{
  "id": "1608.06533",
  "version": "v1",
  "published": "2016-08-23T15:00:58.000Z",
  "updated": "2016-08-23T15:00:58.000Z",
  "title": "Size-Ramsey numbers of cycles versus a path",
  "authors": [
    "Andrzej Dudek",
    "Farideh Khoeini",
    "Paweł Prałat"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1601.02564",
  "categories": [
    "math.CO"
  ],
  "abstract": "The size-Ramsey number $\\hat{R}(\\mathcal{F},H)$ of a family of graphs $\\mathcal{F}$ and a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours, say, red and blue, yields a red copy of a graph from $\\mathcal{F}$ or a blue copy of $H$. In this paper we first focus on $\\mathcal{F} = \\mathcal{C}_{\\le cn}$, where $\\mathcal{C}_{\\le cn}$ is the family of cycles of length at most $cn$, and $H = P_n$. In particular, we show that $2.00365 n \\le \\hat{R}(\\mathcal{C}_{\\le n},P_n) \\le 31n$. Using similar techniques, we also managed to analyze $\\hat{R}(C_n,P_n)$, which was investigated before but only using the regularity method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-23T15:00:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "size-ramsey number",
      "regularity method",
      "smallest integer",
      "blue copy",
      "first focus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}