ≤ 1, we define the notion of (b, η)-sparseness for S above in the analogous way.

Fact 16. For every 0 < η ≤ 1 there exists a constant C = C(η) such that, if p = p(n) ≥ C/\sqrt{n}, then the probability that R ∈ R(n, p) is η-uniform tends to 1 as n → ∞ along the odd integers.

Proof. Since for odd n the graph G_{2R} = G_{2R}(n) may be identified with GR̄ = GR̄(n) for some R̄ ∈ R(n, p), with the map R ↦ R̄ measure preserving, the assertion follows from Fact 15 and the definition of G(n, R).

From Fact 16 one may deduce the following result concerning the uniformity and sparseness of R_{M} ∈ R(n, M) for large M.

Fact 17. For every 0 < η ≤ 1 there exists a constant C = C(η) such that, if M = M(n) ≥ C\sqrt{n}, then the probability that R ∈ R(n, M) is η-uniform and (4, η)-sparse tends to 1 as n → ∞ along the odd integers.

Proof. We start by noticing that, for any 0 < η ≤ 1, if 0 < η_{0} ≤ 1 is small enough, then for any S ⊆ [n] the fact that the graph G = G(n, S) is η_{0}-uniform implies that G is, say, (4, η)-sparse. Therefore we proceed to show that for any 0 < η_{0} ≤ 1, if Cn^{1/2} ≤ M = M(n) ≤ n for some sufficiently large constant C, then G = G(n, R_{M}) is η_{0}-uniform with probability 1 − o(1) as n → ∞ along the odd integers.

Pick η_{1} = η_{0}/6 and ε = η_{1}^{2}/3. Let C = (1 + ε)C_{1}, where C_{1} = C(η_{1}) is as given by Fact 16. and assume that C\sqrt{n} ≤ M = M(n) ≤ n. Set p = p(n) = M/(1 + ε)n. We may generate R_{M} ∈ R(n, M) by picking R_{p} ∈ R(n, p) conditioned on R_{p} satisfying |R_{p}| ≤ M, and then by adding random elements of [n]\R_{p} to R_{p} to obtain a set R_{M} of cardinality M. Since with probability 1 − o(1) as n → ∞, we have |R_{p}| ≥ (1 − ε)pn, we shall assume that our R_{p} does satisfy this condition. Assume also that G = G(n, R_{p}) is η_{1}-uniform, and recall that by Fact 16 this event also holds with probability 1 − o(1). It now suffices to show that, under these two conditions on R_{p}, the set R_{M} is η_{0}-uniform whatever elements were added to R_{p} to generate R_{M}.

Write G_{M} for G(n, R_{M}) and G_{p} for G(n, R_{p}). Let U, W ⊂ V (G_{M}) be two disjoint sets contained in two distinct vertex classes of G_{M} with |U|, |W | ≥ η_{1}n. Put ρ_{M} = M/n and ρ_{p} = |R_{p}|/n. Note that then (1 − 2ε)ρ_{M} ≤ ρ_{p} ≤ ρ_{M} and that |ρ_{p} − p| ≤ εp. Moreover, we have

e_{G_{M}} (U, W) ≥ e_{G_{p}}(U, W) ≥ (1 − η_{1})ρ_{p}|U||W | ≥ (1 − 2η_{1})ρ_{M}|U||W |

and

\begin{aligned}e_{G_{M}} (U, W) ≤ e_{G_{p}}(U, W) + 2εpn|U| & ≤ (1 + η_{1})ρ_{p}|U||W | + 2(εp/η_{1})|U||W | \\ & ≤ (1 + 2η_{1})ρ_{p}|U||W | ≤ (1 + 2η_{1})ρ_{M}|U||W |. \end{aligned}

Now notice that

d_{G_{M}} (U, W) = 3e_{G_{M}} (U, W)/ρ_{M}|U||W |

and the density between two any sets of tripartition of G(n, R_{M}) is 3. Thus,

|d_{G_{M}} (U, W) − 3| ≤ \frac{3|e_{G_{M}} (U, W) − ρ_{M}|U||W | |}{ρ_{M}|U||W |} ≤ 6η_{1} = η_{0} ,

and so G_{M} = G(n, R_{M}) is indeed η_{0}-uniform, as required.

