{
  "id": "1806.08849",
  "version": "v1",
  "published": "2018-06-22T20:47:40.000Z",
  "updated": "2018-06-22T20:47:40.000Z",
  "title": "Finding Certain Arithmetic Progressions in 2-Coloured Cyclic Groups",
  "authors": [
    "Matei Mandache"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We say a pair of integers $(a, b)$ is findable if the following is true. For any $\\delta > 0$ there exists a $p_0$ such that for any prime $p \\ge p_0$ and any red-blue colouring of $\\mathbb{Z} /p\\mathbb{Z}$ in which each colour has density at least $\\delta$, we can find an arithmetic progression of length $a+b$ inside $\\mathbb{Z}/p\\mathbb{Z}$ whose first $a$ elements are red and whose last $b$ elements are blue. Szemer\\'edi's Theorem on arithmetic progressions implies that $(0,k)$ and $(1,k)$ are findable for any $k$. We prove that $(2, k)$ is also findable for any $k$. However, the same is not true of $(3, k)$. Indeed, we give a construction showing that $(3, 30000)$ is not findable. We also show that $(14, 14)$ is not findable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-22T20:47:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cyclic groups",
      "arithmetic progressions implies",
      "szemeredis theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}