{
  "id": "1106.0369",
  "version": "v1",
  "published": "2011-06-02T06:15:54.000Z",
  "updated": "2011-06-02T06:15:54.000Z",
  "title": "Minimum density of union-closed families",
  "authors": [
    "Igor Balla"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let F be a finite union-closed family of sets whose largest set contains n elements. In \\cite{Wojcik92}, Wojcik defined the density of F to be the ratio of the average set size of F to n and conjectured that the minimum density over all union-closed families whose largest set contains n elements is (1 + o(1))\\log_2(n)/(2n) as n approaches infinity. We use a result of Reimer \\cite{Reimer03} to show that the density of F is always at least log_2(n)/(2n), verifying Wojcik's conjecture. As a corollary we show that for n \\geq 16, some element must appear in at least \\sqrt{(\\log_2(n))/n}(|F|/2) sets of F.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-06-02T06:15:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimum density",
      "union-closed family",
      "largest set contains",
      "average set",
      "approaches infinity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.0369B"
    }
  }
}