Evidence for a forbidden configuration conjecture: One more case solved

A simple matrix is a ( 0 , 1 ) -matrix with no repeated columns. Let F and A be ( 0 , 1 ) -matrices. We say that A avoids F if there is no submatrix of A which is a row and column permutation of F . Let ‖ A ‖ denote the number of columns of A . We define forb ( m , F ) = max { ‖ A ‖ : A  is an  m -rowed simple matrix which avoids  F } . o matrices H and K , define [ H ∣ K ] as the concatenation of H and K . Let t ⋅ H denote the concatenation of t copies of H . Given a number t with t ≥ 1 , define F 8 ( t ) = [ 1 0 1 0 0 1 0 1 1 1 0 0 1 1 0 0 t ⋅ [ 1 0 0 1 1 1 0 0 ] ] . We are able to show that forb ( m , F 8 ( t ) ) is Θ ( m 2 ) and that this matrix is “maximal” (in some sense) with respect to this property. A conjecture of Anstee and Sali predicts three “maximal” 4-rowed cases to consider with quadratic bounds, and F 8 ( t ) is one of them. Establishing the quadratic upper bounds for all three cases would establish the veracity of the conjecture for all 4-rowed configurations.