{
  "id": "1204.1927",
  "version": "v2",
  "published": "2012-04-09T17:06:15.000Z",
  "updated": "2013-06-23T22:15:43.000Z",
  "title": "On the co-degree threshold for the Fano plane",
  "authors": [
    "Louis DeBiasio",
    "Tao Jiang"
  ],
  "comment": "12 pages, 2 figures. This version of the paper contains some additional results obtained using Razborov's flag algebra calculus. We did not include these results in the \"European Journal of Combinatorics\" version",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a 3-graph H, let \\ex_2(n, H) denote the maximum value of the minimum codegree of a 3-graph on n vertices which does not contain a copy of H. Let F denote the Fano plane, which is the 3-graph \\{axx',ayy',azz',xyz',xy'z,x'yz,x'y'z'\\}. Mubayi proved that \\ex_2(n,F)=(1/2+o(1))n and conjectured that \\ex_2(n, F)=\\floor{n/2} for sufficiently large n. Using a very sophisticated quasi-randomness argument, Keevash proved Mubayi's conjecture. Here we give a simple proof of Mubayi's conjecture by using a class of 3-graphs that we call rings. We also determine the Tur\\'an density of the family of rings.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-23T22:15:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fano plane",
      "co-degree threshold",
      "mubayis conjecture",
      "minimum codegree",
      "sophisticated quasi-randomness argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.1927D"
    }
  }
}