{
  "id": "1404.5258",
  "version": "v3",
  "published": "2014-04-21T18:05:22.000Z",
  "updated": "2015-11-12T13:40:00.000Z",
  "title": "Maximum-size antichains in random set-systems",
  "authors": [
    "Maurício Collares Neto",
    "Robert Morris"
  ],
  "comment": "14 pages, added important observation to the Introduction",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that, for $pn \\to \\infty$, the largest set in a $p$-random sub-family of the power set of $\\{1, \\ldots, n\\}$ containing no $k$-chain has size $( k - 1 + o(1) ) p \\binom{n}{n/2}$ with high probability. This confirms a conjecture of Osthus, and has been proved independently by Balogh, Mycroft and Treglown.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-08T13:06:09.000Z",
      "title": "Maximum-size antichains in random sets",
      "abstract": "We show that, for $pn \\to \\infty$, the largest set in a $p$-random subset of the power set of $\\{1, \\ldots, n\\}$ containing no $k$-chain has size $( k - 1 + o(1) ) p \\binom{n}{n/2}$ with high probability. In the case $k = 2$, this confirms a conjecture of Osthus, and has been proved independently by Balogh, Mycroft and Treglown.",
      "comment": "13 pages, proof for general k",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-11-12T13:40:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random sets",
      "maximum-size antichains",
      "power set",
      "largest set",
      "random subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.5258C"
    }
  }
}