On Minimum Saturated Matrices

Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M contains no submatrix F ∈ F (as a row and column permutation of F ). A matrix M without repeated columns is F-saturated if M is F-admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat(n,F) which is the minimal number of columns of an F-saturated matrix with n rows. We establish the estimate sat(n,F) = O(nk−1) for any family F of k-row matrices and also compute the sat-function for a few small forbidden matrices.