{
  "id": "1603.08153",
  "version": "v1",
  "published": "2016-03-26T23:15:11.000Z",
  "updated": "2016-03-26T23:15:11.000Z",
  "title": "Rainbow Arithmetic Progressions in Finite Abelian Groups",
  "authors": [
    "Michael Young"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For positive integers $n$ and $k$, the \\emph{anti-van der Waerden number} of $\\mathbb{Z}_n$, denoted by $aw(\\mathbb{Z}_n,k)$, is the minimum number of colors needed to color the elements of the cyclic group of order $n$ and guarantee there is a rainbow arithmetic progression of length $k$. Butler et al. showed a reduction formula for $aw(\\mathbb{Z}_{n},3) = 3$ in terms of the prime divisors of $n$. In this paper, we analagously define the anti-van der Waerden number of a finite abelian group $G$ and show $aw(G,3)$ is determined by the order of $G$ and the number of groups with even order in a direct sum isomorphic to $G$. The \\emph{unitary anti-van der Waerden number} of a group is also defined and determined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-26T23:15:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite abelian group",
      "rainbow arithmetic progression",
      "anti-van der waerden number",
      "direct sum isomorphic",
      "cyclic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160308153Y"
    }
  }
}