{
  "id": "1909.02867",
  "version": "v1",
  "published": "2019-09-06T12:42:28.000Z",
  "updated": "2019-09-06T12:42:28.000Z",
  "title": "On the upper chromatic number and multiplte blocking sets of PG($n,q$)",
  "authors": [
    "Zoltán L. Blázsik",
    "Tamás Héger",
    "Tamás Szőnyi"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\\leq \\frac38p+1$, a small $t$-fold (weighted) $(n-k)$-blocking set of $\\mathrm{PG}(n,p)$, $p$ prime, must contain the weighted sum of $t$ not necessarily distinct $(n-k)$-spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-06T12:42:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upper chromatic number",
      "multiplte blocking sets",
      "double blocking set",
      "trivial lower bound",
      "subspace contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}