The fine structure of \((v,3)\) directed triple systems: \(v\equiv 2\) (mod 3). (Q2715966)

The paper is concerned with \((v,\lambda )\) DTS (directed triple systems of order \(v\) and index \(\lambda \) for \(v \equiv 2 \pmod 3\) and \(\lambda = 3)\) (each ordered pair appears exactly in \(\lambda \) triples of the collection). By \(c_i\) denote the number of triples that appear in the collection \(i\) times. The vector \((c_1,c_2,c_3)\) (the fine structure) is determined by \((t,s)\), where \(c_2 = t\) and \(c_3 = \lfloor v(v-1)/3 \rfloor -s\). The authors prove that the set of all possible values of \((t,s)\) can be described as \(\{(t,s) \mid 0 \leq t \leq s \leq \lfloor v(v-1)/3 \rfloor \), \(s \geq 6\}\). The constructive part of the proof makes use of group divisible designs and of DTS with a hole.