Matrices with forbidden subconfigurations

Abstract We consider matrices with entries from the set {0, 1, …, q −1}. Suppose that S k is a k × q k matrix having all possible k -tuples as columns. We determine the best possible bound f ( m , k ) with the property that if A is any m ×( f ( m , k )+1) matrix of distinct columns, then some row and column permutation of A contains S k as a submatrix. Our result generalizes a number of the results for q = 2 due to Anstee, Furedi, Quinn, Sauer, Perles and Shelah, and is obtained by means of a simple inductive argument. Interesting matrices meeting the bound are constructed.