The Intersection Problem for Steiner Triple Systems

Abstract

A Steiner triple system of order n is a pair (S,T) where T is an edge disjoint partition of the edge set of K_n (the complete undirected graph on n vertices with vertex set S) into triangles (or triples). It is by now well-known that the spectrum for triple systems is precisely the set of all n congruent to 1 or 3 (mod 6) [2]. Let J(n) denote the possible number of triples that two triple systems of order n can have in common. In this thesis, we provide an alternative proof of the following result of C.C. Lindner and A. Rosa [4] that J(n) = {0,1,2,...,x=(n(n-1))/6}\{x-1,x-2,x-3,x-5} for all n congruent to 1 or 3 (mod 6) with the exceptions of n = 9 and n = 13. Our alternative proof of this result makes use of H.L. Fu's solution to the intersection problem for quasigroups [1] which was not available at the time of the original publication.