A note on the extremal number of Berge-$C_4$

We improve the known upper bound for the extremal number of Berge-$C_4$-free $3$-uniform hypergraphs. More precisely, we prove that every $n$-vertex $3$-uniform hypergraph with no Berge cycle of length four has at most $$\frac{n^{3\mathrm{/}2}}{2+\sqrt{2}}+O(n)$$ hyperedges. This improves the previous best-known leading constant $1\mathrm{/}\sqrt{10}$ to $1\mathrm{/}(2+\sqrt{2})$.