Large Sets Of Triple Systems

Acollection (X,β₁), (X,β₂),..., (X,β_q) of q STS(v)s is aλ-fold large set of STS(v) and denoted by LSTS_λ(v) ifevery 3-subset of X is contained in exactlyλSTS(v)s of the collection. It is indecomposable and denoted by IDLSTS_λ(v) if there does not exist an LSTS_λ' (v) contained in the collection for anyλ' <λ. In 1995, Griggs and Rosa posed a problem: for which values ofλ> 1 and orders v≡1 or 3 (mod 6) do there exist an IDLSTS_λ(v)? In the first part of this paper, we use partitionable candelabra systems (PCSs) and holeyλ-fold large set of STS(v) (HLSTS_λ(v)) as auxiliary designs to establish a recursive construction for IDLSTS_λ(v). It shows that there is an IDLSTS_λ(v) forλ= 5,6, v≡1 or 3 (mod 6), and v≠3 with the only possible exception IDLSTS_λ(7).A Hybrid triple system of order v HTS(u,λ) = (X,β). whereβis a collection of cycle and transitive triples of X such that every ordered pair of X belongs toλtriples ofβ. 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 + l)-set, each (Y\{y},A_i) is a HTS(v,λ) and all A_is form a partition of all cycle and transitive triples of Y. In the last part of 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 ifλ= 1,2,4, v≡0,1 (mod 3), and v≥4.