λ-fold indecomposable large sets of Steiner triple systems

A family (X, B1), (X, B2), . . . , (X, Bq) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTSλ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTSλ(v) if there exists no LSTSλ (v) contained in the collection for any λ λ. In 1995, Griggs and Rosa posed a problem: For which values of λ 1 and orders v ≡ 1, 3 (mod 6) do there exist IDLSTSλ(v)? In this paper, we use partitionable candelabra systems (PCSs) and holey λ-fold large set of STS(v) (HLSTSλ(v)) as auxiliary designs to establish a recursive construction for IDLSTSλ(v) and show that there exists an IDLSTSλ(v) for λ = 2, 3, 4 and v ≡ 1, 3 (mod 6).

λ-fold indecomposable large sets of Steiner triple systems

A family (X, B ₁), (X, B ₂), …, (X, B _q ) of q STS(ν)s is a λ-fold large set of STS(v) and denoted by LSTS _λ (ν) if every 3-subset of X is contained in exactly λ STS(ν)s of the collection. It is indecomposable and denoted by IDLSTS _λ (ν) if there exists no LSTSλ′ (ν) contained in the collection for any λ′ < λ. In 1995, Griggs and Rosa posed a problem: For which values of λ > 1 and orders ν ≡ 1,3 (mod 6) do there exist IDLSTS _λ (ν)? In this paper, we use partitionable candelabra systems (PCSs) and holey λ-fold large set of STS(ν) (HLSTS _λ (ν)) as auxiliary designs to establish a recursive construction for IDLSTS _λ (ν) and show that there exists an IDLSTS _λ (ν) for λ = 2, 3,4 and ν ≡ 1,3 (mod 6).