{
  "id": "2405.19113",
  "version": "v1",
  "published": "2024-05-29T14:23:20.000Z",
  "updated": "2024-05-29T14:23:20.000Z",
  "title": "Typical Ramsey properties of the primes, abelian groups and other discrete structures",
  "authors": [
    "Andrea Freschi",
    "Robert Hancock",
    "Andrew Treglown"
  ],
  "comment": "58 pages, 1 figure",
  "categories": [
    "math.CO",
    "math.GR",
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-29T14:23:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian group",
      "typical ramsey properties",
      "discrete structures",
      "long monochromatic arithmetic progressions",
      "contains arbitrarily long monochromatic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}