{
  "id": "1208.5371",
  "version": "v2",
  "published": "2012-08-27T12:15:53.000Z",
  "updated": "2012-10-13T12:30:40.000Z",
  "title": "Union-Closed vs Upward-Closed Families of Finite Sets",
  "authors": [
    "Emanuele Rodaro"
  ],
  "comment": "40 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A finite family $\\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\\in\\mathrsfs{F}$ implies $f\\cup g\\in\\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further the connection between union-closed families and upward-closed families started in Reimer (2003) using rising operators. With these techniques we are able to obtain tight lower bounds to the average of the length of the elements of $\\mathrsfs{F}$ and to prove that the number of joint-irreducible elements of $\\mathrsfs{F}$ can not exceed $2{n\\choose \\lfloor n/2\\rfloor}+{n\\choose \\lfloor n/2\\rfloor+1}$ where $|X| = n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-10-13T12:30:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65"
    ],
    "keywords": [
      "finite set",
      "upward-closed families",
      "tight lower bounds",
      "frankls conjecture",
      "connection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.5371R"
    }
  }
}