# On the cover Tur\'an number of Berge hypergraphs

## TL;DR

This paper introduces the cover Turán number for hypergraphs, providing bounds and exact densities for certain cases, notably when the hypergraph uniformity is three, advancing extremal hypergraph theory.

## Contribution

It defines the cover Turán number for Berge hypergraphs and establishes bounds and exact densities, especially for 3-uniform hypergraphs, extending Turán-type results.

## Key findings

- Established a general upper bound on the cover Turán number.
- Determined the cover Turán density for all graphs when the hypergraph is 3-uniform.
- Provided insights into the structure of Berge-$G$-free hypergraphs.

## Abstract

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this paper, we define a variant of Tur\'an number in hypergraphs, namely the cover Tur\'an number, denoted as $\hat{ex}_R(n, G)$, as the maximum number of edges in the shadow graph of a Berge-$G$ free $R$-graph on $n$ vertices. We show a general upper bound on the cover Tur\'an number of graphs and determine the cover Tur\'an density of all graphs when the uniformity of the host hypergraph equals to $3$.