The spectrum for overlarge sets of hybrid triple systems

A hybrid triple system of order v and index λ, denoted by HTS(v, λ), is a pair (X, B) where X is a v-set and B is a collection of cyclic triples and transitive triples on X, such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint HTS(v, λ), denoted by OLHTS(v, λ), is a collection {(Y\ {y}, A _i )} _i , such that Y is a (v + 1)-set, each (Y \{y}, A _i ) is an HTS(v, λ) and all A _i s form a partition of all cyclic triples and transitive triples on Y. In this paper, we shall discuss the existence problem of OLHTS(v, λ) and give the following conclusion: there exists an OLHTS(v, λ) if and only if λ = 1, 2, 4, v ≡ 0,1 (mod 3) and v ≥ 4.