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

Abstract

Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r \in \mathbb N$, 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 $\mathbb N$ is $(A,r)$-Rado for all $r \in \mathbb N$. 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 $[n]:=\{1,\dots,n\}$ instead of $\mathbb N$. In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence $(S_n)_{n\in\mathbb N}$ 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 $\hat p:=\hat p(n)$ such that $$\lim _{n \rightarrow \infty} \mathbb P [ S_{n,p} \text{ is } (A,r)\text{-Rado}]= \begin{cases} 0 &\text{ if } p=o(\hat p); \\ 1 &\text{ if } p=\omega(\hat p). \end{cases}$$ 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 $S_n:=[n]^d$ 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.