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# Steiner triple systems

A *Steiner triple system* (of order *n*) STS(*n*)
is a 2-(*n*,3,1) design, that is, a Steiner system S(2,3,*n*),
in other words, a collection of 3-subsets of an *n*-set
such that any pair of elements of the *n*-set
is contained in a unique one among these 3-sets.
As was shown by Kirkman, a Steiner triple system of order *n* exists
if and only if either *n* = 0, 1, or *n* congruent to 1 or 3 (mod 6).

The unique Steiner triple system of order 7 is known as the *Fano* plane.
It is the (unique) projective plane of order 2.
The unique Steiner triple system of order 9 is the (unique) affine plane
of order 3.
Up to isomorphism, there are 2 Steiner triple systems of order 13,
and 80 of order 15, and 11084874829 of order 19.

The *block graph* of a Steiner triple system
is the graph with these 3-sets as vertices, where
two 3-sets are adjacent when they have nonempty intersection.
Such a graph is strongly regular, with parameters
*v* = *n*(*n* – 1)/6,
*k* = 3(*n* – 3)/2,
λ = (*n* + 3)/2,
μ = 9.

More generally, the block graph of a
Steiner system S(2,*m*,*n*)
is strongly regular.

Again more generally, the block graph of a quasi-symmetric design
is strongly regular.

## Reference

P. Kaski & P. R. J. Östergård,
*The Steiner Triple Systems of Order 19*,
Math. Comput. **73** (2004) 2075-2092.