{
  "id": "1007.3037",
  "version": "v3",
  "published": "2010-07-18T20:24:11.000Z",
  "updated": "2012-04-17T10:05:20.000Z",
  "title": "When does the K_4-free process stop?",
  "authors": [
    "Lutz Warnke"
  ],
  "comment": "39 pages, 3 figures. Minor edits. To appear in Random Structures and Algorithms",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "The K_4-free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K_4. Let G be the random maximal K_4-free graph obtained at the end of the process. We show that for some positive constant C, with high probability as $n \\to \\infty$, the maximum degree in G is at most $C n^{3/5}\\sqrt[5]{\\log n}$. This resolves a conjecture of Bohman and Keevash for the K_4-free process and improves on previous bounds obtained by Bollob\\'as and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has $\\Theta(n^{8/5}\\sqrt[5]{\\log n})$ edges and is `nearly regular', i.e., every vertex has degree $\\Theta(n^{3/5}\\sqrt[5]{\\log n})$. This answers a question of Erd\\H{o}s, Suen and Winkler for the K_4-free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least $\\Omega(n^{2/5}(\\log n)^{4/5}/\\log \\log n)$, which matches an upper bound obtained by Bohman up to a factor of $\\Theta(\\log \\log n)$. Our analysis of the K_4-free process also yields a new result in Ramsey theory: for a special case of a well-studied function introduced by Erd\\H{o}s and Rogers we slightly improve the best known upper bound.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-04-17T10:05:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60C05"
    ],
    "keywords": [
      "process stop",
      "high probability",
      "upper bound",
      "additional structural property",
      "random maximal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3037W"
    }
  }
}