Typical Ramsey properties of the primes and abelian groups

Given a matrix A with integer entries, a subset S of an abelian group and integer r we say that S is (A,r)-Rado if any r-colouring of S yields a monochromatic solution to the system of equations Ax=0. A classical result of Rado characterises all those matrices A such that the natural numbers are (A,r)-Rado for all integers r. Rödl and Ruciński and Friedgut, Rödl and Schacht proved a random version of Rado’s theorem where one considers a random subset of [n]:={1,..,n}.

We investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence S_n of finite subsets of abelian groups, let S_{n,p} be a random subset of S_n obtained by including each element of S_n independently with probability p. We are interested in determining the probability threshold for S_{n,p} being (A,r)-Rado.

Our main result is a general black box for hypergraphs which we use to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. A consequence of the Green-Tao theorem is the van der Waerden theorem for the primes: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for [n]^d and use it to prove an integer lattice generalisation of the random version of Rado's theorem.

Joint work with Andrea Freschi (Alfréd Rényi Institute of Mathematics) and Andrew Treglown (University of Birmingham)