Forbidden Configurations and Product Constructions

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F , we define that a (0,1)-matrix A has F as a configuration if there is a submatrix of A which is a row and column permutation of F (trace is the set system version of a configuration). Let |A| denote the number of columns of A. We define forb(m,F ) = max{|A| : A is m-rowed simple matrix and has no configuration F}. We extend this to a family F = {F1, F2, . . . , Ft} and define forb(m,F) = max{|A| : A is m-rowed simple matrix and has no configuration F ∈ F}. We consider products of matrices. Given an m1×n1 matrix A and an m2×n2 matrix B, we define A × B as the (m1 + m2) × n1n2 matrix whose columns are obtained from placing every column of A on top of every column of B. Let Ik denote the k × k identity matrix, let Ic k denote the (0,1)-complement of Ik and let Tk denote the k × k (0,1)-matrix with a 1 in position i, j if and only if i ≤ j. We show forb(m, {I2 × I2, I2 × T2, T2 × T2}) is Θ(m3/2) while obtaining a linear bound when forbidding all 2-fold products of all 2 × 2 (0,1)simple matrices. For two matrices F, P , where P is m-rowed, let f(F, P ) = max{|A| : A is m-rowed submatrix of P with no configuration F}. We establish f(I2 × I2, Im/2 × Im/2) is Θ(m3/2) whereas f(I2 × T2, Im/2 × Tm/2) and f(T2 × ∗Research supported in part by NSERC and Renyi Institute †Research supported in part by NSERC of first author