{
  "id": "1810.01210",
  "version": "v1",
  "published": "2018-10-02T12:49:32.000Z",
  "updated": "2018-10-02T12:49:32.000Z",
  "title": "On non-repetitive sequences of arithmetic progressions:the cases $k \\in \\{4,5,6,7,8\\}$",
  "authors": [
    "Borut Lužar",
    "Martina Mockovčiaková",
    "Pascal Ochem",
    "Alexandre Pinlou",
    "Roman Soták"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A $d$-subsequence of a sequence $\\varphi = x_1\\dots x_n$ is a subsequence $x_i x_{i+d} x_{i+2d} \\dots$, for any positive integer $d$ and any $i$, $1 \\le i \\le n$. A \\textit{$k$-Thue sequence} is a sequence in which every $d$-subsequence, for $1 \\le d \\le k$, is non-repetitive, i.e. it contains no consecutive equal subsequences. In 2002, Grytczuk proposed a conjecture that for any $k$, $k+2$ symbols are enough to construct a $k$-Thue sequences of arbitrary lengths. So far, the conjecture has been confirmed for $k \\in \\{1,2,3,5\\}$. Here, we present two different proving techniques, and confirm it for all $k$, with $2 \\le k \\le 8$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-02T12:49:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68R15"
    ],
    "keywords": [
      "arithmetic progressions",
      "non-repetitive sequences",
      "thue sequence",
      "conjecture",
      "consecutive equal subsequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}