In the sequel, it will be necessary for us to look at R(n, M) in a way that it resembles a product of a large number of spaces. Let us assume that the integer m divides M, and put M_{0} = M/m. We then define the space R̃(n, m, M_{0}) as the uniform space of m-tuples R̃ = (R_{i})_{i=1}^{m} of pairwise disjoint M_{0}-subsets R_{i} ⊆ [n]. Thus all mtuples R̃ ∈ R̃(n, m, M_{0}) are equiprobable, and the map R̃ = (R_{i})_{i=1}^{m} ∈ R̃(n, m, M_{0}) ↦ ⋃_{i=1}^{m} R_{i} ∈ R(n, M) is measure-preserving. We shall also consider the probability space G̃ = G̃(n, m, M_{0}) of the balanced 3-decomposable m-edge-coloured graphs G̃ = G̃(n, R̃) = (G_{i})_{i=1}^{m} determined by the G_{i} = G(n, R_{i}) (1 ≤ i ≤ m), where R̃ = (R_{i})_{i=1}^{m} is a random element of R̃(n, m, M_{0}). In this space we consider the event A(b, η) that a graph from G̃ should be (b, η)-sparse, and denote the conditional probability space obtained from G̃ by conditioning on A(b, η) by G̃(n, m, M_{0} | b, η).

Lemma 18. Let m ≥ 3, k_{0} ≥ 1, b > 4, 0 < ξ ≤ 1 and 0 < δ ≤ 1 be given. Then there exist constants 0 < ε = ε(m, ξ, δ) ≤ 1 and C = C(m, k_{0}, b, ξ, δ) for which the following holds for any sufficiently large n. Suppose Cn^{1/2} ≤ M = mM_{0} = mM_{0}(n) ≤ n/(log log n)^{2}, let G̃ = (G_{i})_{i=1}^{m} ∈ G̃(n, m, M_{0} | b, η) and let Π̃_{1}, …, Π̃_{m} be the (\underline{ε}, k_{0})- canonical sequence of partitions associated to G̃, where as usual \underline{ε} = \underline{ε}(m, b, ε, k_{0}). Then the probability that there exists a (δ, k_{0}, \underline{ε}; G̃)-flower which contains no spontaneous triangles is smaller than ξ^{M}.

Proof. Put β = ξ^{10^{8}mδ^{−2}}, and let 0 < ε = ε(β) = ε(m, ξ, δ) ≤ 1 be as given by Lemma 11. Furthermore, let η and K_{0} ≥ 1/2ε be such that Lemma 5 holds and let C_{1} = C(η) be as in Fact 17. Finally, set C = max\{C_{1}, 10^{7}m\sqrt{K_{0}}/δ\sqrt{ε}\}. We shall show that for such a choice of ε, η and C the assertion holds. We also remark that in the sequel we tacitly assume that n is sufficiently large whenever it is needed.

Let us first restate our result in terms of m-coloured graphs G̃ = (G_{i})_{i=1}^{m} from G̃(n, m, M_{0}). Let B(δ) be the event that G̃ should be (b, η)-sparse and moreover it should contain a (δ, k_{0}, \underline{ε}; G̃)-flower without a spontaneous triangle. We have to show that

Prob(B(δ) | A(b, η)) ≤ ξ^{M} .

Suppose we show that

Prob(B(δ)) ≤ ξ^{M}/2.  (7)

Then, by our choice of C, we have that for any large enough n we have Prob(A(b, η)) > 1/2. Therefore

Prob(B(δ) | A(b, η)) = \frac{Prob(B(δ) ∩ A(b, η))}{Prob(A(b, η))} ≤ \frac{Prob(B(δ))}{Prob A(b, η))} ≤ ξ^{M},

as required. Hence it only remains to prove (7).

Let us estimate first the probability that a fixed (δ, G̃)-flower should contain no spontaneous triangle. Thus, let the colours 1 ≤ w(1) < w(2) < w(3) ≤ m, the vertex v ∈ V (G), and the family {(X^{(i)}, Y^{(i)}): 1 ≤ i ≤ g}, where

u = |X^{(i)}| = |Y^{(i)}| ≥ n/2K_{0} ,

and 2gu ≥ δn, form a (δ, G̃)-flower. We may further assume that all the X^{(i)} (1 ≤ i ≤ g′ = ⌈g/2⌉) are contained in a single vertex class of the tripartition of G, as are all the Y^{(i)} (1 ≤ i ≤ g′). Below we shall only consider X^{(i)} and Y^{(i)} for 1 ≤ i ≤ g′. Also, let us mention for completeness that G has 3n vertices and e(G) = |E(G)| = 3nM = 3nmM_{0} edges.

