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:
“Words that are saturated with lies or atrocity, do not easily resume life.”
—George Steiner (b. 1929)
“We recognize caste in dogs because we rank ourselves by the familiar dog system, a ladderlike social arrangement wherein one individual outranks all others, the next outranks all but the first, and so on down the hierarchy. But the cat system is more like a wheel, with a high-ranking cat at the hub and the others arranged around the rim, all reluctantly acknowledging the superiority of the despot but not necessarily measuring themselves against one another.”
—Elizabeth Marshall Thomas. “Strong and Sensitive Cats,” Atlantic Monthly (July 1994)