Partitioning Steiner triple systems into complete arcs

Abstract

For a Steiner triple system of order v to have a complete s-arc one must have s( s + 1)/2⩾ v with equality only if s = 1 or 2 mod 4. To partition a Steiner triplesystem of order s(s + 1) 2 into complete s-arcs, one must have s = 1 mod 4. In this paper wegive constructions of Steiner triple systems of order s(s + 1) 2 which can be partitioned into complete s-arcs for all s = 1 mod 4. For s = 1 or 5 mod 12, we construct cyclic Steiner triple systems having this property. For s = 9 mod 12 we use Kirkman triple systems of order s having one additional property to construct these Steiner triple systems. We further establish that Kirkman triple systems having this additional property exist at least for s = 9 mod 24 and s = 21 mod 120.