# Tur\'an numbers of Berge trees

## TL;DR

This paper investigates the Turán numbers of Berge trees in hypergraphs, establishing bounds and extremal configurations for hypergraphs avoiding Berge copies of certain trees, extending classical results to hypergraph settings.

## Contribution

It provides new bounds for the Turán number of Berge trees in hypergraphs and characterizes extremal hypergraphs for specific parameters.

## Key findings

- Established a bound of rac{n(k-1)}{r+1} hyperedges for hypergraphs to contain Berge trees.
- Proved the bound is sharp when r+1 divides n.
- Determined extremal hypergraphs in the sharp cases.

## Abstract

A classical conjecture of Erd\H{o}s and S\'os asks to determine the Tur\'an number of a tree. We consider variants of this problem in the settings of hypergraphs and multi-hypergraphs. In particular, for all $k$ and $r$, with $r \ge k (k-2)$, we show that any $r$-uniform hypergraph $\mathcal{H}$ with more than $\frac{n(k-1)}{r+1}$ hyperedges contains a Berge copy of any tree with $k$ edges different from the $k$-edge star. This bound is sharp when $r+1$ divides $n$ and for such values of $n$ we determine the extremal hypergraphs.