A note on the uniformity threshold for Berge hypergraphs

Abstract

A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. Grósz, 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 $th(F)\le R(F,F^{\prime })$, where $R$ denotes the off-diagonal Ramsey number and $F^{\prime }$ is any graph obtained form $F$ by deleting an edge. We improve this bound to $th(F)\le R(K_{\chi (F)},F^{\prime })$, and use the new bound to determine $th(F)$ exactly for several classes of graphs.

8 pages