{
  "id": "2210.12699",
  "version": "v1",
  "published": "2022-10-23T11:12:00.000Z",
  "updated": "2022-10-23T11:12:00.000Z",
  "title": "Subdigraphs of prescribed size and outdegree",
  "authors": [
    "Raphael Steiner"
  ],
  "comment": "short note, 3 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In 2006, Noga Alon raised the following open problem: Does there exist an absolute constant $c>0$ such that every $2n$-vertex digraph with minimum out-degree at least $s$ contains an $n$-vertex subdigraph with minimum out-degree at least $\\frac{s}{2}-c$ ? In this note, we answer this natural question in the negative, by showing that for arbitrarily large values of $n$ there exists a $2n$-vertex tournament with minimum out-degree $s=n-1$, in which every $n$-vertex subdigraph contains a vertex of out-degree at most $\\frac{s}{2}-\\left(\\frac{1}{2}+o(1)\\right)\\log_3(s)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-23T11:12:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C20"
    ],
    "keywords": [
      "minimum out-degree",
      "prescribed size",
      "vertex subdigraph contains",
      "open problem",
      "absolute constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}