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An S 3,4, n is called a Steiner quadruple system. The abbreviation SQS n is often used for these systems.
Up to isomorphism, SQS 8 and SQS 10 are unique, there are 4 SQS 14 s and 1,, SQS 16 s. An S 4,5, n is called a Steiner quintuple system.
Sufficient conditions are not known. There is a unique Steiner quintuple system of order 11, but none of order 15 or order The smallest order for which the existence is not known as of is An S 2,3, n is called a Steiner triple system , and its blocks are called triples.
It is common to see the abbreviation STS n for a Steiner triple system of order n. The fact that this condition on n is sufficient for the existence of an S 2,3, n was proved by Raj Chandra Bose  and T.
Up to isomorphism, the STS 7 and STS 9 are unique, there are two STS 13 s, 80 STS 15 s, and 11,,, STS 19 s. This is called resolvable and such systems are called Kirkman triple systems after Thomas Kirkman , who studied such resolvable systems before Steiner.
Dale Mesner, Earl Kramer, and others investigated collections of Steiner triple systems that are mutually disjoint i. The corresponding question for finding thirteen different disjoint S 2,3,15 systems was asked by James Sylvester in and answered by RHF Denniston in There is at least one such set of S 2,3,15 but its isomorphism is not known.
This makes S an idempotent , commutative quasigroup. Commutative idempotent quasigroups satisfying this additional property are called Steiner quasigroups.
Equalities, while technically possible, lead to trivial systems. It is a necessary condition for the existence of S t , k , n that b and r are integers.
Steiner triple systems were defined for the first time by Wesley S. Woolhouse in in the Prize question of Lady's and Gentlemen's Diary.
In Kirkman posed a variation of the problem known as Kirkman's schoolgirl problem , which asks for triple systems having an additional property resolvability.
Several examples of Steiner systems are closely related to group theory. Agostini Akrapovic Busso Conti Lafranconi Mistral QD Exhaust Zard.
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