Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-15T02:35:59.673Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction of Steiner Triple Systems Having Exactly One Triple in Common

Published online by Cambridge University Press:  20 November 2018

Charles C. Lindner
Charles C. Lindner*
Affiliation:
Auburn University, Auburn, Alabama
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Steiner triple system is a pair (Q, t) where Q is a set and t a collection of three element subsets of Q such that each pair of elements of Q belong to exactly one triple of t. The number |Q| is called the order of the Steiner triple system (Q, t). It is well-known that there is a Steiner triple system of order n if and only if n ≡ 1 or 3 (mod 6). Therefore in saying that a certain property concerning Steiner triple systems is true for all n it is understood that n ≡ 1 or 3 (mod 6). Two Steiner triple systems (Q, t1) and (Q, t2) are said to be disjoint provided that t1 ∩ t2 = Ø. Recently, Jean Doyen has shown the existence of a pair of disjoint Steiner triple systems of order n for every n ≧ 7 [1].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 1 , February 1974 , pp. 225 - 232
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Doyen, Jean, Construction of disjoint Steiner triple systems, Proc. Amer. Math. Soc. 32 (1972), 409–416.Google Scholar
2. Hilton, A. J. W., A simplification of Moore's proof of the existence of Steiner triple systems, J. Combinatorial Theory Ser. A 13 (1972), 422–425.Google Scholar
3. Lindner, C. C., Construction of quasigroups using the singular direct product, Proc. Amer. Math. Soc. 29 (1971), 263–266.Google Scholar
4. Lindner, C. C., Identities preserved by the singular direct product, Algebra Universalis 1 (1971), 86–89.Google Scholar
5. Lindner, C. C., Identities preserved by the singular direct product. II, Algebra Universalis 2 (1972), 113–117.Google Scholar
6. Lindner, C. C. and Mendelsohn, N. S., Construction of perpendicular Steiner quasigroups, Aequationes Math. 9 (1973), 150–156.Google Scholar
7. Sade, A., Produit direct-singulier de quasigroupes orthogonaux et anti-abéliens, Ann. Soc. Sci. Bruxelles Sér. I 74 (1960), 91–99.Google Scholar
8. Stein, S. K., On the foundations of quasigroups, Trans. Amer. Math. Soc. 85 (1957), 228–256.Google Scholar