Typical Ramsey properties of the primes, abelian groups and other discrete structures

Given a matrix A with integer entries, a subset S of an abelian group and r N, 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 N is (A, r) -Rado for all r N. R\"odl and Ruci\'nski and Friedgut, R\"odl and Schacht proved a random version of Rado's theorem where one considers a random subset of n: =\1, , n\ instead of N. In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence (Sₙ) ₍ ₍ of finite subsets of abelian groups, let S₍, ₏ be a random subset of Sₙ obtained by including each element of Sₙ independently with probability p. We are interested in determining the probability threshold p: = p (n) such that ₍ P S₍, ₏ is (A, r) -Rado= cases 0 \\ 1 & if p= (p). cases Our main result, which we coin the random Rado lemma, is a general black box 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 Sₙ: =nᵈ and use it to prove an integer lattice generalisation of the random version of Rado's theorem. Various threshold results for abelian groups are also given.