  An $r$-matrix is a matrix with symbols in $\{0,1,\ldots,r-1\}$. A matrix is
simple if it has no repeated columns. Let ${\cal F}$ be a finite set of
$r$-matrices. Let $\hbox{forb}(m,r,{\cal F})$ 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
$F\in{\cal F}$. Many investigations have involved $r=2$. For general $r$,
$\hbox{forb}(m,r,{\cal F})$ is polynomial in $m$ if and only if for every pair
$i,j\in\{0,1,\ldots,r-1\}$ there is a matrix in ${\cal F}$ whose entries are
only $i$ or $j$.
  Let ${\cal T}_{\ell}(r)$ denote the following $r$-matrices. For a pair
$i,j\in\{0,1,\ldots,r-1\}$ we form four $\ell\times\ell$ 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
$\hbox{forb}(m,r,{\cal T}_{\ell}(r))$ is a constant. Let ${\cal F}$ be a finite
set of 2-matrices. We ask if $\hbox{forb}(m,r,{\cal T}_{\ell}(3)\backslash
{\cal T}_{\ell}(2)\cup {\cal F})$ is $\Theta(\hbox{forb}(m,2,{\cal F}))$ and
settle this in the affirmative for some cases including most 2-columned $F$.
