On Distance Sets in the Triangular Lattice

A point set X in the Euclidean plane is called a k-distance set if there are exactly k different distances between distinct points in X. Let g(k) denote the maximum number of points in a k-distance set. For example, it is well known that g(1) = 3, realized by the three vertices of an equilateral triangle. Erdős and Fishburn [2] introduce the problem of determining g(k). To date only six values of g(k) are known: g(1) = 3, g(2) = 5, g(3) = 7, g(4) = 9, g(5) = 12, and g(6) = 13. The first five of these were found by Erdős and Fishburn [2] and the last value was recently determined by Wei [7]. The triangular lattice is defined as follows: