Small Embeddings of Partial Steiner Triple Systems. Issue 8 (28th June 2013) - Horsley, Daniel
Abstract It was proved in 2009 that any partial Steiner triple system of order u has an embedding of order v for each admissible v≥2u+1. This result is best possible in the sense 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. Many partial Steiner triple systems do have embeddings of orders smaller than 2u+1, but much less is known about when these embeddings exist. In this paper, we detail a method for constructing such embeddings. We use this method to show that each member of a wide class of partial Steiner triple systems has an embedding of order v for at least half (or nearly half) of the orders v<2u+1 for which an embedding could exist. For some members of this class we are able to completely determine the set of all orders for which the member has an embedding.