Quasigroup formulation of the Brown–Erdős–Sós conjecture

Let $Q$ be a quasigroup of order $n$, with multiplication written multiplicatively, and let $S$ be a subset of triples of the form $(a,b,ab)$ for $a,b\in Q$. A subset of points spans a triple in $S$ when all three entries of that triple belong to the subset.

Quasigroup formulation of the Brown–Erdős–Sós conjecture. For every $c>0$, there is a threshold $n_0$ such that if $Q$ has order $n\geq n_0$, then for every such subset $S$, whenever $|S|\geq cn^2$, some seven-element subset of $Q$ spans at least four triples from $S$.

The source states that this formulation is equivalent to the Brown–Erdős–Sós conjecture. It is the first open case of that conjecture, while the paper proves the assertion for finite groups.