Tight bounds on the maximum size of a set of permutations with bounded VC-dimension

  • Authors:

  • Josef Cibulka;Jan KynčL

  • Affiliations:

  • Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic;Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Repu ...

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2012

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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^@Q^(^n^l^o^g^@a^(^n^)^) and for every t=1, we have r"2"t"+"2(n)=2^@Q^(^n^@a^(^n^)^^^t^) and r"2"t"+"3(n)=2^O^(^n^@a^(^n^)^^^t^l^o^g^@a^(^n^)^). We also study the maximum number p"k(n) of 1-entries in an nxn(0,1)-matrix with no (k+1)-tuple of columns containing all (k+1)-permutation matrices. We determine that, for example, p"3(n)=@Q(n@a(n)) and p"2"t"+"2(n)=n2^(^1^/^t^!^)^@a^(^n^)^^^t^+/-^O^(^@a^(^n^)^^^t^^^-^^^1^) for every t=1. We also show that for every positive s there is a slowly growing function @z"s(n) (for example @z"2"t"+"3(n)=2^O^(^@a^^^t^(^n^)^) for every t=1) satisfying the following. For all positive integers n and B and every nxn(0,1)-matrix M with @z"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.