# Tur\'an numbers for Berge-hypergraphs and related extremal problems

## TL;DR

This paper investigates the maximum size of hypergraphs avoiding Berge-embeddings of a given graph, establishing bounds and exploring connections with other extremal functions, with specific focus on bipartite graphs and girth conditions.

## Contribution

It provides new bounds on extremal functions for Berge-hypergraphs for general graphs and explores their relationships with other extremal problems, including specific cases like bipartite graphs.

## Key findings

- Established new upper and lower bounds for Berge-F hypergraphs.
- Connected Berge-hypergraph extremal functions with other graph and hypergraph extremal problems.
- Proved a counting result for hypergraphs of girth five, complementing known asymptotic formulas.

## Abstract

Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The maximum number of edges in an $n$-vertex $r$-graph with no subhypergraph isomorphic to any Berge-$F$ is denoted $\ex_r(n,\textrm{Berge-}F)$. In this paper we establish new upper and lower bounds on $\ex_r(n,\textrm{Berge-}F)$ for general graphs $F$, and investigate connections between $\ex_r(n,\textrm{Berge-}F)$ and other recently studied extremal functions for graphs and hypergraphs. One case of specific interest will be when $F = K_{s,t}$. Additionally, we prove a counting result for $r$-graphs of girth five that complements the asymptotic formula $\textup{ex}_3 (n , \textrm{Berge-}\{ C_2 , C_3 , C_4 \} ) = \frac{1}{6} n^{3/2} + o( n^{3/2} )$ of Lazebnik and Verstra\"{e}te [{\em Electron.\ J. of Combin}. {\bf 10}, (2003)].