The existence spectrum for overlarge sets of pure directed triple systems

Abstract

A directed triple system of order v, denoted by DTS(v), is a pair (X, $\mathcal{B}$) where X is a v-set and $\mathcal{B}$ is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of $\mathcal{B}$. A DTS(v) (X, $\mathcal{A}$) is called pure and denoted by PDTS(v) if (a, b, c) ∈ $\mathcal{A}$ implies (c, b, a) ∉ $\mathcal{A}$. An overlarge set of PDTS(v), denoted by OLPDTS(v), is a collection {(Y {y _i }, $\mathcal{A}_i^j$): y _i ∈ Y, j ∈ Z ₃}, where Y is a (v + 1)-set, each (Y {y _i }, $\mathcal{A}_i^j$) is a PDTS(v) and these $\mathcal{A}_i s$ form a partition of all transitive triples on Y. In this paper, we shall discuss the existence problem of OLPDTS(v) and give the following conclusion: there exists an OLPDTS(v) if and only if v ≡ 0,1 (mod 3) and v > 3.