{
  "id": "2109.06110",
  "version": "v1",
  "published": "2021-09-13T16:35:07.000Z",
  "updated": "2021-09-13T16:35:07.000Z",
  "title": "Disproof of a conjecture of Erdős and Simonovits on the Turán number of graphs with minimum degree 3",
  "authors": [
    "Oliver Janzer"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In 1981, Erd\\H{o}s and Simonovits conjectured that for any bipartite graph $H$ we have $\\mathrm{ex}(n,H)=O(n^{3/2})$ if and only if $H$ is $2$-degenerate. Later, Erd\\H{o}s offered 250 dollars for a proof and 100 dollars for a counterexample. In this paper, we disprove the conjecture by finding, for any $\\varepsilon>0$, a $3$-regular bipartite graph $H$ with $\\mathrm{ex}(n,H)=O(n^{4/3+\\varepsilon})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-13T16:35:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "turán number",
      "minimum degree",
      "simonovits",
      "conjecture",
      "regular bipartite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}