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.
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 or (recently) 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.[1] 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.[2][3][4]
^"Encyclopaedia of Design Theory: t-Designs". Designtheory.org. 2004-10-04. Retrieved 2012-08-17.
^Keevash, Peter (2014). "The existence of designs". arXiv:1401.3665 [math.CO].
^Erica Kleirrach (2015-06-09). "A Design Dilemma Solved, Minus Designs". Quanta Magazine. Retrieved 2015-06-27.
a Steinersystem (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and t = 2 or (recently) t ≥ 2. A Steiner system...
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matrix (table) is given by: The Fano plane, as a block design, is a Steiner triple system. As such, it can be given the structure of a quasigroup. This quasigroup...
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which under the Witt design W 24 {\displaystyle \mathrm {W} _{24}} of Steinersystem S ( 5 , 8 , 24 ) {\displaystyle \operatorname {S(5,8,24)} } emerges...
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