Let us now suppose that the R_{i} (1 ≤ i < w(2)) have been chosen. Thus, in particular, the graphs H_{i} = G_{w(1)}[X^{(i)}, Y^{(i)}] (1 ≤ i ≤ g′), namely the bipartite subgraphs of G_{w(1)} with bipartition V (H_{i}) = X^{(i)} ∪ Y^{(i)} and edge set E_{G_{w(1)}}(X^{(i)}, Y^{(i)}), have been fixed. By the definition of a (δ, G̃)-flower, each H_{i} (1 ≤ i ≤ g′) is ε-uniform, and for all 1 ≤ i ≤ g′ we have e(H_{i}) = e_{G_{w(1)}}(X^{(i)}, Y^{(i)}) ≥ ρ_{0}u^{2}, where ρ_{0} = 10^{−6}δM_{0}/n.

Let us now pick R_{w(2)}, and let us study the neighbourhood N_{w(2)}^{(i)}(v) ⊂ X^{(i)} (1 ≤ i ≤ g′) of the vertex v in the graph G_{w(2)} inside the set X^{(i)}. Put d = ρ_{0}u. Again by the definition of a (δ, G̃)-flower, we know that d_{1}^{(i)} = |N_{w(2)}^{(i)}(v)| ≥ d = ρ_{0}u. Now let us condition on the values of the d_{1}^{(i)} (1 ≤ i ≤ g′). Once these g′ numbers are fixed, for every 1 ≤ i ≤ g′, all the subsets of cardinality d_{1}^{(i)} of the set X^{(i)} \ S_{1}^{(i)} are equally likely to be chosen as N_{w(2)}^{(i)}(v), where S_{1}^{(i)} is the neighbourhood of v within X^{(i)} in the graph ⋃_{1≤k<w(2)} G_{k}. Furthermore, we make the simple but important observation that, because we choose R_{w(2)} randomly and uniformly from all the M_{0}-subsets of [n] \ ⋃_{1≤k<w(2)} R_{k}, and because we have decided on the cardinalities d_{1}^{(i)} (1 ≤ i ≤ g′) in advance, the sets N_{w(2)}^{(i)}(v) (1 ≤ i ≤ g′) are all selected independently.

Let us now suppose that the R_{i} (w(2) ≤ i < w(3)) have also been chosen, and pick R_{w(3)}. Let N_{w(3)}^{(i)}(v) ⊂ Y^{(i)} be the G_{w(3)}-neighbourhood of v within Y^{(i)}. Put d_{2}^{(i)} = |N_{w(3)}^{(i)}(v)| (1 ≤ i ≤ g′), and note that again d_{2}^{(i)} ≥ d = ρ_{0}u. We now condition on the values of the d_{2}^{(i)} (1 ≤ i ≤ g′). As above, under this conditioning, for every 1 ≤ i ≤ g′, all the subsets of cardinality d_{2}^{(i)} of the set Y^{(i)} \ S_{2}^{(i)} are equally likely to be chosen as N_{w(3)}^{(i)}(v), where S_{2}^{(i)} is the neighbourhood of v within Y^{(i)} in the graph ⋃_{1≤k<w(3)} G_{k}. As before, the sets N_{w(3)}^{(i)}(v) (1 ≤ i ≤ g′) are all selected independently.

We now apply Lemma 11 to all the H_{i} (1 ≤ i ≤ g′). Recall that d = ρ_{0}u and notice that, for all 1 ≤ i ≤ g′, the density e(H_{i})u^{−2} of H_{i} is at least ρ_{0} = d/u and d_{j}^{(i)} ≥ d (j ∈ {1, 2}). Also, by the choice of C, we have that d ≥ 2(u/ε)^{1/2}. Moreover, since M = mM_{0} ≤ n/(log log n)^{2}, for large enough n we have that |S_{j}^{(i)}| ≤ u/ log log u for all 1 ≤ i ≤ g′, j ∈ {1, 2}. We now recall that the sets N_{w(2)}^{(i)}(v) (1 ≤ i ≤ g′) are all selected independently, as are all the N_{w(3)}^{(i)}(v) (1 ≤ i ≤ g′). Thus, applying Lemma 11 simultaneously to all the H_{i} (1 ≤ i ≤ g′), we see that the probability that we do not have a spontaneous triangle in G is at most β^{dg′} ≤ ξ^{2M}.

