{
  "id": "math/0507588",
  "version": "v1",
  "published": "2005-07-28T14:06:14.000Z",
  "updated": "2005-07-28T14:06:14.000Z",
  "title": "On the degree of regularity of generalized van der Waerden triples",
  "authors": [
    "Nikos Frantzikinakis",
    "Bruce Landman",
    "Aaron Robertson"
  ],
  "comment": "5 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $1 \\leq a \\leq b$ be integers. A triple of the form $(x,ax+d,bx+2d)$, where $x,d$ are positive integers is called an {\\em (a,b)-triple}. The {\\em degree of regularity} of the family of all $(a,b)$-triples, denoted dor($a,b)$, is the maximum integer $r$ such that every $r$-coloring of $\\mathbb{N}$ admits a monochromatic $(a,b)$-triple. We settle, in the affirmative, the conjecture that dor$(a,b) < \\infty$ for all $(a,b) \\neq (1,1)$. We also disprove the conjecture that dor($a,b) \\in \\{1,2,\\infty\\}$ for all $(a,b)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-07-28T14:06:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "generalized van der waerden triples",
      "regularity",
      "maximum integer",
      "conjecture",
      "denoted dor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7588F"
    }
  }
}