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 Θ ( n log α ( n ) ) and for every t ⩾ 1 , we have r 2 t + 2 ( n ) = 2 Θ ( n α ( n ) t ) and r 2 t + 3 ( n ) = 2 O ( n α ( n ) t log α ( n ) ) . o study the maximum number p k ( n ) of 1-entries in an n × 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 ) = Θ ( n α ( n ) ) and p 2 t + 2 ( n ) = n 2 ( 1 / t ! ) α ( n ) t ± O ( α ( n ) t − 1 ) for every t ⩾ 1 . o show that for every positive s there is a slowly growing function ζ s ( n ) (for example ζ 2 t + 3 ( n ) = 2 O ( α t ( n ) ) for every t ⩾ 1 ) satisfying the following. For all positive integers n and B and every n × n ( 0 , 1 ) -matrix M with ζ s ( n ) B n 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.