Evidence for a forbidden configuration conjecture: One more case solved

AbstractA 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}.For two 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 F8(t)=[1010010111001100t⋅[10011100]].We are able to show that forb(m,F8(t)) is Θ(m2) 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 F8(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