  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.
