{
  "id": "2406.10745",
  "version": "v1",
  "published": "2024-06-15T21:53:31.000Z",
  "updated": "2024-06-15T21:53:31.000Z",
  "title": "Strong Brandt-Thomassé Theorems",
  "authors": [
    "Tomasz Łuczak",
    "Joanna Polcyn",
    "Christian Reiher"
  ],
  "comment": "34 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Solving a long standing conjecture of Erd\\H{o}s and Simonovits, Brandt and Thomass\\'e proved that the chromatic number of each triangle-free graph $G$ such that $\\delta(G)>|V(G)|/3$ is at most four. In fact, they showed the much stronger result that every maximal triangle-free graph $G$ satisfying this minimum degree condition is a blow-up of either an Andr\\'asfai or a Vega graph. Here we establish the same structural conclusion on $G$ under the weaker assumption that for $m\\in\\{2, 3, 4\\}$ every sequence of $3m$ vertices has a subsequence of length $m+1$ with a common neighbour. In forthcoming work this will be used to solve an old problem of Andr\\'asfai in Ramsey-Tur\\'an theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-15T21:53:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C07",
      "05C15"
    ],
    "keywords": [
      "minimum degree condition",
      "maximal triangle-free graph",
      "weaker assumption",
      "structural conclusion",
      "vega graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}