The spectrum of orthogonal Steiner triple systems

Abstract Two Steiner triple systems (V, 𝓑 ) and (V, 𝓓 ) are orthogonal if they have no triples in common, and if for every two distinct intersecting triples { x,y,z } and { x, y, z } of 𝓑, the two triples { x,y,a } and { u, v, b } in (𝓓 satisfy a ≠ b. It is shown here that if v ≡ 1,3 (mod 6), v ≥ 7 and v ≠ 9, a pair of orthogonal Steiner triple systems of order v exist. This settles completely the question of their existence posed by O'Shaughnessy in 1968.