Forbidden Configurations and Product Constructions

Abstract

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F, we say 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 ${{\rm forb}(m, F) = {\rm max}\{\|A\| \,:\, A}$ is m-rowed simple matrix and has no configuration F. We extend this to a family ${\mathcal{F} = \{F_1, F_2, \ldots , F_t\}}$ and define ${{\rm forb}(m, \mathcal{F}) = {\rm max}\{\|A\| \,:\, A}$ is m-rowed simple matrix and has no configuration ${F \in \mathcal{F}\}}$ . We consider products of matrices. Given an m ₁ × n ₁ matrix A and an m ₂ × n ₂ matrix B, we define the product A × B as the (m ₁ + m ₂) × n ₁ n ₂ matrix whose columns consist of all possible combinations obtained from placing a column of A on top of a column of B. Let I _k denote the k × k identity matrix, let ${I_k^{c}}$ denote the (0,1)-complement of I _k and let T _k denote the k × k upper triangular (0,1)-matrix with a 1 in position i, j if and only if i ≤ j. We show forb(m, {I ₂ × I ₂, T ₂ × T ₂}) is ${\Theta(m^{3/2})}$ 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 ${f(F, P) = {\rm max}_{A} \{\|A\| \,:\,A}$ is m-rowed submatrix of P with no configuration F}. We establish f(I ₂ × I ₂, I _m/2 × I _m/2) is ${\Theta(m^{3/2})}$ whereas f(I ₂ × T ₂, I _m/2 × T _m/2) and f(T ₂ × T ₂, T _m/2 × T _m/2) are both ${\Theta(m)}$. Additional results are obtained. One of the results requires extensive use of a computer program. We use the results on patterns due to Marcus and Tardos and generalizations due to Klazar and Marcus, Balogh, Bollobás and Morris.