### Very small embeddings of some partial Steiner triple systems

Darryn Bryant
*Department of Mathematics, University of Queensland,
Brisbane, Australia*

A Steiner triple system of order v is a partition of the edges
of the complete graph on v vertices into triangles. A partial
Steiner triple system of order v is a partition of some subset
of the edges of the complete graph of order v into triangles.
Not every partial Steiner triple system of order v can be
completed to a Steiner triple system of order v, but it is
known that any partial Steiner triple system can be completed
to a Steiner triple system of larger order. Such completions are
called embeddings of the partial Steiner triple systems. The
Lindner conjecture states that any partial Steiner triple system
of order u can be embedded in a Steiner triple system of order v
whenever v is equivalent to 1 or 3 (mod 6) and v is at least 2u+1.
Although in one sense this is the best possible result on the
embedding problem, some partial triple systems of order u can be
embedded in Steiner triple systems of order less than 2u+1. Such
embeddings will be topic of this talk.