Steiner triple system
A Steiner system , that is, a set of size n with a distinguished collection of subsets of size 3 ("blocks") such that every subset of size 2 is contained in exactly one block; denoted \mathrm{STS}(n). Such a system exists if and only if n \equiv 1,3 \pmod 6: this was already established by Revd T.P. Kirkman in 1846.
The projective plane \mathrm{P}(2,2) of order 2, consisting of 7 points and 7 lines each containing 3 points, in which any two points determine a unique line, is an \mathrm{STS}(7).
A resolution of a design is a partition of its blocks into "parallel" classes, such that element of the underlying set is contained in just one block of each class: a resolvable design is one with a resolution. A resolvable Steiner triple system is a Kirkman triple system \mathrm{KTS}(n). Such systems exist if and only if n \equiv 3 \pmod 6. The Kirkman schoolgirls problem, of finding a \mathrm{KTS}(15), was one of the classical combinatorial problems, solved by T.P. Kirkman in 1850.
A Steiner triple system gives rise to a quasi-group structure on the underlying set, defined by the binary operation x \cdot x = x and x \cdot y = z when x\ne y and \{x,y,z\} is the unique block containing \{x,y\}.
References
- Thomas Beth, Dieter Jungnickel, Hanfried Lenz, "Design theory", Cambridge University Press (1986) Zbl 0602.05001
- Anne Penfold Street, Deborah J. Street, "Combinatorics of experimental design", Clarendon Press (1987) ISBN 0-19-853255-5 Zbl 0622.05001
Steiner triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steiner_triple_system&oldid=54266