A second infinite family of Steiner triple systems without almost parallel classes

For each positive integer n, we construct a Steiner triple system of order v= 2(3(n)) + 1 with no almost parallel class; that is, with no set of v-1/3 disjoint triples. In fact, we construct families of (v, k, lambda)-designs with an analogous property. The only previously known examples of Steiner triple systems of order congruent to 1 (mod 6) without almost parallel classes were the projective triple systems of order 2(n)-1 with n odd, and 2 of the 11,084,874,829 Steiner triple systems of order 19. Crown Copyright (C) 2013 Published by Elsevier Inc. All rights reserved.