{
  "id": "2008.07297",
  "version": "v1",
  "published": "2020-08-17T13:27:43.000Z",
  "updated": "2020-08-17T13:27:43.000Z",
  "title": "On monochromatic solutions to $x-y=z^2$",
  "authors": [
    "Tom Sanders"
  ],
  "comment": "6pp; for Endre Szemeredi's 80th Birthday volume",
  "journal": "Acta Math. Hungar. 161 (2020), no. 2, 550-556",
  "doi": "10.1007/s10474-020-01079-6",
  "categories": [
    "math.CO"
  ],
  "abstract": "For $k \\in \\mathbb{N}$, write $S(k)$ for the largest natural number such that there is a $k$-colouring of $\\{1,\\dots,S(k)\\}$ with no monochromatic solution to $x-y=z^2$. That $S(k)$ exists is a result of Bergelson, and a simple example shows that $S(k) \\geq 2^{2^{k-1}}$. The purpose of this note is to show that $S(k)\\leq 2^{2^{2^{O(k)}}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-17T13:27:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "monochromatic solution",
      "largest natural number",
      "simple example"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}