On determining when small embeddings of partial Steiner triple systems exist

Abstract

AbstractA partial Steiner triple system of order is a pair , where is a set of elements and is a set of triples of elements of such that any two elements of occur together in at most one triple. If each pair of elements occur together in exactly one triple it is a Steiner triple system. An embedding of a partial Steiner triple system is a (complete) Steiner triple system such that and . For a given partial Steiner triple system of order it is known that an embedding of order exists whenever satisfies the obvious necessary conditions. Determining whether “small” embeddings of order exist is a more difficult task. Here we extend a result of Colbourn on the ‐completeness of these problems. We also exhibit a family of counterexamples to a conjecture concerning when small embeddings exist.