On the existence of dense substructures in finite groups

2019/02/21 by Wong, Ching
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.07819

Abstract

Fix 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 special case of a conjecture of Brown, Erdős and Sós. We remark that the result was recently discovered independently by Nenadov, Sudakov and Tyomkyn.

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