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.