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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 Erdős and Fishburn (1996). Maximum planar sets that determine k distances for k less than 5 has been identified. The 6-distance conjecture of Erdős 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.

A Proof of Erdős-Fishburn's Conjecture for $g(6)=13$