Evidence for a  Forbidden Configuration Conjecture; one more case
solved.
by R. Anstee, M. Raggi and A. Sali
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A \emph{simple} matrix is a (0,1)-matrix with no repeated columns. For
a (0,1)-matrix $F$, we define that a (0,1)-matrix $A$ has \emph{no}
$F$ as a \emph{configuration} if there is \emph{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 and with no configuration $F \}.$
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For two matrices $H,K$ we define $[H\,|\,K]$ as the concatenation of
$H$ and $K$. We let $t\cdot H$ denote the concatenation of $t$ copies
of $H$. Given $t\ge 1$, we define
\vspace{-0.4cm}
$$F_{8}(t)=\left[\begin{array}{rrrr}
1 & 0 & 1 & 0\\
0 & 1 & 0 & 1\\
1 & 1 & 0 & 0\\
1 & 1 & 0 & 0\end{array}
   t\cdot\left[\begin{array}{rr}
1 & 0\\
0 & 1\\
1 & 1\\
0 & 0\end{array}\right]\right].$$\vspace{-0.5cm}
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\noindent We are able to show that $\forb(m,F_8(t))$ is $\Theta(m^2)$
while for any column $\alpha$ not contained in $F_8(1)$, we show that
$\forb(m,[F_8(t)\,|\,\alpha])$ is $\Omega(m^3)$. A conjecture of
Anstee and Sali predicts three 4-rowed cases to consider with
quadratic bounds and $F_8(t)$ is one of them. Establishing the
quadratic bounds for all 3 cases would establish the conjecture for
all 4-rowed configurations.
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