Large forbidden configurations and design theory

Abstract

Let forb(m, F ) denote the maximum number of columns possible in a (0,1)-matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F . We consider cases where the configuration F has a number of columns that grows with m. For a k × matrix G, define s · G to be the concatenation of s copies of G. In a number of cases we determine forb(m, m α · G) is Θ(m k+α ). Results of Keevash on the existence of designs provide constructions that provide asymptotic lower bounds. An induction idea of Anstee and Lu is useful in obtaining upper bounds.