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

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 t >= 1, we have r(2t+2) (n) = 2(Theta (n alpha(n)t)) and r(2t+3) (n) = 2 (O(n alpha(n)t log 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, for example, p(3)(n) = Theta(n alpha(n)) and p(2t+2)(n) = n2((1/t)alpha(n)t +/- O(alpha(n)t-1)) for every t >= 1. We also show that for every positive s there is a slowly growing function zeta(s)(n) (for example zeta(2t+3)(n) = 2(O(alpha t(n))) for every t >= 1) 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. (C) 2012 Elsevier Inc. All rights reserved.