Location of Repository

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

Topics:
Mathematics - Combinatorics, Computer Science - Discrete Mathematics

Year: 2012

DOI identifier: 10.1016/j.jcta.2012.04.004

OAI identifier:
oai:arXiv.org:1104.5007

Provided by:
arXiv.org e-Print Archive

Download PDF:To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.