Forbidden Berge Hypergraphs
Richard Anstee, Santiago Salazar

TL;DR
This paper investigates the maximum size of simple (0,1)-matrices avoiding certain Berge hypergraph configurations, providing asymptotic results for matrices with 3 to 5 rows, linking to extremal graph theory problems.
Contribution
It determines the asymptotic behavior of the extremal function for Berge hypergraphs in matrices with 3 to 5 rows, extending understanding of hypergraph containment and related extremal problems.
Findings
Asymptotic formulas for $Bh(m,F)$ for all 3- and 4-rowed $F$.
Most 5-rowed $F$ cases are resolved.
Connections established with extremal graph theory problems like $K_r$ in $K_{s,t}$-free graphs.
Abstract
A \emph{simple} matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix , we say that a (0,1)-matrix has as a \emph{Berge hypergraph} if there is a submatrix of and some row and column permutation of , say , with . Letting denote the number of columns in , we define the extremal function . We determine the asymptotics of for all - and -rowed and most -rowed . For certain , this becomes the problem of determining the maximum number of copies of in a -vertex graph that has no subgraph, a problem studied by Alon and Shinkleman.
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