Dániel Gerbner: On the weight of Berge-$F$-free hypergraphs

For a graph $F$, we say a hypergraph is a Berge-$F$ if it can be obtained from

$F$ by replacing each edge of $F$ with a hyperedge containing it. The weight of

a non-uniform hypergraph $H$ is the quantity $\sum_{h \in E(H)} |h|$.

 

Suppose $H$ is a Berge-$F$-free hypergraph on $n$ vertices. We prove that as long

as every edge of $H$ has size at least the Ramsey number of $F$ and at most $o(n)$,

the weight of $H$ is $o(n^2)$. We show that this is close to best possible in some sense.

Moreover, we give close to tight bounds for most weight functions.