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 ( m 2 ) and prove forb ( m , F ) is Ω ( m 3 ) for all other 6 -rowed F . We also prove that forb ( m , G ) is O ( m 2 ) for a particular 5 × 6 simple G and the addition of any column α to G makes forb ( m , [ G α ] ) to be Ω ( m 3 ) . 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 .