Decomposition of Steiner Triple Systems into Triangles

Abstract

A triangle in a Steiner triple system $S$ is a triple of blocks from $S$ which meet pairwise and whose intersection is empty. If $S$ contains $b$ blocks, and $b=3q+8$, where $0\le 8\le 2$, then a triangulation of $S$ is a collection of $q$ triangles $\{T_1,T_2,\dots ,T_q\}$ in $S$ such that no two distinct triangles share a common block. It is shown that, for $v\equiv 1$ or $3(\mathrm{mod}6)$, there exists a Steiner triple system which admits a triangulation. Moreover, if $8=2$, there is a triangulated triple system in which the pair of blocks not occurring in a triangle are disjoint, and a triangulated triple system in which they intersect.