### A Survey of Forbidden Configuration Results

Richard Anstee
2013 Electronic Journal of Combinatorics
Let $F$ be a $k\times \ell$ (0,1)-matrix. We say a (0,1)-matrix $A$ has $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a trace and in the language of hypergraphs a configuration is a subhypergraph.Let $F$ be a given $k\times \ell$ (0,1)-matrix. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. The matrix $F$ need not be simple. We define $\hbox{forb}(m,F)$ as the
more » ... }(m,F)$as the maximum number of columns of any simple$m$-rowed matrix$A$which do not contain$F$as a configuration. Thus if$A$is an$m\times n$simple matrix which has no submatrix which is a row and column permutation of$F$then$n\le\hbox{forb}(m,F)$. Or alternatively if$A$is an$m\times (\hbox{forb}(m,F)+1)$simple matrix then$A$has a submatrix which is a row and column permutation of$F$. We call$F$a forbidden configuration. The fundamental result is due to Sauer, Perles and Shelah, Vapnik and Chervonenkis. For$K_k$denoting the$k\times 2^k$submatrix of all (0,1)-columns on$k$rows, then$\hbox{forb}(m,K_k)=\binom{m}{k-1}+\binom{m}{k-2}+\cdots \binom{m}{0}$. We seek asymptotic results for$\hbox{forb}(m,F)$for a fixed$F$and as$m$tends to infinity . A conjecture of Anstee and Sali predicts the asymptotically best constructions from which to derive the asymptotics of$\hbox{forb}(m,F)$. The conjecture has helped guide the research and has been verified for$k\times \ellF$with$k=1,2,3$and for simple$F$with$k=4$as well as other cases including$\ell=1,2$. We also seek exact values for$\hbox{forb}(m,F)\$.