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

Enumerations of maximum partial triple systems on 16 and 17 points