Combinatorial Theory

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 \((2n-1)^{d/3}\).

Mathematics Subject Classifications: 05B10, 20E05

Keywords: Difference sets, distinct difference configurations, free groups, combinatorial designs