  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. A
hypergraph is Berge-$F$-free if it does not contain a subhypergraph that is a
Berge-$F$. The weight of a non-uniform hypergraph $\mathcal{H}$ is the quantity
$\sum_{h \in E(\mathcal{H})} |h|$.
  Suppose $\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this
short note, we prove that as long as every edge of $\mathcal{H}$ has size at
least the Ramsey number of $F$ and at most $o(n)$, the weight of $\mathcal{H}$
is $o(n^2)$. This result is best possible in some sense. Along the way, we
study other weight functions, and strengthen results of Gerbner and Palmer; and
Gr\'osz, Methuku and Tompkins.
