On Tur\'{a}n problems for Berge forests

TL;DR

This paper investigates the maximum size of r-uniform hypergraphs avoiding Berge forests, providing exact and general Turán number results for these structures.

Contribution

It introduces new bounds and exact results for Turán numbers related to Berge forests in r-uniform hypergraphs.

Findings

Derived exact Turán numbers for specific Berge forests.

Established bounds for Turán numbers in general cases.

Extended previous results to broader classes of Berge forests.

Abstract

For a graph $F$, an $r$-uniform hypergraph $H$ is a Berge-$F$ if there is a bijection $\phi:E(F)\rightarrow E(H)$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph is $\mathcal{F}$-free if it does not contain any member in $\mathcal{F}$ as a subhypergraph. The Tur\'an number of $\mathcal{F}$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-uniform hypergraph on $n$ vertices. In this paper, some exact and general results on the Tur\'{a}n numbers for several types of Berge forests are obtained.