{
  "id": "2409.14213",
  "version": "v1",
  "published": "2024-09-21T18:17:05.000Z",
  "updated": "2024-09-21T18:17:05.000Z",
  "title": "Avoiding secants of given size in finite projective planes",
  "authors": [
    "Tamás Héger",
    "Zoltán Lóránt Nagy"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $q$ be a prime power and $k$ be a natural number. What are the possible cardinalities of point sets ${S}$ in a projective plane of order $q$, which do not intersect any line at exactly $k$ points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this paper we show a series of results which point out the existence of all or almost all possible values $m\\in [0, q^2+q+1]$ for $|S|=m$, provided that $k$ is not close to the extremal values $0$ or $q+1$. Moreover, using polynomial techniques we show the existence of a point set $S$ with the following property: for every prescribed list of numbers $t_1, \\ldots t_{q^2+q+1}$, $|S\\cap \\ell_i|\\neq t_i$ holds for the $i$th line $\\ell_i$, $\\forall i \\in \\{1, 2, \\ldots, q^2+q+1\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-21T18:17:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite projective planes",
      "avoiding secants",
      "point set",
      "natural number",
      "extremal values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}