A note on the uniformity threshold for Berge hypergraphs

TL;DR

This paper refines the upper bound on the uniformity threshold for Berge hypergraphs, leading to exact determinations for certain graph classes, advancing understanding of hypergraph extremal properties.

Contribution

The authors improve the existing bound on the uniformity threshold from a Ramsey number to a chromatic number-based bound and determine this threshold exactly for specific graph classes.

Findings

Improved upper bound: th(F) ≤ R(K_{χ(F)},F')

Exact thresholds determined for several graph classes

Enhanced understanding of Berge hypergraph extremal properties

Abstract

A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. Gr\'osz, Methuku and Tompkins in 2020 showed that for any graph $F$, there is an integer $r_0=r_0(F)$, such that for any $r\ge r_0$, any $r$-uniform hypergraph without a Berge copy of $F$ has $o(n^2)$ hyperedges. The smallest such $r_0$ is called the uniformity threshold of $F$ and is denoted by $th(F)$. They showed that $th(F)\le R(F,F')$, where $R$ denotes the off-diagonal Ramsey number and $F'$ is any graph obtained form $F$ by deleting an edge. We improve this bound to $th(F)\le R(K_{\chi(F)},F')$, and use the new bound to determine $th(F)$ exactly for several classes of graphs.