{
  "id": "1810.04746",
  "version": "v1",
  "published": "2018-10-10T20:47:16.000Z",
  "updated": "2018-10-10T20:47:16.000Z",
  "title": "Stability and Erdős--Stone type results for $F$-free graphs with a fixed number of edges",
  "authors": [
    "Jamie Radcliffe",
    "Andrew Uzzell"
  ],
  "comment": "15 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "A fundamental problem of extremal graph theory is to ask, 'What is the maximum number of edges in an $F$-free graph on $n$ vertices?' Recently Alon and Shikhelman proposed a more general, subgraph counting, version of this question. They considered the question of determining the maximum number of copies of a fixed graph $T$ in an $F$-free graph on $n$ vertices. In this more general context, where we are no longer counting edges, it is also natural to ask what is the maximum number of copies of $T$ in an $F$-free graph with $m$ edges and no restriction on the number of vertices. Frohmader, in a different context, determined the answer when $T$ and $F$ are both complete graphs. We prove results for this problem analogous to the Erd\\H{o}s--Stone theorem, the Erd\\H{o}s--Simonovits theorem, and the stability theorem of Erd\\H{o}s--Simonovits.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-10T20:47:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free graph",
      "erdős-stone type results",
      "fixed number",
      "maximum number",
      "extremal graph theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}