Cameron and Erdős [6] asked whether the number of maximal sum-free sets in {1, . . . , n} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2 ⌊n/4⌋ for the number of maximal sum-free sets. Here, we prove the following: For each 1 ≤ i ≤ 4, there is a constant C i such that, given any n ≡ i mod 4, {1, . . . , n} contains (C i + o(1))2 n/4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [11,12], a structural result of Deshouillers, Freiman, Sós and Temkin [7] and a recent bound on the number of subsets of integers with small sumset by Green and Morris [13]. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.

The classical Corrádi‐Hajnal theorem claims that every n‐vertex graph G with δ(G)≥2n/3 contains a triangle factor, when 3|n. In this paper we present two related results that both use the absorbing technique of Rödl, Ruciński and Szemerédi. Our main result determines the minimum degree condition necessary to guarantee a triangle factor in graphs with sublinear independence number. In particular, we show that if G is an n‐vertex graph with α(G)=o(n) and δ(G)≥(1/2+o(1))n, then G has a triangle factor and this is asymptotically best possible. Furthermore, it is shown for every r that if every linear size vertex set of a graph G spans quadratically many edges, and δ(G)≥(1/2+o(1))n, then G has a Kr‐factor for n sufficiently large. We also propose many related open problems whose solutions could show a relationship with Ramsey‐Turán theory. Additionally, we also consider a fractional variant of the Corrádi‐Hajnal Theorem, settling a conjecture of Balogh‐Kemkes‐Lee‐Young. Let t∈(0,1) and w:E(Kn)→[0,1]. We call a triangle t‐heavy if the sum of the weights on its edges is more than 3t. We prove that if 3|n and w is such that for every vertex v the sum of w(e) over edges e incident to v is at least (1+2t3+o(1))n, then there are n/3 vertex disjoint heavy triangles in G. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 669–693, 2016

Cameron and Erdős [6] raised the question of how many maximal sum-free sets there are in {1, . . . , n}, giving a lower bound of 2 ⌊n/4⌋ . In this paper we prove that there are in fact at most 2 (1/4+o(1))n maximal sum-free sets in {1, . . . , n}. Our proof makes use of container and removal lemmas of Green [8,9] as well as a result of Deshouillers, Freiman, Sós and Temkin [7] on the structure of sum-free sets.A fundamental notion in combinatorial number theory is that of a sum-free set: A set S of integers is sum-free if x + y ∈ S for every x, y ∈ S (note x and y are not necessarily distinct here). The topic of sum-free sets of integers has a long history. Indeed, in 1916 Schur [19] proved that, if n is sufficiently large, then any r-colouring of [n] := {1, . . . , n} yields a monochromatic triple x, y, z such that x + y = z.Note that both the set of odd numbers in [n] and the set {⌊n/2⌋ + 1, . . . , n} are maximal sum-free sets. (A sum-free subset of [n] is maximal if it is not properly contained in another sum-free subset of [n].) By considering all possible subsets of one of these maximal sum-free sets, we see that [n] contains at least 2 ⌈n/2⌉ sum-free sets. Cameron and Erdős [5] conjectured that in fact [n] contains only O(2 n/2 ) sum-free sets. The conjecture was proven independently by Green [8] and Sapozhenko [16]. Recently, a refinement of the Cameron-Erdős conjecture was proven in [1], giving an upper bound on the number of sum-free sets in [n] of size m (for each 1 ≤ m ≤ ⌈n/2⌉).Let f (n) denote the number of sum-free subsets of [n] and f max (n) denote the number of maximal sum-free subsets of [n]. Recall that the sum-free subsets of [n] described above lie in *

Addressing a question of Cameron and Erdős, we show that, for infinitely many values of n, the number of subsets of {1, 2, . . . , n} that do not contain a k-term arithmetic progression is at most 2 O(r k (n)) , where r k (n) is the maximum cardinality of a subset of {1, 2, . . . , n} without a k-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all values of n, we prove a weaker bound, which is nevertheless sufficient to transfer the current best upper bound on r k (n) to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size Θ(r k (n)) contain superlinearly many k-term arithmetic progressions.For integers r and k, Erdős asked whether there is a set of integers S with no (k+1)term arithmetic progression, but such that any r-coloring of S yields a monochromatic k-term arithmetic progression. Nešetřil and Rödl, and independently Spencer, answered this question affirmatively. We show the following density version: for every k ≥ 3 and δ > 0, there exists a reasonably dense subset of primes S with no (k+1)-term arithmetic progression, yet every U ⊆ S of size |U | ≥ δ|S| contains a k-term arithmetic progression.Our proof uses the hypergraph container method, which has proven to be a very powerful tool in extremal combinatorics. The idea behind the container method is to have a small certificate set to describe a large independent set. We give two further applications in the appendix using this idea.

The study of intersecting structures is central to extremal combinatorics. A family of permutations F ⊂ S n is t-intersecting if any two permutations in F agree on some t indices, and is trivial if all permutations in F agree on the same t indices. A k-uniform hypergraph is t-intersecting if any two of its edges have t vertices in common, and trivial if all its edges share the same t vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for n sufficiently large with respect to t, the largest t-intersecting families in S n are the trivial ones. The classic Erdős-Ko-Rado theorem shows that the largest t-intersecting k-uniform hypergraphs are also trivial when n is large. We determine the typical structure of t-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollobás set-pairs inequality to bound the number of maximal intersecting families, which can then 225 be combined with known stability theorems. We also obtain similar results for vector spaces.

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r ∈ N, almost every n-vertex K r+1 -free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r (log n) 1/4 . The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.

It is known that a sequence {Π i } i∈N of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Π i converges to 1∕24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Π i converges to |S|∕24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.

Recently, settling a question of Erdős, Balogh, and Petříčková showed that there are at most 2 n 2 /8+o(n 2 ) n-vertex maximal triangle-free graphs, matching the previously known lower bound. Here, we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph G admits a vertex partition X ∪ Y such that G[X ] is a perfect matching and Y is an independent set.Our proof uses the Ruzsa-Szemerédi removal lemma, the Erdős-Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint P 3 s, which is of independent interest.