In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and t ≥ 2.
A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsets of S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In an alternate notation for block designs, an S(t,k,n) would be a t-(n,k,1) design.
This definition is relatively modern, generalizing the classical definition of Steiner systems which in addition required that k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple system, while an S(3,4,n) was called a Steiner quadruple system, and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.
As of 2012, an outstanding problem in design theory is if any nontrivial Steiner systems have t ≥ 6. It is also unknown if infinitely many have t = 5.
Read more about Steiner System: Properties, History, Mathieu Groups, The Steiner System S(5, 6, 12), The Steiner System S(5, 8, 24)
Famous quotes containing the words steiner and/or system:
“Language can only deal meaningfully with a special, restricted segment of reality. The rest, and it is presumably the much larger part, is silence.”
—George Steiner (b. 1929)
“Those words freedom and opportunity do not mean a license to climb upwards by pushing other people down. Any paternalistic system that tries to provide for security for everyone from above only calls for an impossible task and a regimentation utterly uncongenial to the spirit of our people.”
—Franklin D. Roosevelt (1882–1945)