The maximum number of odd integral distances between points in the plane (Q1921342)

P. Erdös asked for the maximum number of odd integral distances occurring between \(n\) points in the plane [Oral communication, Conference on Graph Theory, Combinatorics and Compuratation (1994), Boca Raton, FL]. The author here proves that this number is \(n^2/3 + r(r-3)/6\) for all \(n\), where \(r=1, 2, 3\) and \(n \equiv r \pmod 3\).