{
  "id": "1910.10876",
  "version": "v1",
  "published": "2019-10-24T01:47:53.000Z",
  "updated": "2019-10-24T01:47:53.000Z",
  "title": "The asymptotic behavior of the covering number of a graph on two layers of the Boolean lattice",
  "authors": [
    "József Balogh",
    "William Linz"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $n > k > \\ell \\ge 1$ be positive integers, and define a bipartite graph $G_{k, \\ell}=(V, E)$ by $V(G_{k, \\ell})=(\\binom{[n]}{k}, \\binom{[n]}{\\ell})$ and $(A, B) \\in E(G_{k, \\ell})$ if and only if $A \\in \\binom{[n]}{\\ell}$, $B\\in \\binom{[n]}{k}$, and $A\\subset B$. We confirm a conjecture of Badakhshian, Katona and Tuza for the two-sided covering number $\\gamma(G_{k, 2})$, for fixed $k$ and $n\\rightarrow\\infty$. Additionally, we prove a general lower bound for $\\gamma(G_{k, \\ell})$, with $k$ and $\\ell$ fixed and $n\\rightarrow \\infty$. Our proof uses the graph removal lemma and a Frankl-R\\\"odl nibble type theorem of Pippenger.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-24T01:47:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behavior",
      "boolean lattice",
      "general lower bound",
      "graph removal lemma",
      "nibble type theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}