The spectrum of \(d\)-cyclic oriented triple systems (Q2715955)

A partition of a \(v\)-element complete digraph into triples is known as directed triple system \(\text{DTS}(v)\) if triples are transitive and as \(\text{OTS}(v)\) if the triples are transitive or cyclic. An \(\text{OTS}(v)\) is said to be \(d\)-cyclic, if it admits an automorphism that fixes \(f = v-d\) points and cyclically permutes \(d\) points. The authors solve the existence problem when \(f \geq 2\) (the case \(f=1\) had been solved before). A necessary condition is \(v \geq 2f+1\), and \(\text{OTS}(v)\) then exists, if \(v,f \equiv 0,1 \pmod 3\), while \(\text{DTS}(v)\) exists, if \(\{v,f\} = \{x,y\}\) for \(x \equiv 0 \pmod 3\) and \(y \equiv 1 \pmod 3\).