{
  "id": "math/9910092",
  "version": "v1",
  "published": "1999-10-19T13:12:25.000Z",
  "updated": "1999-10-19T13:12:25.000Z",
  "title": "On Generalized Van der Waerden Triples",
  "authors": [
    "Bruce Landman",
    "Aaron Robertson"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of {1,2,...,w(k,r)} must contain a monochromatic k-term arithmetic progression {x,x+d,x+2d,...,x+(k-1)d}. We investigate the following generalization of w(3,r). For fixed positive integers a and b with a <= b, define N(a,b;r) to be the least positive integer, if it exists, such that any r-coloring of {1,2,...,N(a,b;r)} must contain a monochromatic set of the form {x,ax+d,bx+2d}. We show that N(a,b;2) exists if and only if b <> 2a, and provide upper and lower bounds for it. We then show that for a large class of pairs (a,b), N(a,b;r) does not exist for r sufficiently large. We also give a result on sets of the form {x,ax+d,ax+2d,...,ax+(k-1)d}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1999-10-19T13:12:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "generalized van der waerden triples",
      "positive integer",
      "monochromatic k-term arithmetic progression",
      "van der waerdens classical theorem",
      "arithmetic progressions states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math.....10092L"
    }
  }
}