Extremal functions of excluded tensor products of permutation matrices

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.