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 s >= 4, we have almost tight upper and lower bounds of the form 2^{n poly(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 p_3(n) = Theta(n alpha(n)) and that p_s(n) can be bounded by functions of the form n 2^poly(alpha(n)) for every fixed s >= 4. We also show that for every positive s there is a slowly growing function zeta_s(m) (of the form 2^poly(alpha(m)) for every fixed s >= 5) 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.Comment: 22 pages, 4 figures, correction of the bound on r_3 in the abstract and other minor change

Maximum size of reverse-free sets of permutations

Two words have a reverse if they have the same pair of distinct letters on the same pair of positions, but in reversed order. A set of words no two of which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size of a reverse-free set of words from [n]^k where no letter repeats within a word. We show the following lower and upper bounds in the case n >= k: F(n,k) \in n^k k^{-k/2 + O(k/log k)}. As a consequence of the lower bound, a set of n-permutations each two having a reverse has size at most n^{n/2 + O(n/log n)}.Comment: 10 page

Shattering Thresholds for Random Systems of Sets, Words, and Permutations

This paper considers a problem that relates to the theories of covering arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability thresholds. Specifically, we want to find the number of subsets of [n]:={1,2,....,n} we need to randomly select, in a certain probability space, so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to words, we ask for the number of n-letter words on a q-letter alphabet that are needed to shatter all t-subwords of the q^n words of length n. Finally, we explore the number of random permutations of [n] needed to shatter (specializing to t=3), all length 3 permutation patterns in specified positions. We uncover a very sharp zero-one probability threshold for the emergence of such shattering; Talagrand's isoperimetric inequality in product spaces is used as a key tool.Comment: 25 page