{
  "id": "1511.07541",
  "version": "v1",
  "published": "2015-11-24T02:20:16.000Z",
  "updated": "2015-11-24T02:20:16.000Z",
  "title": "Ramsey numbers of trees versus odd cycles",
  "authors": [
    "Matthew Brennan"
  ],
  "comment": "10 pages. arXiv admin note: text overlap with arXiv:1511.07306",
  "categories": [
    "math.CO"
  ],
  "abstract": "Burr, Erd\\H{o}s, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \\ge 3$ and $n \\ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \\ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \\ge 3$ and $n \\ge 50m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-24T02:20:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C55",
      "05C05",
      "05C38"
    ],
    "keywords": [
      "odd cycle",
      "ramsey numbers",
      "constant factor",
      "result holds",
      "schelp yields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151107541B"
    }
  }
}