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.

Orthogonal group divisible designs of type hn for h ≤ 6

“…The problem was completely settled in [2]. That is, Theorem 1.1 [2]. A pair of orthogonal Steiner triple systems of order v exists if and only if v ≡ 1, 3 (mod 6), v ≥ 7 and v = 9.…”

“…The problem of the existence of orthogonal Steiner triple systems has been given quite a lot of attention by various authors (see [1,2,5,6,8]) since it was posed by O'Shaughnessy [7] in 1968. The problem was completely settled in [2].…”

Skew‐orthogonal steiner triple systems

“…The existence question for orthogonal pairs of STSv, or OSTSv out-lived this application and was only recently solved by Colbourn, Gibbons, Mathon, Mullin and Rosa [3]. Their main result is that, apart from v 3 or CCC 1063 8539/99/060431-10 9 for which they do not exist, an OSTSv exists whenever the necessary conditions v 1Y 3 mod 6 for STSv are ful®lled.…”

“…Assume v P K. Then there is a PBDvY K. Now for every k P K, there exists a pair S and the given PBD. The well-known fact [3] that OSTS has a closed spectrum implies S 1 c S 2 . Suppose uvaY xyb are blocks of S 1 and uvbY xya are blocks of S 2 .…”

“…We should mention that certain inductive OSTS constructions in [3] and [22] do not preserve the skew property simply because of the nonexistence of their base case, SOSTS(15), shown earlier. Despite the failure of these constructions, we propose the following conjecture.…”