EMBEDDING PARTIAL STEINER TRIPLE SYSTEMS WITH FEW TRIPLES.

In 2009 it was established that any partial Steiner triple system of order u has an embedding of order v for each v &#8805: 2u+1 such that v = 1, 3 (mod 6), in accordance with a conjecture of Lindner. It is known that for each u ≥ 9, there exists a partial Steiner triple system of order u that does not have an embedding of order v for any v < 2u + 1, so this result is best possible in one sense. Many partial Steiner triple systems do have embeddings of orders smaller than 2u + 1, however, although little is known about when such embeddings exist. In this paper we construct embeddings of orders less than 2u+1 for partial Steiner triple systems with few triples. In particular, we show that a partial Steiner triple system of order u ≥ 62 with at most u²/50-11u/100-116/75 triples has an embedding of order v for each admissible integer v ≥ 8u+17/5. [ABSTRACT FROM AUTHOR]