# Steiner system

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A pair , where is a finite set of elements and is a set of -subsets in (called blocks) such that every -subset from is contained in exactly one block of  . The number is called the order of the Steiner system . A Steiner system is a particular case of a block design and of a tactical configuration. A Steiner system for is a balanced incomplete block design (BIBD), and for , it is a finite projective plane. A necessary condition for the existence of a Steiner system is that the number is an integer for all satisfying . The sufficiency of this condition was proved for , , , (see , ).

In 1844 W. Woolhouse stated the existence problem for Steiner systems, and P. Kirkman solved it in 1847 for (Steiner triple systems). In 1853 J. Steiner examined the case of .

Problems usually considered for Steiner systems are: 1) the determination of the maximum number of mutually non-isomorphic Steiner systems of a given order ; 2) the existence of Steiner systems with a given group of automorphisms; 3) the imbedding of partial Steiner systems (not containing some -subsets of ) in a finite Steiner system; 4) the existence of resolvable Steiner systems (with representable as a union of partitions of ); 5) the maximal packing (minimal covering) of a complete set of -subsets of by disjoint (using Steiner systems).

The majority of results on Steiner systems are obtained for small values of and (see ).

How to Cite This Entry:
Steiner triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steiner_triple_system&oldid=42968