A Variation of the Friendship Theorem

Abstract

A friendship set is a finite set with a symmetric nonreflexive binary relation, called on, satisfying : (i) if a is not on b, then there exists a unique element which is on both a and b ; (ii) if a is on b, then there exists at most one element which is on both a and b. It is shown that for any nontrivial friendship set, there exists a least integer m, called the degree of the set, such that each element is on exactly m or $m - 1$ elements. If every element is on the same number of elements, the set is said to be homogeneous. The number of elements in a friendship set of degree m is either $m^2 + 1$ or $m^2 - m + 1$, depending on whether it is homogeneous or not. There are at most four distinct homogeneous friendship sets. Necessary conditions for the existence of nonhomogeneous friendship sets are given in terms of the irreducibility of certain polynomials.