A proof of Erdős-Fishburn's conjecture for \(g(6)=13\) (Q1953348)

Summary: 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 a \(k\)-distance set. This problem was introduced by Erdős and Fishburn (1996). Maximum planar sets that determine \(k\) distances for \(k\) less than 5 have 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.