{
  "id": "1812.06448",
  "version": "v1",
  "published": "2018-12-16T11:46:09.000Z",
  "updated": "2018-12-16T11:46:09.000Z",
  "title": "Partitioning the power set of $[n]$ into $C_k$-free parts",
  "authors": [
    "Eben Blaisdell",
    "András Gyárfás",
    "Robert A. Krueger",
    "Ronen Wdowinski"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that for $n \\geq 3, n\\ne 5$, in any partition of $\\mathcal{P}(n)$, the set of all subsets of $[n]=\\{1,2,\\dots,n\\}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,C\\subseteq [n]$ such that $A\\cap B$, $A\\cap C$, and $B\\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp (for fixed $k$) when $G=C_k, P_k, S_k$, a cycle, path, or star with $k$ edges. Additional bounds are given for $G=C_4$ and $G=S_3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-16T11:46:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "power set",
      "general ramsey-type problem",
      "color class contains",
      "additional bounds",
      "complementary pairs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}