Multivalued matrices and forbidden configurations

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An r-matrix is a matrix with symbols in {0,1,…,r−1}. A matrix is simple if it has no repeated columns. Let F be a finite set of r-matrices. Let forb(m,r,F) denote the maximum number of columns possible in a simple r-matrix A that has no submatrix which is a row and column permutation of any F∈F. Many investigations have involved r=2. For general r, Füredi and Sali proved that forb(m,r,F) is polynomial in m if and only if for every pair i,j∈{0,1,…,r−1} there is a matrix in F whose entries are only i or j. Let Tℓ(r) denote the set of 4(r2)r-matrices of the following form. For a pair i,j∈{0,1,…,r−1} we form four ℓ×ℓ matrices, namely the matrix with i's on the diagonal and j's off the diagonal and the matrix with i's on and above the diagonal and j's below the diagonal and the two matrices with the roles of i,j reversed. Anstee and Lu determined that forb(m,r,Tℓ(r)) is a constant. Let F be a finite set of 2-matrices. In the present paper we investigate the question whether forb(m,r,Tℓ(r)﹨Tℓ(2)∪F) is Θ(forb(m,2,F)) and settle this in the affirmative for some cases with F={F}, including most 2-columned F.