Forbidden configurations and Steiner designs

Abstract

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