Twin Steiner triple systems
A set of blocks of the form \(\{a,x,y\}, \{a,z,w\}, \{b,x,w\}, \{b,z,y\}\) in a Steiner triple system STS\((v)\) is called a Pasch configuration. Replacing these blocks by \(\{b,x,y\}, \{b,z,w\}, \{a,x,w\}, \{a,z,y\}\) produces a new STS\((v)\), which may or may not be isomorphic to the original design. This paper contributes to an investigation of the sets of designs that can be produced from these so-called Pasch switches. Of particular interest is determination of the equivalence classes produced when Pasch switching is performed on the class of all pairwise non-isomorphic STS\((v)\) containing at least one Pasch configuration, for fixed \(v\). The unique STS(9) has no Pasch configuration (i.e. is anti-Pasch). However for each of the cases \(v = 7, 13\) and \(15\), there is exactly one equivalence class of non anti-Pasch systems. But in general there are more, and this is reinforced by the results of this paper. A pair of twin Steiner triple systems of order \(v\) is a set of two STS\((v)\)s, each of which contains precisely one Pasch configuration which, when switched, produces the other system. Twin STS\((v)\)s have been exhibited for \(v = 19, 21, 25, 27\) and \(31\) in \textit{M. J. Grannell, T. S. Griggs} and \textit{J. P. Murphy} [Congr. Numerantium 86, 19-25 (1992; Zbl 0773.05023)]. The authors present a pair of twin STS\((33)\)s, and further show that there exists a pair of twin STS\((w)\)s for \(w = 3u(2s+1), u=19, 21, 25, 27, 31\) and \(33\). That is, there is an infinite family of equivalence classes each containing at most two systems. But, could some of these equivalence classes contain just one system? That is, can we construct identical twins, where the twins are isomorphic? The base twins constructed for \(v=19, 21, 25, 27, 31\) and \(33\) are all non-isomorphic. However, using hill-climbing, which seems to work so well with all kinds of triple systems, the authors were able to construct one pair of identical twin STS\((21)\)s, five pairwise non-isomorphic pairs of identical twin STS\((27)\)s, and three pairwise non-isomorphic pairs of identical twin STS\((33)\)s. However, they were unable to produce pairs of identical twin STS\((25)\)s or STS\((31)\)s. Furthermore, the recursive construction used for twins unfortunately does not necessarily produce identical twins, so we do not yet have an infinite family of identical twins. There is clearly much work still to be done in this area.
scientific article; zbMATH DE number 1018525
scientific article