{
  "id": "1501.03129",
  "version": "v1",
  "published": "2015-01-13T19:53:48.000Z",
  "updated": "2015-01-13T19:53:48.000Z",
  "title": "A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity",
  "authors": [
    "Zoltán Füredi"
  ],
  "comment": "4 pages plus references",
  "categories": [
    "math.CO"
  ],
  "abstract": "The following sharpening of Tur\\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there exists an (at most) $p$-chromatic subgraph $H_0$ such that $e(H_0)\\geq e(G)-t$. Using this result we present a concise, contemporary proof (i.e., one applying Szemer\\'edi's regularity lemma) for the classical stability result of Simonovits.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-13T19:53:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "extremal graphs",
      "szemerédis regularity",
      "simonovits",
      "applying szemeredis regularity lemma",
      "classical stability result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}