{
  "id": "1104.5007",
  "version": "v3",
  "published": "2011-04-26T19:55:51.000Z",
  "updated": "2012-04-13T16:34:33.000Z",
  "title": "Tight bounds on the maximum size of a set of permutations with bounded VC-dimension",
  "authors": [
    "Josef Cibulka",
    "Jan Kyncl"
  ],
  "comment": "22 pages, 4 figures, correction of the bound on r_3 in the abstract and other minor changes",
  "journal": "Journal of Combinatorial Theory, Series A 119 (7), 1461-1478 (2012)",
  "doi": "10.1016/j.jcta.2012.04.004",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let r_k(n) be the maximum size of a set of n-permutations with VC-dimension k. Raz showed that r_2(n) grows exponentially in n. We show that r_3(n)=2^Theta(n log(alpha(n))) and for every s >= 4, we have almost tight upper and lower bounds of the form 2^{n poly(alpha(n))}. We also study the maximum number p_k(n) of 1-entries in an n x n (0,1)-matrix with no (k+1)-tuple of columns containing all (k+1)-permutation matrices. We determine that p_3(n) = Theta(n alpha(n)) and that p_s(n) can be bounded by functions of the form n 2^poly(alpha(n)) for every fixed s >= 4. We also show that for every positive s there is a slowly growing function zeta_s(m) (of the form 2^poly(alpha(m)) for every fixed s >= 5) satisfying the following. For all positive integers n and B and every n x n (0,1)-matrix M with zeta_s(n)Bn 1-entries, the rows of M can be partitioned into s intervals so that at least B columns contain at least B 1-entries in each of the intervals.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-04-13T16:34:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximum size",
      "tight bounds",
      "bounded vc-dimension",
      "columns contain",
      "n-permutations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.5007C"
    }
  }
}