# On the weight of Berge-$F$-free hypergraphs

## TL;DR

This paper establishes an upper bound on the weight of Berge-$F$-free hypergraphs with edges of size between the Ramsey number of $F$ and a sublinear function of $n$, extending previous results in hypergraph theory.

## Contribution

It proves that such hypergraphs have weight $o(n^2)$ under specified conditions, improving and generalizing earlier results in the field.

## Key findings

- Weight of Berge-$F$-free hypergraphs is $o(n^2)$ when edge sizes are between the Ramsey number of $F$ and $o(n)$.
- The result is optimal in some cases.
- The paper also explores other weight functions and strengthens existing theorems.

## Abstract

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.