Abstract
Let k≥6. We prove that any large enough finite group G contains k elements which span quadratically many triples of the form (a,b,ab)∈S×G, given any dense set S⊆G×G. The quadratic bound is asymptotically optimal. In particular, this provides an elementary proof of a conjecture of Brown, Erdős and Sós, in finite groups and when k is large.
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