Subsets of free groups with distinct differences

Let F_n be a free group of rank n, with free generating set X. A subset D of F_n is a Distinct Difference Configuration if the differences g^{-1}h are distinct, where g and h range over all (ordered) pairs of distinct elements of D. The subset D has diameter at most d if these differences all have word length at most d. When n is fixed and d is large, the paper shows that the largest distinct difference configuration in F_n of diameter at most d has size approximately (2n-1)^{d/3}.