{
  "id": "1802.03387",
  "version": "v1",
  "published": "2018-02-09T18:50:03.000Z",
  "updated": "2018-02-09T18:50:03.000Z",
  "title": "Zero-sum Analogues of van der Waerden's Theorem on Arithmetic Progressions",
  "authors": [
    "Aaron Robertson"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $r$ and $k$ be positive integers with $r \\mid k$. Denote by $w_{\\mathrm{\\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\\chi:[1,w_{\\mathrm{\\mathfrak{z}}}(k;r)] \\rightarrow \\{0,1,\\dots,r-1\\}$ admits a $k$-term arithmetic progression $a,a+d,\\dots,a+(k-1)d$ with $\\sum_{j=0}^{k-1} \\chi(a+jd) \\equiv 0 \\,(\\mathrm{mod }\\,r)$. We investigate these numbers as well as a \"mixed\" monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{\\mathrm{\\mathfrak{z}}}(k;r)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-09T18:50:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "van der waerdens theorem",
      "zero-sum analogues",
      "van der waerden numbers",
      "term arithmetic progression",
      "minimum integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}