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Construction techniques for anti-Pasch Steiner triple systems. (English) Zbl 0956.05023
A Pasch configuration, also known as a quadrilateral, comprises four triples of a Steiner triple system (STS) whose union has cardinality six. An STS without any Pasch configurations is called anti-Pasch. It has long been conjectured that there exists an anti-Pasch STS of order \(v\) for all \(v\equiv 1\) or \(3\pmod 6\), apart from the values \(v=3\) and 7. This paper presents four methods for constructing anti-Pasch STSs. The first generalises a construction of Stinson and Wei to obtain a general singular direct product construction. The second generalises the Bose construction. The third employs a construction due to Lu. The fourth uses Wilson-type inflation techniques and Latin squares having no subsquares of order two. These constructions produce anti-Pasch systems of order \(v\) for \(v\equiv 1\) or \(7\pmod{18}\), for \(v\equiv 49\pmod{72}\), and for many other values of \(v\). To complete a proof of the anti-Pasch conjecture, it would now suffice to construct anti-Pasch STSs for all \(v\) of the forms \(6p+ 1\) where \(p\equiv 5\pmod 6\) is a prime, and \(12p+ 1\) where \(p\equiv 1\pmod 6\) is a prime.

MSC:
05B07 Triple systems
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