Steiner Systems

Steiner triples Several examples from the recent posts on finite geometry are special cases of Steiner triples. A Steiner triple system, written STS(n), is a set S of n elements and a collection of three-element subsets of S called blocks, such that every two-element subset of S belongs to exactly one block. STS(7) is the smallest finite projective plane, the Fano plane, and STS(9) is the smallest finite affine plane. In both cases you interpret the elements of S as points and the blocks as lines. Several questions come up naturally. First, for which values of n does an STS(n) exist? For a given n, can there be more than one STS(n), i.e. are there Steiner triple systems of the same size that are not isomorphic? When does (some instance of) STS(n) correspond to a projective or affine space? An instance of STS(n) corresponds to a projective space iff n = 2k − 1 for k ≥ 3. For n ≥ 13 there are multiple isomorphism classes of STS(n), so for k > 3 there will be an instance of STS(2k − 1)…

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