{
  "id": "1705.02166",
  "version": "v1",
  "published": "2017-05-05T10:42:38.000Z",
  "updated": "2017-05-05T10:42:38.000Z",
  "title": "Lines in Euclidean Ramsey theory",
  "authors": [
    "David Conlon",
    "Jacob Fox"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we construct a red/blue-coloring of $\\mathbb{E}^n$ containing no red copy of $\\ell_2$ and no blue copy of $\\ell_m$ for any $m \\geq 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erd\\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number $n$, there is a set $K \\subset \\mathbb{E}^1$ and a red/blue-coloring of $\\mathbb{E}^n$ containing no red copy of $\\ell_2$ and no blue copy of $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-05T10:42:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euclidean ramsey theory",
      "blue copy",
      "natural number",
      "red copy",
      "consecutive points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}