Enumerations of maximum partial triple systems on 14, 16 and 17 points

Fatih Demirkale, Diane Donovan, Mike Grannell

Published 2017-08-25Version 1

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 11,084,874,829 non-isomorphic STS(19)s. In this paper we complete enumeration of non-isomorphic maximum partial triple systems for $v\le 17$. It is shown that there are 35,810,097 systems on 17 points, 47,744,568 on 16 points, and 787 on 14 points. We also establish that there are precisely 157,151 non-isomorphic pairwise balanced designs, PBD($17,\{3,5\}$)s, having a single block of size 5. Structural properties of all 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.