{
  "id": "1207.6717",
  "version": "v1",
  "published": "2012-07-28T18:09:19.000Z",
  "updated": "2012-07-28T18:09:19.000Z",
  "title": "On the triangle space of a random graph",
  "authors": [
    "Bobby DeMarco",
    "Arran Hamm",
    "Jeff Kahn"
  ],
  "comment": "20 pages",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its edges lies in a triangle (which happens (w.h.p.) when $p$ is at least about $\\sqrt{(3/2)\\ln n/n}$, and not below this unless $p$ is very small.) We give two related proofs of this statement, together with a relatively simple proof of a fundamental \"stability\" theorem for triangle-free subgraphs of $G_{n,p}$, originally due to Kohayakawa, \\L uczak and R\\\"odl, that underlies the first of our proofs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-28T18:09:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C35",
      "05D40",
      "55U10",
      "60C05"
    ],
    "keywords": [
      "random graph",
      "triangle space",
      "clique complex",
      "triangle-free subgraphs",
      "cycle space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.6717D"
    }
  }
}