{
  "id": "2411.14566",
  "version": "v1",
  "published": "2024-11-21T20:29:59.000Z",
  "updated": "2024-11-21T20:29:59.000Z",
  "title": "A canonical Ramsey theorem for even cycles in random graphs",
  "authors": [
    "José D. Alvarado",
    "Y. Kohayakawa",
    "Patrick Morris",
    "Guilherme O. Mota"
  ],
  "comment": "24 pages + 4 pages of appendix, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "The celebrated canonical Ramsey theorem of Erd\\H{o}s and Rado implies that for $2\\leq k\\in \\mathbb{N}$, any colouring of the edges of $K_n$ with $n$ sufficiently large gives a copy of $C_{2k}$ which has one of three canonical colour patterns: monochromatic, rainbow or lexicographic. In this paper we show that if $p=\\omega(n^{-1+1/(2k-1)}\\log n)$, then ${\\mathbf{G}}(n,p)$ will asymptotically almost surely also have the property that any colouring of its edges induces canonical copies of $C_{2k}$. This determines the threshold for the canonical Ramsey property with respect to even cycles, up to a $\\log$ factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-21T20:29:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random graphs",
      "edges induces canonical copies",
      "canonical ramsey property",
      "rado implies",
      "canonical colour patterns"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}