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

Andrea Freschi, Robert Hancock, Andrew Treglown

TL;DR

The paper develops a comprehensive framework for typical Ramsey phenomena in random discrete structures by introducing the random Rado lemma, a hypergraph-container–driven tool that yields $1$- and $0$-statements for $(A,r)$-Rado properties in random subsets of abelian groups. It unifies and extends classical results on partition regularity, supersaturation, and random Ramsey theory to primes, lattices, finite groups, and vector spaces, yielding random analogues of van der Waerden, Green–Tao, and Rado-type theorems. Central to the approach are notions of rank and projected solutions, and a simplified version of the lemma that applies cleanly to finite groups and finite-field powers with explicit thresholds like $|S_n|^{- rac{1}{m_S(A)}}$. The results have broad implications for how typical Ramsey properties endure under random thinning, with concrete applications to arithmetic progressions in primes, random lattices, and resilience phenomena in the primes, and point to rich future directions in supersaturation and container-method techniques in combinatorics.

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 $r \in \mathbb N$. Rödl and Ruciński and Friedgut, Rödl and Schacht proved a random version of Rado's theorem where one considers a random subset of $[n]:=\{1,\dots,n\}$ 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 $$\lim _{n \rightarrow \infty} \mathbb P [ S_{n,p} \text{ is } (A,r)\text{-Rado}]= \begin{cases} 0 &\text{ if } p=o(\hat p); \\ 1 &\text{ if } p=ω(\hat p). \end{cases}$$ 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.

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

TL;DR

The paper develops a comprehensive framework for typical Ramsey phenomena in random discrete structures by introducing the random Rado lemma, a hypergraph-container–driven tool that yields - and -statements for -Rado properties in random subsets of abelian groups. It unifies and extends classical results on partition regularity, supersaturation, and random Ramsey theory to primes, lattices, finite groups, and vector spaces, yielding random analogues of van der Waerden, Green–Tao, and Rado-type theorems. Central to the approach are notions of rank and projected solutions, and a simplified version of the lemma that applies cleanly to finite groups and finite-field powers with explicit thresholds like . The results have broad implications for how typical Ramsey properties endure under random thinning, with concrete applications to arithmetic progressions in primes, random lattices, and resilience phenomena in the primes, and point to rich future directions in supersaturation and container-method techniques in combinatorics.

Abstract

Given a matrix with integer entries, a subset of an abelian group and , we say that is -Rado if any -colouring of yields a monochromatic solution to the system of equations . A classical result of Rado characterises all those matrices such that is -Rado for all . Rödl and Ruciński and Friedgut, Rödl and Schacht proved a random version of Rado's theorem where one considers a random subset of instead of . In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence of finite subsets of abelian groups, let be a random subset of obtained by including each element of independently with probability . We are interested in determining the probability threshold such that 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 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.
Paper Structure (43 sections, 51 theorems, 160 equations, 1 figure)

This paper contains 43 sections, 51 theorems, 160 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Arithmetic progressions in the primes
  4. The Green--Tao theorem
  5. Monochromatic arithmetic progressions in the primes
  6. Partition regularity and supersaturation
  7. Partition regularity
  8. Supersaturation
  9. The random Ramsey problem for graphs and the integers
  10. Ramsey properties of random graphs
  11. Ramsey properties of random sets of integers
  12. Ramsey properties of random sets of $\mathbb Z_n$
  13. Further applications of the random Rado lemma
  14. The random Rado lemma
  15. Definitions and main statement
  16. ...and 28 more sections

Key Result

Theorem 2.1

Let $k \geq 2$. The set of primes $\mathbf P$ contains arithmetic progressions of arbitrary length. Moreover, given any subset $S \subseteq \mathbf P_n$ of size $|S|=\Omega (|\mathbf P_n|)$, the number of $k$-APs in $S$ is $\Theta (n^2/\log ^k n)$.Note that the moreover part of Theorem gt is implici

Figures (1)

  • Figure 1: A visualisation of the main new (blue) and previously known (red) results covered in this paper.

Theorems & Definitions (114)

  • Theorem 2.1: Green and Tao greentao
  • Theorem 2.2: Random Green--Tao theorem
  • Lemma 2.3
  • Theorem 2.4: Balogh, Liu and Sharifzadeh bls
  • Theorem 2.5: van der Waerden vdw
  • Theorem 2.6: van der Waerden theorem for the primes greentao
  • Theorem 2.7: Random van der Waerden theorem for the primes
  • Definition 3.1: The columns condition over $R$
  • Theorem 3.2: Rado rado
  • Theorem 3.3: Bergelson, Deuber and Hindman berg2
  • ...and 104 more