A note on the Brown--Erd\H{o}s--S\'os conjecture in groups

We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some $k$, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with $t$ triples spanning $\mathcal{O}(\sqrt{t})$ vertices, which is the best possible up to the implied constant. We confirm that for all $t$ we can find a collection of $t$ triples spanning at most $t+3$ vertices, resolving the Brown--Erd\H os--S\'os conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemer\'edi's theorem and the density Hales--Jewett theorem.