Forbidden configurations and Steiner designs

Let F be a (0, 1) matrix. A (0, 1) matrix M is said to have F as a configuration if there is a submatrix of M which is a row and column permutation of F . We say that a matrix M is simple if it has no repeated columns. For a given v ∈ N, we shall denote by forb(v,F) the maximum number of columns in a simple (0, 1) matrix with v rows for which F does not occur as a configuration. We say that a matrix M is maximal for F if M has forb(v,F) columns. In this paper we show that for certain natural choices of F , forb(v,F) ≤ ( v t) t+1 . In particular this gives an extremal characterization for Steiner t-designs as maximal (0, 1) matrices in terms of certain forbidden configurations.