page_title: Steiner system
hierarchical_section_title: Steiner system
context_page_description: In combinatorial mathematics, a Steiner system 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, is an n-element set S together with a set of k-element subsets of S 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 would be a t- design.
  This definition is relatively new. The classical definition of Steiner systems also required that k = t + 1. An S was called a Steiner triple system, while an S is called a Steiner quadruple system, and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.
  Long-standing problems in design theory were whether there exist any nontrivial Steiner systems with t ≥ 6; also whether infinitely many have t = 4 or 5. Both existences were proved by Peter Keevash in 2014. His proof is non-constructive and, as of 2019, no actual Steiner systems are known for large values of t.
context_section_description: 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 new.  The classical definition of Steiner systems also required that k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple (or triad) system, while an S(3,4,n) is called a Steiner quadruple system, and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.
  Long-standing problems in design theory were whether there exist any nontrivial Steiner systems (nontrivial meaning t < k < n) with t ≥ 6; also whether infinitely many have t = 4 or 5. Both existences were proved by Peter Keevash in 2014. His proof is non-constructive and, as of 2019, no actual Steiner systems are known for large values of t.
caption: The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.