On the number of Mendelsohn and transitive triple systems
It is a great pity that the authors have chosen to rename as transitive triple systems designs which have appeared in the literature as directed designs (see for example \textit{J. Seberry} and \textit{D. Skillicorn} [J. Comb. Theory, Ser. A 29, 244-248 (1980; Zbl 0439.05011)], \textit{D. J. Street} and \textit{J. Seberry} [Util. Math. 18, 27-34 (1980; Zbl 0455.05015)], \textit{C. J. Colbourn} and \textit{M. J. Colbourne} [J. Comb. Theory, Ser. A 34, 375-378 (1983; Zbl 0508.05019)]) with applications in statistics and computer science. This paper discusses cyclic triple systems, from which Mendelsohn triple systems derive (MTS), where each block \(\{\) a,b,c\(\}\) is a collection of three ordered pairs \(\{\) (a,b),(b,c),(c,a)\(\}\) and directed or transitive triple systems (TTS) where each block \(\{\) d,e,f\(\}\) is a collection of three ordered pairs \(\{\) (d,e),(e,f),(d,f)\(\}\) where, in both cases, each ordered pair occurs once. The authors find the best possible asymptotic bounds on the large number of nonisomorphic MTSs and TTSs by showing that for large n the number N(n) of non-isomorphic MTSs or TTSs is approximately \(n^{n^ 2/3}\) in the sense that \(\lim_{n\to \infty}(\ln N(n))(n^ 2\ln n)=1/3.\)
scientific article; zbMATH DE number 3891381