{
  "id": "2004.02800",
  "version": "v1",
  "published": "2020-04-06T16:45:40.000Z",
  "updated": "2020-04-06T16:45:40.000Z",
  "title": "Large induced trees in dense random graphs",
  "authors": [
    "Nemanja Draganić"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Erd\\H{o}s and Palka initiated the study of the maximal size of induced trees in random graphs in 1983. They proved that for every fixed $0<p<1$ the size of a largest induced tree in $G_{n,p}$ is concentrated around $2\\log_q (np)$ with high probability, where $q=(1-p)^{-1}$. De la Vega showed concentration around the same value for $p=C/n$ where $C$ is a large constant, and his proof also works for all larger $p$. We show that for any given tree $T$ with bounded maximum degree and of size $(2-o(1))\\log_q(np)$, $G_{n,p}$ contains an induced copy of $T$ with high probability for $n^{-1/2}\\ln^{10/9}n\\leq p\\leq 0.99$. This is asymptotically optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-06T16:45:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dense random graphs",
      "large induced trees",
      "high probability",
      "largest induced tree",
      "vega showed concentration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}