{
  "id": "0706.4420",
  "version": "v1",
  "published": "2007-06-29T13:56:47.000Z",
  "updated": "2007-06-29T13:56:47.000Z",
  "title": "Bounds on Van der Waerden Numbers and Some Related Functions",
  "authors": [
    "Tom Brown",
    "Bruce M. Landman",
    "Aaron Robertson"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\\{1,2,...,n\\} \\to \\{1,2,...,s\\}$ admits a $k_i$-term arithmetic progression of color $i$ for some $i$, $1 \\leq i \\leq s$. In the case when $k_1=k_2=...=k_s=k$ we simply write $w(k;s)$. That such a minimum integer exists follows from van der Waerden's theorem on arithmetic progressions. In the present paper we give a lower bound for $w(k,m)$ for each fixed $m$. We include a table with values of $w(k,3)$ which match this lower bound closely for $5 \\leq k \\leq 16$. We also give an upper bound for $w(k,4)$, an upper bound for $w(4;s)$, and a lower bound for $w(k;s)$ for an arbitrary fixed $k$. We discuss a number of other functions that are closely related to the van der Waerden function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-06-29T13:56:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "van der waerden numbers",
      "related functions",
      "lower bound",
      "upper bound",
      "van der waerdens theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.4420B"
    }
  }
}