Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers

Abstract

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.

The Maker--Breaker Rado Game on a Random Set of Integers

“…We will use the following simplification of Theorem 4.7 in [10], a container result, which is itself a consequence of the general container theorems of Balogh, Morris and Samotij [1] and Saxton and Thomason [21]. Note that in [10] the theorem is stated for the homogeneous case Ax = 0 only. However, the result easily generalises to the inhomogeneous case Ax = b.…”

“…Let p > Cn −1/m(A) . Note that m(A) > 1 (see Proposition 4.3 in [10]) and thus pn tends to infinity as n tends to infinity. Let R be as in the statement of the theorem and set X :…”

“…Theorem 5 ( [10,23]). Let A be a fixed integer-valued matrix of dimension ℓ × k. Given A is irredundant and satisfies ( * ), for all ε > 0, there exists a positive constant C such that if p > Cn −1/m(A) , then for any…”

On the optimality of the uniform random strategy

“…Hence the dimension of these rows must be rk(A) − rk(A Q ) = r Q . We identify a set of r Q linearly independent rows and permute them to the top of the matrix, hence obtaining the promised block decomposition (13). Note that the rank of B is r Q by construction.…”

“…To prove (iii), note that from (13) it follows that for any solution x ∈ S(A, b) = S(P⋅A, P ⋅ b) we S(B, c). The second statement follows by noting that c(P, A, Q, b) = 0 provided that b = 0.…”