Linear algebra methods for forbidden configurations

Abstract

We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m,F) as the maximum number of columns in a simple m-rowed matrix A for which no k×l submatrix of A is a row and column permutation of F. In set theory notation, F is a forbidden trace. For all k-rowed F (simple or non-simple) Füredi has shown that forb(m,F) is O(m ^k). We are able to determine for which k-rowed F we have that forb(m,F) is O(m ^k−1) and for which k-rowed F we have that forb(m,F) is Θ(m ^k).

We need a bound for a particular choice of F. Define D ₁₂ to be the k×(2^k−2^k−2−1) (0,1)-matrix consisting of all nonzero columns on k rows that do not have [ ¹₁ ] in rows 1 and 2. Let 0 denote the column of k 0’s. Define F _k (t) to be the concatenation of 0 with t+1 copies of D ₁₂. We are able to show that forb(m,F _k (t)) is Θ(m ^k−1). Linear algebra methods and indicator polynomials originated in this context in a paper of the authors and Füredi and Sali. We provide a novel application of these methods.

The results are further evidence for the conjecture of Anstee and Sali on the asymptotics for fixed F of forb(m,F).