{
  "id": "2203.12735",
  "version": "v1",
  "published": "2022-03-23T21:36:58.000Z",
  "updated": "2022-03-23T21:36:58.000Z",
  "title": "Integer colorings with no rainbow $k$-term arithmetic progression",
  "authors": [
    "Hao Lin",
    "Guanghui Wang",
    "Wenling Zhou"
  ],
  "journal": "European Journal of Combinatorics (2022)",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we study the rainbow Erd\\H{o}s-Rothschild problem with respect to $k$-term arithmetic progressions. For a set of positive integers $S \\subseteq [n]$, an $r$-coloring of $S$ is \\emph{rainbow $k$-AP-free} if it contains no rainbow $k$-term arithmetic progression. Let $g_{r,k}(S)$ denote the number of rainbow $k$-AP-free $r$-colorings of $S$. For sufficiently large $n$ and fixed integers $r\\ge k\\ge 3$, we show that $g_{r,k}(S)<g_{r,k}([n])$ for any proper subset $S\\subset [n]$. Further, we prove that $\\lim_{n\\to \\infty}g_{r,k}([n])/(k-1)^n= \\binom{r}{k-1}$. Our result is asymptotically best possible and implies that, almost all rainbow $k$-AP-free $r$-colorings of $[n]$ use only $k-1$ colors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-23T21:36:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "term arithmetic progression",
      "integer colorings",
      "proper subset",
      "sufficiently large",
      "positive integers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}