Forbidden Families of Minimal Quadratic and Cubic Configurations

A matrix is simple if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix F , we say a matrix A has F as a configuration, denoted F ≺ A, if there is a submatrix of A which is a row and column permutation of F . Let |A| denote the number of columns of A. Let F be a family of matrices. We define the extremal function forb(m,F) = max{|A| : A is an m−rowed simple matrix and has no configuration F ∈ F}. We consider pairs F = {F1, F2} such that F1 and F2 have no common extremal construction and derive that individually each forb(m,Fi) has greater asymptotic growth than forb(m,F), extending research started by Anstee and Koch.