The friendship theorem

Abstract

The “Friendship Theorem” states that, in a party of n persons, if every pair of persons has exactly one common friend, then there is someone in the party who is everyone else's friend. (It is assumed that “friendship” is a symmetric, irreflexive relation). This theorem has been proved by Erdös, Rényi and Sós; by G. Higman, and by Wilf. Each of these proofs has relied, at some point, on sophisticated mathematics, and Wilf has stated that no wholly elementary proof is known. The purpose of this note is to give an elementary graph theoretic proof in which the key step is a simple counting argument due to Ball. The Friendship Theorem is then applied to obtain a recent result of Ryser on intersections of finite sets.