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 rk(n) be the maximum size of a set of n-permutations with VCdimension k. Raz showed that r2(n) grows exponentially in n. We show that r3(n) = 2 Θ(n logα(n)) and for every t ≥ 1, we have r2t+2(n) = 2 Θ(nα(n)) and r2t+3(n) = 2 O(nα(n) . We also study the maximum number pk(n) of 1entries in an n × n (0, 1)-matrix with no (k + 1)-tuple of columns containing all (k+1)-permutation matrices. We determine that p3(n) = Θ(nα(n)) and p2t+2(n) = n2 ±O(α(n)) for every t ≥ 1. We also show that for every positive s there is a slowly growing function ζs(m) (for example ζs(m) = 2 (m)) for every odd s ≥ 5) satisfying the following. For all positive integers m,n,B and every m×n (0, 1)-matrix M with ζs(m)Bn 1-entries, the rows of M can be partitioned into s intervals so that some ⌊Bn/m⌋-tuple of columns contains at least B 1-entries in each of the intervals.