Sparse Distance Sets in the Triangular Lattice

A planar point-set X in Euclidean plane is called a k-distance set if there are exactly k different distances among the points in X. The function g(k) denotes the maximum number of points in the Euclidean plane that is a k-distance set. In 1996, Erdős and Fishburn conjectured that for k > 7, every g(k)-point subset of the plane that determines k different distances is similar to a subset of the triangular lattice. We believe that if g(k) is an increasing function of k, then the conjecture is false. We present data that supports our claim and a method of construction that unifies known optimal point configurations for k > 3.