  Fix $k \geq 6$. We prove that any large enough finite group $G$ contains $k$
elements which span quadratically many triples of the form $(a,b,ab) \in S
\times G$, given any dense set $S \subseteq G \times G$. The quadratic bound is
asymptotically optimal. In particular, this provides an elementary proof of a
special case of a conjecture of Brown, Erd\H{o}s and S\'{o}s. We remark that
the result was recently discovered independently by Nenadov, Sudakov and
Tyomkyn.
