General lemmas for Berge-Tur\'an hypergraph problems

TL;DR

This paper introduces two general lemmas to analyze the maximum size of Berge-$F$-free hypergraphs, leading to new bounds and improvements for various graph structures like paths, cycles, and cliques.

Contribution

The paper develops general lemmas that unify and extend existing results on Berge-$F$-free hypergraphs, providing new bounds for multiple graph classes.

Findings

Bounds established for Berge-$F$-free hypergraphs when $F$ is a path or cycle.

Improved results for Berge-$F$-free hypergraphs with $F$ as a theta graph or $K_{2,t}$.

New bounds for hypergraphs avoiding Berge copies of cliques and general trees.

Abstract

For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$. We denote the maximum number of hyperedges in an $n$-vertex $r$-uniform Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F).$ In this paper we prove two general lemmas concerning the maximum size of a Berge-$F$-free hypergraph and use them to establish new results and improve several old results. In particular, we give bounds on $\mathrm{ex}_r(n,\textrm{Berge-}F)$ when $F$ is a path (reproving a result of Gy\H{o}ri, Katona and Lemons), a cycle (extending a result of F\"uredi and \"Ozkahya), a theta graph (improving a result of He and Tait), or a $K_{2,t}$ (extending a result of Gerbner, Methuku and Vizer). We also establish new bounds when $F$ is a clique (which implies extensions of results by Maherani and Shahsiah and by Gy\'arf\'as) and when $F$ is a general tree.