Now, to complete the proof, it suffices to estimate the number of all possible candidates for (δ, G̃)-flowers in our (b, η)-sparse m-edge-coloured graph G̃. Clearly, the vertex v can be selected in at most n ways, there are at most \binom{m}{3} ≤ m^{3} possible choices for the indices w(1), w(2) and w(3) and, since |Π̃_{w(1)}| ≤ K_{0} +1 (cf. Lemma 5) and g ≤ (|Π̃_{w(1)}|−1)/2 < K_{0}, the number of possible choices for the set of pairs {(X^{(i)}, Y^{(i)}): 1 ≤ i ≤ g} can be estimated very generously by K_{0} × K_{0}! ≤ (K_{0} + 1)!. Thus, since M grows to infinity at least as fast as \sqrt{n}, we have

Prob(B(δ)) ≤ nm^{3}(K_{0} + 1)!ξ^{2M} ≤ ξ^{M}/2

whenever n is large enough. Thus (7) holds and Lemma 18 is proved.

As an almost immediate consequence of the above lemma and Lemma 10, we get the following result, which will be crucial for the proof of Theorem 1.

Let R(n, M | b, η) denote the uniform probability space whose elements are the (b, η)- sparse M-subsets R of [n]. Clearly, R(n, M | b, η) may be obtained from R(n, M) by conditioning on the event that G(n, R) should be (b, η)-sparse. Let the associated probability space of the G(n, R) (R ∈ R(n, M | b, η)) be denoted by G(n, M | b, η). Thus to pick an element G from G(n, M | b, η) we simply generate R ∈ R(n, M | b, η) and let G = G(n, R). Suppose the integer m divides M. Clearly, since all graphs from G(n, M | b, η) are generated by M-subsets of [n] and each such subset can be decomposed into m subsets of size M_{0} = M/m in the same number of ways, one can generate an element of G(n, M | b, η) by choosing a graph from G̃(n, m, M_{0} | b, η) and ignoring the colouring of its edges.

Now our next result can be stated as follows.

Lemma 19. For every b ≥ 4 and 0 < ξ ≤ 1, there exist constants 0 < η(b, ξ) ≤ 1, C = C(b, ξ), and N = N(b, ξ) such that, for every n ≥ N and C\sqrt{n} ≤ M = mM_{0} = mM_{0}(n) ≤ n/(log log n)^{2}, where m = m(b) is as given in Lemma 10, the probability that G ∈ G(n, M | b, η) contains no spontaneous triangle is at most ξ^{M}.

Proof. Let b ≥ 4 and 0 < ξ ≤ 1 be given. Choose m = m(b) ≥ 3, 0 < δ = δ(b) ≤ 1 and k_{0} = k_{0}(b) ≥ 1 as in Lemma 10, and let 0 < ε = ε(m, ξ, δ) ≤ 1 be as in Lemma 18. Finally, let 0 < η = η(b, ε) ≤ 1 be as given by Lemma 10, and let C = C(m, k_{0}, b, ξ, δ) be as given by Lemma 18. As always, we shall always assume that n is sufficiently large whenever it is needed.

Let us now observe that, because of the choice of m, k_{0}, and δ, every element of G̃(n, m, M_{0} | b, η) contains a (\underline{ε}(m, b, ε, k_{0}), k_{0})-flower and, because of Lemma 18, with probability at least 1 − ξ^{M} every such flower must contain a spontaneous triangle. Thus the probability that an element of G̃(n, m, M_{0} | b, η) should contain no spontaneous triangle is smaller than ξ^{M}. As we have already mentioned, the graphs from G̃(n, m, M_{0} | b, η) naturally correspond to elements from G(n, mM_{0} | b, η), and hence Lemma 19 follows.

We may now finally prove Theorem 1.

Proof of Theorem 1. Clearly, it suffices to prove that, for any given 0 < α ≤ 1, there is a suitable choice for C = C(α) such that

(†) if Cn^{1/2} ≤ M = M(n) ≤ n, then lim_{n→∞} Prob(R →_{α} 3) = 1, where

the limit is taken along odd values of n. Thus henceforth we may and shall assume that n is odd. Also, below we assume that n

is sufficiently large whenever it is needed. For convenience, let us say that a property holds almost surely if it holds with probability tending to 1 as n tends to infinity along odd integers.

Let a constant 0 < α ≤ 1 be given, and assume that 1 ≤ M = M(n) ≤ n. Note that the Heath-Brown–Szemerédi result mentioned in §0 implies that any set A ⊂ [n] with |A| ≥ n/(log log n)^{2} contains a 3-term arithmetic progression provided n is sufficiently large. Thus we may and shall assume that αM ≤ n/(log log n)^{2}, since otherwise R →_{α} 3 for any set R ⊆ [n] with |R| = M.

