Extremal functions of excluded tensor products of permutation matrices

Abstract

For a 0–1 matrix Q, ex(n,Q) is the maximum number of 1s in an n×n 0–1 matrix of which no submatrix majorizes Q. We show that if P is a permutation matrix and Q is arbitrary, then the order of growth of ex(n,P⊗Q) is almost the same as that of ex(n,Q), extending a result used in Marcus and Tardos’s proof of the Stanley–Wilf conjecture.