# t-wise Berge and t-heavy hypergraphs

## TL;DR

This paper introduces new hypergraph types called t-heavy and t-wise Berge hypergraphs, extending Turán number bounds for paths, cycles, and cliques, with detailed results for 3-uniform hypergraphs.

## Contribution

It defines t-heavy and t-wise Berge hypergraphs and extends Turán number bounds to these new classes, providing asymptotic and exact results for various graph structures.

## Key findings

- Extended Turán bounds to t-heavy and t-wise Berge hypergraphs.
- Asymptotic determination of Turán numbers for paths and cycles.
- Exact Turán numbers for cliques in these hypergraph classes.

## Abstract

In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph $\mathcal{H}$ is a $t$-heavy copy of a graph $F$ if there is a copy of $F$ on its vertex set such that each edge of $F$ is contained in at least $t$ hyperedges of $\mathcal{H}$. $\mathcal{H}$ is a $t$-wise Berge copy of $F$ if additionally for distinct edges of $F$ those $t$ hyperedges are distinct. We extend known upper bounds on the Tur\'an number of Berge hypergraphs to the $t$-wise Berge hypergraphs case. We asymptotically determine the Tur\'an number of $t$-heavy and $t$-wise Berge copies of long paths and cycles and exactly determine the Tur\'an number of $t$-heavy and $t$-wise Berge copies of cliques. In the case of 3-uniform hypergraphs, we consider the problem in more details and obtain additional results.