SOLVED

Let $G$ be a $3$-uniform hypergraph with $6$ vertices and $3$ $3$-edges. Is it true that
\[\mathrm{ex}_3(n,G)=o(n^2)?\]

A conjecture of Brown, Erdős, and Sós. The answer is yes, proved by Ruzsa and Szemerédi [RuSz78] (this is known as the Ruzsa-Szemerédi problem).

In [Er81] Erdős asks whether the same is true for any $3$-uniform hypergraph on $k$ vertices with $k-3$ $3$-edges.