The existence spectrum for overlarge sets of pure directed triple systems

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.