  For $v\equiv 1$ or 3 (mod 6), maximum partial triple systems on $v$ points
are Steiner triple systems, STS($v$)s. The 80 non-isomorphic STS(15)s were
first enumerated around 100 years ago, but the next case for Steiner triple
systems was unresolved until around 2004 when it was established that there are
precisely 11084874829 non-isomorphic STS(19)s. In this paper we complete
enumeration of non-isomorphic maximum partial triple systems for $v\le 19$. It
is shown that there are 35810097 systems on 17 points and 47744568 on 16
points. We also establish that there are precisely 157151 non-isomorphic
pairwise balanced designs, PBD($17,\{3,5\}$)s, having a single block of size 5.
Structural properties of these systems are determined, including their
automorphism groups, and the numbers of Pasch configurations, mitres and Fano
planes contained in them. The systems themselves are available from the
authors.
