Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij [3], and independently Saxton and Thomason [62] developed very general container theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán properties) of (hyper)graphs and the integers. In particular:• We generalise the random Ramsey theorem of Rödl and Ruciński [54, 55, 56] by providing a resilience analogue. Our result unifies and generalises several fundamental results in the area including the random version of Turán's theorem due to Conlon and Gowers [14] and Schacht [64]. • The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter [40]. • All of the above results in fact hold for uniform hypergraphs.• For a (hyper)graph H, we determine, up to an error term in the exponent, the number of nvertex (hyper)graphs G that have the Ramsey property with respect to H (that is, whenever G is r-coloured, there is a monochromatic copy of H in G). • We strengthen the random Rado theorem of Friedgut, Rödl and Schacht [24] by proving a resilience version of the result. • For partition regular matrices A we determine, up to an error term in the exponent, the number of subsets of {1, . . . , n} for which there exists an r-colouring which contains no monochromatic solutions to Ax = 0. Along the way a number of open problems are posed. MSC2000: 5C30, 5C55, 5D10, 11B75.

A probability measure on the subsets of the edge set of a graph G is a 1-independent probability measure (1-ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1-ipm , denote by G the associated random graph model. Let  1,⩾p (G) denote the collection of 1-ipms on G for which each edge is included in G with probability at least p. For G = Z 2 , Balister and Bollobás asked for the value of the least p ⋆ such that for all p > p ⋆ and all ∈  1,⩾p (G), (G) almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p ⋆. We also determine the 1-independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f 1,G (p), the infimum over all ∈  1,⩾p (G) of the probability that G is connected. We determine f 1,G (p) exactly when G is a path, a complete graph and a cycle of length at most 5.

Abstract. Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any 'non-trivial' solutions to L. In this paper we consider the following three general questions:

We develop a limit theory of Latin squares, paralleling the recent limit theories of dense graphs and permutations. We introduce a notion of density, an appropriate version of the cut distance, and a space of limit objects -so-called Latinons. Key results of our theory are the compactness of the limit space and the equivalence of the topologies induced by the cut distance and the left-convergence. Last, using Keevash's recent results on combinatorial designs, we prove that each Latinon can be approximated by a finite Latin square.

Abstract. Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any 'non-trivial' solutions to L. We determine the precise size of the largest L-free subset of [n] for several general classes of linear equations L of the form px + qy = rz for fixed p, q, r ∈ N where p ≥ q ≥ r. Further, for all such linear equations L, we give an upper bound on the number of maximal L-free subsets of [n]. In the case when p = q ≥ 2 and r = 1 this bound is exact up to an error term in the exponent. We make use of container and removal lemmas of Green [12] to prove this result. Our results also extend to various linear equations with more than three variables.MSC2010: 11B75, 05C69.