is an S(2,3,7) Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.
In Combinatorics|combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a Block design#Generalization:t-design|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 (mathematics)|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 block design|design theory is if any nontrivial (t ) Steiner systems have t ≥ 6. It is also unknown if infinitely many have t = 5.
is an S(2,3,7) Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.
In Combinatorics|combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a Block design#Generalization:t-design|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 (mathematics)|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 ...