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

• Authors: Josef Cibulka;Jan Kynčl
• Venue: Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2012

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 VC-dimension k. Raz showed that r2(n) grows exponentially in n. We show that r3(n) = 2θ(n log α (n)) and for every t ≥ 1, we have [EQUATION] and [EQUATION]. We also study the maximum number pk(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 p3(n) = θ(nα(n)) and [EQUATION] for every t ≥ 1. We also show that for every positive s there is a slowly growing function ζs(m) (for example [EQUATION]) for every odd s ≥ 5) satisfying the following. For all positive integers m, n, B and every m x 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.