{
  "id": "2205.02275",
  "version": "v1",
  "published": "2022-05-04T18:23:03.000Z",
  "updated": "2022-05-04T18:23:03.000Z",
  "title": "Poset Ramsey Number $R(P,Q_n)$. II. Antichains",
  "authors": [
    "Christian Winter"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For two posets $(P,\\le_P)$ and $(P',\\le_{P'})$, we say that $P'$ contains a copy of $P$ if there exists an injective function $f\\colon P'\\to P$ such that for every two $X,Y\\in P$, $X\\le_P Y$ if and only if $f(X)\\le_{P'} f(Y)$. Given two posets $P$ and $Q$, let the poset Ramsey number $R(P,Q)$ be the smallest integer $N$ such that any coloring of the elements of an $N$-dimensional Boolean lattice in blue or red contains either a copy of $P$ where all elements are blue or a copy of $Q$ where all elements are red. We determine the poset Ramsey number $R(A_t,Q_n)$ of an antichain versus a Boolean lattice for small $t$ by showing that $R(A_t,Q_n)=n+3$ for $3\\le t\\le \\log \\log n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-04T18:23:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "05D10"
    ],
    "keywords": [
      "poset ramsey number",
      "dimensional boolean lattice",
      "smallest integer",
      "injective function",
      "red contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}