Let us put α′ = α/2. Set b = 6/α′ ≥ 4 and 0 < ξ = α′/4 ≤ 1, and let m = m(b) ≥ 3 be as given in Lemma 10. Moreover, let 0 < η = η(b, ξ) ≤ 1 and C_{1} = C(b, ξ) be as given in Lemma 19, and let C_{2} = C(η) be as given in Fact 17. We let C = C(α′) = max{(4/3α′)C_{1}, C_{2}}. We shall now show that (†) holds with this choice of C.

Our first aim is to verify that R ∈ R(n, M) almost surely has the property that any subset A′ ⊆ R with at least α′|R| elements contains an arithmetic triple. For simplicity, let us write R →′_{α′} 3 if R has this property.

Let us start by picking an integer multiple M′ = M′(n) of m such that (3α′/4)M ≤ M′ ≤ α′M holds for any sufficiently large n. Put M_{0} = M_{0}(n) = M′/m. We now consider the spaces R(n, M′ | b, η) and G(n, M′ | b, η). Note that C_{1}n^{1/2} ≤ M′ = mM_{0} ≤ n/(log log n)^{2}, and hence that by Lemma 19 the probability that G ∈ G(n, M′ | b, η) contains no spontaneous triangle is at most ξ^{M′}. Thus, the probability that R ∈ R(n, M′ | b, η) contains no arithmetic triple is at most ξ^{M′} for all large enough n. Therefore, the number of (b, η)-sparse subsets A′ of [n] with M′ elements that do not contain arithmetic triples is at most ξ^{M′} \binom{n}{M′}.

Let D be the event that R ∈ R(n, M) should contain a (b, η)-sparse subset A′ with M′ elements which is free of arithmetic triples. Then the probability that D holds is at most

ξ^{M′} \binom{n}{M′}\binom{n − M′}{M − M′}\binom{n}{M}^{−1} ≤ (\frac{eξn}{M′})^{M′} (\frac{M}{n} )^{M′} ≤ (\frac{4eξ}{3α′} )^{M′} = o(1).

Let now S be the event that R ∈ R(n, M) should be (4, η)-sparse. Then, by Fact 17, we have that almost surely S holds. We now note that if R is (4, η)-sparse, then any subset A′ ⊆ R with |A′| = M′ is (b, η)-sparse. Therefore, if D fails and S holds, then R →′_{α′} 3. Since almost surely D does fail and S does hold, we conclude that a random set R ∈ R(n, M) satisfies R →′_{α′} 3 almost surely.

Now recall that n is odd, and write n = 2k + 1. Observe that if A is a subset of R with at least α|R| elements, then at least one of the subsets A_{1} = A ∩ {0, …, k} and A_{2} = A ∩ {k, …, 2k} must have at least α′|R| = α|R|/2 elements, and that A_{i} (i ∈ {1, 2}) contains an arithmetic triple if and only if it contains an arithmetic progression of length three. Thus (†) does hold and Theorem 1 is proved.

Corollary 3 may be deduced from Theorem 1 in a routine manner.

Sketch of the Proof of Corollary 3. Let s = s(n), g = g(n), and α be as in the statement of our corollary. Pick C_{0} sufficiently large so that with p = p(n) = C_{0}n^{−1/2} and R = R_{p} ∈ R(n, p) we have R →_{α/2} 3 with probability 1 − o(1) as n → ∞.

The probability that a fixed set as in (i) of the corollary meets R in more than 3 elements is O(s^{4}n^{−2}). Hence the expected number of such sets is O(s^{4}) = o(\sqrt{n}). The probability that a fixed set as in (ii) meets R in more than Cm/\sqrt{n} elements is at most exp\{−2m/\sqrt{n}\} for large enough C. Therefore the probability that such a set exists is o(1). Now, let G^{(3)} = G_{n}^{(3)} be the 3-uniform hypergraph on [n] whose hyperedges are the 3-term arithmetic progressions contained in [n]. One can easily check that for every ℓ ≥ 2 the number of cycles of length ℓ in G^{(3)} is at most (3n)^{ℓ}. For any such cycle of G^{(3)} the probability that it appears in F = F(R) is p^{2ℓ}. Therefore the expected number of cycles in F shorter than g is at most ∑_{ℓ=2}^{g}(3np^{2})^{ℓ} = o(\sqrt{n}).

