Linear algebra methods for forbidden configurations

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 ).