Matches in SemOpenAlex for { <https://semopenalex.org/work/W1502643917> ?p ?o ?g. }
- W1502643917 abstract "Let $F$ be a $ktimes ell$ (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 $ktimes ell$ (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 maximum number of columns of any simple $m$-rowed matrix $A$ which do not contain $F$ as a configuration. Thus if $A$ is an $mtimes n$ simple matrix which has no submatrix which is a row and column permutation of $F$ then $nlehbox{forb}(m,F)$. Or alternatively if $A$ is an $mtimes (hbox{forb}(m,F)+1)$ 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 $ktimes 2^k$ 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 $ktimes ell$ $F$ 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 $hbox{forb}(m,F)$." @default.
- W1502643917 created "2016-06-24" @default.
- W1502643917 creator A5031068407 @default.
- W1502643917 date "2013-01-29" @default.
- W1502643917 modified "2023-09-25" @default.
- W1502643917 title "A Survey of Forbidden Configuration Results" @default.
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- W1502643917 doi "https://doi.org/10.37236/2379" @default.
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