Steiner System

A Steiner system, named after Jakob Steiner, is a type of block design: an n-element set together with a collection of k-element subsets, called blocks, chosen so that every t-element subset of the underlying set is contained in exactly one block, written S(t, k, n). Classically the definition also required k to equal t plus 1, with the t=2 case called a Steiner triple system and the t=3 case a Steiner quadruple system, though the modern generalized definition no longer strictly follows that naming pattern. Whether nontrivial Steiner systems exist for t of 6 or more was a long-standing open problem, resolved by Peter Keevash in 2014 with a proof that is non-constructive, so as of a few years later no actual Steiner system was known to be constructible for large values of t.