{
  "id": "1505.01426",
  "version": "v1",
  "published": "2015-05-06T16:42:08.000Z",
  "updated": "2015-05-06T16:42:08.000Z",
  "title": "Upper bounds on the smallest size of a saturating set in a projective plane and the Birthday problem",
  "authors": [
    "Daniele Bartoli",
    "Alexander A. Davydov",
    "Massimo Giulietti",
    "Stefano Marcugini",
    "Fernanda Pambianco"
  ],
  "comment": "6 pages, 17 references",
  "categories": [
    "math.CO"
  ],
  "abstract": "The following upper bound on the smallest size $s(2,q)$ of a saturating set in a projective plane $\\Pi _{q}$ (not necessary Desarguesian) of order $q$ is proven. \\begin{equation*} s(2,q)\\leq 2\\sqrt{(q+1)\\ln q}+2\\thicksim 2\\sqrt{q\\ln q}. \\end{equation*} Our approach is probabilistic rather than geometric. The proof uses some known results on the classical Birthday problem, and also provides the following result. In $\\Pi _{q}$ any point $k$-set, with $ k\\leq 2c\\sqrt{(q+1)\\ln q}+2$ and an universal constant $c\\geq 1,$ is a saturating set with probability \\begin{equation*} p>1-\\frac{1}{q^{2c^{2}-2}}. \\end{equation*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-06T16:42:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E21",
      "51E22",
      "94B05"
    ],
    "keywords": [
      "saturating set",
      "upper bound",
      "projective plane",
      "smallest size",
      "universal constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}