Better upper bounds on the Füredi-Hajnal limits of permutations

A binary matrix is a matrix with entries from the set {0, 1}. We say that a binary matrix A contains a binary matrix S if S can be obtained from A by removal of some rows, some columns, and changing some 1-entries to 0-entries. If A does not contain S, we say that A avoids S. A k-permutation matrix P is a binary k× k matrix with exactly one 1-entry in every row and one 1-entry in every column. The Füredi–Hajnal conjecture, proved by Marcus and Tardos, states that for every permutation matrix P , there is a constant cP such that for every n ∈ N, every n × n binary matrix A with at least cPn 1-entries contains P . We show that cP ≤ 2 2/3 log k/(log log k)) asymptotically almost surely for a random k-permutation matrix P . We also show that cP ≤ 2 for every k-permutation matrix P , improving the constant in the exponent of a recent upper bound on cP by Fox. We also consider a higher-dimensional generalization of the Stanley–Wilf conjecture about the number of d-dimensional n-permutation matrices avoiding a fixed d-dimensional k-permutation matrix, and prove almost matching upper and lower bounds of the form (2) · (n!)d−1−1/(d−1) and n−O(k)kΩ(n) · (n!)d−1−1/(d−1), respectively.