In view of the above remarks, with probability 1 − o(1) as n → ∞, there exists a set S ⊆ R with |S| ≥ |R|/2 satisfying (i), (ii), and (iii). Finally, note that if R →_{α/2} 3, then S →_{α} 3.

REFERENCES

[ADLRY 94] Alon, N., Duke, R.A., Lefmann, H., Rödl, V. and Yuster, R., The algorithmic aspects of the regularity

lemma, J. Algorithms, 16 (1994), pp. 80–109.

[EFR 86] Erdős, P., Frankl, P. and Rödl, V., The asymptotic number of graphs not containing a fixed subgraph

and a problem for hypergraphs having no exponent, Graphs Combin., 2 (1986), pp. 113–121.

[ET 36] Erdős, P. and Turán, P., On some sequences of integers, J. London Math. Soc., 11 (1936), pp.

261–264.

[FRW 88] Frankl, P., Rödl, V. and Wilson, R., The number of submatrices of a given type in a Hadamard

matrix and related results, J. Combin. Th. B, 44 (1988), pp. 317–328. [Fu 77] Furstenberg, H., Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic

progressions, J. Analyse Math., 31 (1977), pp. 204–256.

[GN 86] Graham, R.L. and Nešetřil, J., Large minimal sets which force long arithmetic progressions, J.

Combin. Th. A, 42 (1986), pp. 270–276.

[GR 87] Graham, R.L. and Rödl, V., Numbers in Ramsey theory, in “Surveys in Combinatorics 1987,” (Whitehead, C., ed.), London Mathematical Society Lecture Note Series 123, Cambridge University Press, Cambridge, 1987, pp. 111–153.

[HKŁ ⋆⋆] Haxell, P.E., Kohayakawa, Y. and Łuczak, T., The induced size-Ramsey number of cycles, Combin.

Probab. Comput., to appear.

[H-B 87] Heath-Brown, D.R., Integer sets containing no arithmetic progressions, J. London Math. Soc., 35

(1987), pp. 385–394.

[JŁR 90] Janson, S., Łuczak, T. and Ruciński, A., An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, in “Random Graphs ’87,” (Karoński, M., Jaworski, J. and Ruciński, A., eds), Wiley, 1990, pp. 73–87.

[McD 89] McDiarmid, C.J.H., On the method of bounded differences, in “Surveys in Combinatorics 1989,” (Siemons, J., ed.), London Mathematical Society Lecture Notes Series 141, Cambridge University Press, Cambridge, 1989, pp. 148–188.

[NR 87] Nešetřil, J. and Rödl, V., Partite construction and Ramseyan theorems for sets, numbers and spaces,

Comm. Math. Univ. Carol., 28 (1987), pp. 569-580.

[PV 88] Prömel, H.-J. and Voigt, B., A sparse Graham–Rothschild theorem, Trans. Amer. Math. Soc., 309

(1988), pp. 113–137.

[Rö 90] Rödl, V., On Ramsey families of sets, Graphs Combin., 6 (1990), pp. 187–195. [Ro 53] Roth, K.F., On certain sets of integers, J. London Math. Soc., 28 (1953), pp. 104–109. [RSz 78] Ruzsa, I.Z. and Szemerédi, E., Triple systems with no six points carrying three triangles, Colloq.

Math. Soc. J. Bolyai, 18 (1978), pp. 939–945. [Sp 75] Spencer, J., Restricted Ramsey configurations, J. Combin. Th. A, 19 (1975), pp. 278–286. [Sz 75] Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arith.,

27 (1975), pp. 299–345.

[Sz 78] Szemerédi, E., Regular partitions of graphs, in “Problèmes Combinatoires et Théorie des Graphes, Proc. Colloque Inter. CNRS,” (Bermond, J.-C., Fournier, J.-C., Las Vergnas, M., Sotteau, D., eds), CNRS, Paris, 1978, pp. 399–401.

[Th 87a] Thomason, A., Pseudo-random graphs, in “Random Graphs ’85,” (Karoński, M., Palka, Z., eds),

Annals of Discrete Math. 33, North-Holland, Amsterdam, 1987, pp. 307–331.

[Th 87b] Thomason, A., Random graphs, strongly regular graphs and pseudo-random graphs, in “Surveys in Combinatorics 1987,” (Whitehead, C., ed.), London Mathematical Society Lecture Note Series 123, Cambridge University Press, Cambridge, 1987, pp. 173–195.