# Hypergraph based Berge hypergraphs

## TL;DR

This paper explores extremal properties of Berge hypergraphs, generalizing classical graph results to hypergraphs, and establishes bounds on hyperedge sums for Berge-$$-free hypergraphs, linking to hypergraph Turán problems.

## Contribution

It extends Berge hypergraph theory from graphs to hypergraphs, providing new bounds and connecting to Turán problems in hypergraph extremal combinatorics.

## Key findings

- Sum of hyperedge sizes in Berge-$$-free hypergraphs is o(n^r) for large hyperedges.
- Generalizes several graph-based Berge hypergraph results.
- Establishes a connection between Berge hypergraphs and hypergraph Turán problems.

## Abstract

Fix a hypergraph $\mathcal{F}$. A hypergraph $\mathcal{H}$ is called a {\it Berge copy of $\mathcal{F}$} or {\it Berge-$\mathcal{F}$} if we can choose a subset of each hyperedge of $\mathcal{H}$ to obtain a copy of $\mathcal{F}$. A hypergraph $\mathcal{H}$ is {\it Berge-$\mathcal{F}$-free} if it does not contain a subhypergraph which is Berge copy of $\mathcal{F}$. This is a generalization of the usual, graph based Berge hypergraphs, where $\mathcal{F}$ is a graph. In this paper, we study extremal properties of hypergraph based Berge hypergraphs and generalize several results from the graph based setting. In particular, we show that for any $r$-uniform hypregraph $\mathcal{F}$, the sum of the sizes of the hyperedges of a (not necessarily uniform) Berge-$\mathcal{F}$-free hypergraph $\mathcal{H}$ on $n$ vertices is $o(n^r)$ when all the hyperedges of $\mathcal{H}$ are large enough. We also give a connection between hypergraph based Berge hypergraphs and generalized hypergraph Tur\'an problems.