THE SPECTRUM OF ORTHOGONAL STEINER TRIPLE-SYSTEMS

Two Steiner triple systems (V, B) and (V, D) are orthogonal if they ha ve no triples in common, and if for every two distinct intersecting tr iples {x, y, z} and {u, v. z} of B, the two triples {x. y, a} and {u. v, b} in D satisfy a not-equal b. It is shown here that if v = 1, 3 (m od 6), v greater-than-or-equal-to 7 and v not-equal 9, a pair of ortho gonal Steiner triple systems of order v exist. This settles completely the question of their existence posed by O'Shaughnessy in 1968.