{
  "id": "2112.03893",
  "version": "v2",
  "published": "2021-12-07T18:38:49.000Z",
  "updated": "2022-11-08T16:16:58.000Z",
  "title": "Ramsey numbers of cycles versus general graphs",
  "authors": [
    "John Haslegrave",
    "Joseph Hyde",
    "Jaehoon Kim",
    "Hong Liu"
  ],
  "comment": "20 pages, 3 figures. Final version to appear in Forum of Mathematics, Sigma",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n,H)=(n-1)(\\chi(H)-1)+\\sigma(H)$, where $\\sigma(H)$ is the minimum possible size of a colour class in a $\\chi(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the same should be true already when $n\\geq |H|\\chi(H)$. We improve this 40-year-old result of Burr by giving quantitative bounds of the form $n\\geq C|H|\\log^4\\chi(H)$, which is optimal up to the logarithmic factor. In particular, this proves a strengthening of the Allen-Brightwell-Skokan conjecture for all graphs $H$ with large chromatic number.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-11-08T16:16:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C55",
      "05C38",
      "05C48"
    ],
    "keywords": [
      "general graphs",
      "natural lower bound construction",
      "large chromatic number",
      "correct ramsey number",
      "minimum number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}