Altman, E., On a problem of P. Erdős. Amer. Math. Monthly 70 (1963), no. 2, 148--157, JSTOR 2312883; the planar Theorem, printed p. 149, with its proof, pp. 149--153. Library home: Altman 1963, result page Theorem, p. 149. The paper treats the plane only and prints no statement about polyhedra or three dimensions.

Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.