{
  "id": "1305.6715",
  "version": "v1",
  "published": "2013-05-29T08:09:46.000Z",
  "updated": "2013-05-29T08:09:46.000Z",
  "title": "The minimum number of disjoint pairs in set systems and related problems",
  "authors": [
    "Shagnik Das",
    "Wenying Gan",
    "Benny Sudakov"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let F be a set system on [n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdos-Ko-Rado from 1961 says that any such system has size at most ${n-1 \\choose k-1}$. A natural question, which was asked by Ahlswede in 1980, is how many disjoint pairs must appear in a set system of larger size. Except for the case k=2, solved by Ahlswede and Katona, this problem has remained open for the last three decades. In this paper, we determine the minimum number of disjoint pairs in small k-uniform families, thus confirming a conjecture of Bollobas and Leader in these cases. Moreover, we obtain similar results for two well-known extensions of the Erdos-Ko-Rado theorem, determining the minimum number of matchings of size q and the minimum number of t-disjoint pairs that appear in set systems larger than the corresponding extremal bounds. In the latter case, this provides a partial solution to a problem of Kleitman and West.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-29T08:09:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimum number",
      "disjoint pairs",
      "related problems",
      "small k-uniform families",
      "set systems larger"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6715D"
    }
  }
}