{
  "id": "2404.19233",
  "version": "v1",
  "published": "2024-04-30T03:30:31.000Z",
  "updated": "2024-04-30T03:30:31.000Z",
  "title": "Avoiding short progressions in Euclidean Ramsey theory",
  "authors": [
    "Gabriel Currier",
    "Kenneth Moore",
    "Chi Hoi Yip"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if $\\ell_m$ denotes $m$ collinear points with consecutive points of distance one apart, we say that $\\mathbb{E}^n \\not \\to (\\ell_r,\\ell_s)$ if there is a red/blue coloring of $n$-dimensional Euclidean space that avoids red congruent copies of $\\ell_r$ and blue congruent copies of $\\ell_s$. We show that $\\mathbb{E}^n \\not \\to (\\ell_3, \\ell_{20})$, improving the best-known result $\\mathbb{E}^n \\not \\to (\\ell_3, \\ell_{1177})$ by F\\\"uhrer and T\\'oth, and also establish $\\mathbb{E}^n \\not \\to (\\ell_4, \\ell_{18})$ and $\\mathbb{E}^n \\not \\to (\\ell_5, \\ell_{10})$ in the spirit of the classical result $\\mathbb{E}^n \\not \\to (\\ell_6, \\ell_{6})$ due to Erd{\\H{o}}s et. al. We also show a number of similar $3$-coloring results, as well as $\\mathbb{E}^n \\not \\to (\\ell_3, \\alpha\\ell_{6889})$, where $\\alpha$ is an arbitrary positive real number. This final result answers a question of F\\\"uhrer and T\\'oth in the positive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-30T03:30:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10",
      "52C10",
      "11B25"
    ],
    "keywords": [
      "euclidean ramsey theory",
      "avoiding short progressions",
      "colorings avoiding short monochromatic arithmetic",
      "construct colorings avoiding short monochromatic",
      "avoiding short monochromatic arithmetic progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}