Trinal Decompositions of Steiner Triple Systems into Triangles

It is well known that when or , there exists a Steiner triple system (STS) of order n decomposable into triangles (three pairwise intersecting triples whose intersection is empty). A triangle in an STS determines naturally two more triples: the triple of “vertices” , and the triple of “midpoints” . The number of these triples in both cases, that of “vertex” triples (inner) or that of “midpoint triples” (outer), equals one‐third of the number of triples in the STS. In this paper, we consider a new problem of trinal decompositions of an STS into triangles. In this problem, one asks for three distinct decompositions of an STS of order n into triangles such that the union of the three collections of inner triples (outer triples, respectively) from the three decompositions form the set of triples of an STS of the same order. These decompositions are called trinal inner and trinal outer decompositions, respectively. We settle the existence question for trinal inner decompositions completely, and for trinal outer decompositions with two possible exceptions.