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

TL;DR

This paper extends classical Ramsey theory to random subsets of abelian groups, establishing probability thresholds for monochromatic solutions to linear systems, and applies these results to primes, lattices, and other structures.

Contribution

It introduces the random Rado lemma as a new tool to analyze threshold probabilities for Ramsey properties in random abelian groups and related structures.

Findings

Determined probability thresholds for (A,r)-Rado properties in various settings.

Established a random version of the Green-Tao theorem for primes.

Proved a supersaturation result for integer lattices.

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.