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\(\lambda \)-fold indecomposable large sets of Steiner triple systems. (English) Zbl 1233.05049

Summary: A family \((X, \mathcal B_{1}), (X, \mathcal B_{2}), \dots , (X, \mathcal B_{q})\) of \(q\) STS\((v)\)s is a \(\lambda \)-fold large set of STS\((v)\) and denoted by \(\text{LSTS}_{\lambda }(v)\) if every 3-subset of \(X\) is contained in exactly \(\lambda\) STS\((v)\)s of the collection. It is indecomposable and denoted by \(\text{IDLSTS}_{\lambda}(v)\) if there exists no \(\text{LSTS}\lambda^{\prime} (v)\) contained in the collection for any \(\lambda^{\prime} < \lambda\). In 1995, T. S. Griggs and A. Rosa [Lond. Math. Soc. Lect. Note Ser. 218, 25–39 (1995; Zbl 0831.05014)] posed a problem: For which values of \(\lambda > 1\) and orders \(v \equiv 1,3 \pmod 6\) do there exist \(\text{IDLSTS}_{\lambda }(v)\)? In this paper, we use partitionable candelabra systems (PCSs) and holey \(\lambda \)-fold large set of \(\text{STS}(v)\) \((\text{HLSTS}_{\lambda }(v))\) as auxiliary designs to establish a recursive construction for \(\text{IDLSTS}_{\lambda }(v)\) and show that there exists an \(\text{IDLSTS}_{\lambda }(v)\) for \(\lambda = 2, 3,4\) and \(v \equiv 1,3 \pmod 6\).

MSC:

05B07 Triple systems
05B05 Combinatorial aspects of block designs

Citations:

Zbl 0831.05014
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References:

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