{
  "id": "2104.02002",
  "version": "v1",
  "published": "2021-04-05T16:47:06.000Z",
  "updated": "2021-04-05T16:47:06.000Z",
  "title": "Ramsey numbers of Boolean lattices",
  "authors": [
    "Dániel Grósz",
    "Abhishek Methuku",
    "Casey Tompkins"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The poset Ramsey number $R(Q_m,Q_n)$ is the smallest integer $N$ such that any blue-red coloring of the elements of the Boolean lattice $Q_N$ has a blue induced copy of $Q_m$ or a red induced copy of $Q_n$. The weak poset Ramsey number $R_w(Q_m,Q_n)$ is defined analogously, with weak copies instead of induced copies. It is easy to see that $R(Q_m,Q_n) \\ge R_w(Q_m,Q_n)$. Axenovich and Walzer showed that $n+2 \\le R(Q_2,Q_n) \\le 2n+2$. Recently, Lu and Thompson improved the upper bound to $\\frac{5}{3}n+2$. In this paper, we solve this problem asymptotically by showing that $R(Q_2,Q_n)=n+O(n/\\log n)$. In the diagonal case, Cox and Stolee proved $R_w(Q_n,Q_n) \\ge 2n+1$ using a probabilistic construction. In the induced case, Bohman and Peng showed $R(Q_n,Q_n) \\ge 2n+1$ using an explicit construction. Improving these results, we show that $R_w(Q_m,Q_n) \\ge n+m+1$ for all $m \\ge 2$ and large $n$ by giving an explicit construction; in particular, we prove that $R_w(Q_2,Q_n)=n+3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-05T16:47:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean lattice",
      "weak poset ramsey number",
      "explicit construction",
      "upper bound",
      "blue induced copy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}