Forbidden configurations: Boundary cases

A simple matrix is a {0,1}-matrix with no repeated columns. For a {0,1}-matrix F, define F≺A if there is a submatrix of A which is a row and column permutation of F. Let ‖A‖ denote the number of columns of A. Define forb(m,F)=max{‖A‖:A  is  m-rowed simple matrix and  F⊀A}. We classify all 6-rowed configurations F for which forb(m,F) is Θ(m2) and prove forb(m,F) is Ω(m3) for all other 6-rowed F. We also prove that forb(m,G) is O(m2) for a particular 5×6 simple G and the addition of any column α to G makes forb(m,[Gα]) to be Ω(m3). The results are evidence for a conjecture of Anstee and Sali which predicts the growth rate of forb(m,F) as a function of F.