{
  "id": "2204.03010",
  "version": "v1",
  "published": "2022-04-06T18:00:17.000Z",
  "updated": "2022-04-06T18:00:17.000Z",
  "title": "Poset Ramsey number $R(P,Q_n)$. I. Complete multipartite posets",
  "authors": [
    "Christian Winter"
  ],
  "comment": "8 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A poset $(P',\\le_{P'})$ contains a copy of some other poset $(P,\\le_P)$ if there is an injection $f\\colon P'\\to P$ where for every $X,Y\\in P$, $X\\le_P Y$ if and only if $f(X)\\le_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest integer $N$ such that any blue/red coloring of a Boolean lattice of dimension $N$ contains either a copy of $P$ with all elements blue or a copy of $Q$ with all elements red. We denote by $K_{t_1,\\dots,t_\\ell}$ a complete $\\ell$-partite poset, i.e.\\ a poset consisting of $\\ell$ pairwise disjoint sets $A^i$ of size $t_i$, $1\\le i\\le \\ell$, such that for any $i,j\\in\\{1,\\dots,\\ell\\}$ and any two $X\\in A^{i}$ and $Y\\in A^{j}$, $X<Y$ if and only if $i<j$. In this paper we show that $R(K_{t_1,\\dots,t_\\ell},Q_n)\\le n+\\frac{(2+o_n(1))\\ell n}{\\log n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-06T18:00:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "05D10"
    ],
    "keywords": [
      "poset ramsey number",
      "complete multipartite posets",
      "smallest integer",
      "boolean lattice",
      "elements blue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}