The (7, 4)-Conjecture in Finite Groups

Source abstract

The first open case of the Brown–Erdős–Sós conjecture is equivalent to the following: for every c > 0, there is a threshold n 0 such that if a quasigroup has order n ⩾ n 0 , then for every subset S of triples of the form ( a, b, ab ) with | S | ⩾ cn 2 , there is a seven-element subset of the quasigroup which spans at least four triples of S . In this paper we prove the conjecture for finite groups.