A proof of Erdos-Fishburn's conjecture for g(6)=13

A planar point set X in the Euclidean plane is called a k-distance set if there are exactly k distances between two distinct points in X. An interesting problem is to find the largest possible cardinality of k-distance sets. This problem was introduced by Erdos and Fishburn (1996). Maximum planar sets that determine k distances for k less than 5 has been identified. The 6-distance conjecture of Erdos and Fishburn states that 13 is the maximum number of points in the plane that determine exactly 6 different distances. In this paper, we prove the conjecture.