{
  "id": "0801.0798",
  "version": "v2",
  "published": "2008-01-05T10:10:59.000Z",
  "updated": "2016-09-28T01:08:53.000Z",
  "title": "On the monochromatic Schur Triples type problem",
  "authors": [
    "Thotsaporn \"Aek\" Thanatipanonda"
  ],
  "comment": "10 pages, 3 fugures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\\{x,y,x+ay\\}$ triples for $a \\geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum number of such triples is at most $\\frac{n^2}{2a(a^2+2a+3)} + O(n)$ when $a \\geq 2$. We also find a new upper bound for the minimum number, over all $r$-colorings of $[1,n]$, of monochromatic Schur triples, for $r \\geq 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-01-05T10:10:59.000Z",
      "title": "A 2-coloring of $[1,n]$ can have $\\frac{n^2}{2a(a^2+2a+3)} + O(n)$ monochromatic triples of the form $\\{x,y,x+ay\\}, a \\geq 2$, but not less!",
      "abstract": "We solve a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $(x,y,x+ay)$ triples, $a \\geq 2$. We show that the minimum number of such triples is $\\frac{n^2}{2a(a^2+2a+3)} + O(n)$. We also find a new upper bound for the minimum number, over all $r$-colorings of $[1,n]$, of monochromatic Schur triples, for $r \\geq 3$.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-09-28T01:08:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "monochromatic triples",
      "minimum number",
      "monochromatic schur triples",
      "ronald graham"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0801.0798A"
    }
  }
}