Forbidden Berge Hypergraphs

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 $F$, we say that a (0,1)-matrix $A$ has $F$ as a \emph{Berge hypergraph} if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $||A||$ denote the number of columns in $A$, we define the extremal function $Bh(m,{ F})=\max\{||A||\,:\, A \hbox{ is }m\hbox{-rowed simple matrix with no Berge hypergraph }F\}$. We determine the asymptotics of $Bh(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determining the maximum number of copies of $K_r$ in a $m$-vertex graph that has no $K_{s,t}$ subgraph, a problem studied by Alon and Shinkleman.