Editing Steiner system (section)

{{Short description|Block design in combinatorial mathematics}}
[[image:Fano plane.svg|250px|right|thumbnail|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 [[Combinatorics|combinatorial]] [[mathematics]], a '''Steiner system''' (named after [[Jakob Steiner]]) is a type of [[block design]], specifically a [[Block design#Generalization: t-designs|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 (mathematics)|set]] ''S'' together with a set of ''k''-element [[subset]]s of ''S'' (called '''blocks''') with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternative 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 [[block design|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.<ref>{{cite web|url=http://designtheory.org/library/encyc/tdes/g |title=Encyclopaedia of Design Theory: t-Designs |publisher=Designtheory.org |date=2004-10-04 |access-date=2012-08-17}}</ref> 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''.<ref>{{cite arXiv |last=Keevash |first=Peter |eprint=1401.3665 |title=The existence of designs |date=2014|class=math.CO }}</ref><ref>{{cite web|author=Erica Kleirrach|url=https://www.quantamagazine.org/150-year-old-math-design-problem-solved-20150609/ |title=A Design Dilemma Solved, Minus Designs |publisher=Quanta Magazine |date=2015-06-09 |access-date=2015-06-27}}</ref><ref>{{cite web|last1=Kalai|first1=Gil|title=Designs exist!|url=http://www.bourbaki.ens.fr/TEXTES/1100.pdf|publisher=Séminaire Bourbaki}}</ref>