Skew‐orthogonal steiner triple systems

Two Steiner triple systems, S 1 =( V ,ℬ 1 ) and S 2 =( V ,ℬ 2 ), are orthogonal ( S 1 ⟂ S 2 ) if ℬ 1 ∩ ℬ 2 =∅ and if { u ,ν} ≠ { x,y }, uνw,xyw ∈ ℬ 1 , uνs , xyt ∈ ℬ 2 then s ≠ t . The solution to the existence problem for orthogonal Steiner triple systems, ( OSTS ) was a major accomplishment in design theory. Two orthogonal triple systems are skew‐orthogonal ( SOSTS , written S 1 ∼ S 2 ) if, in addition, we require uνw , xys ∈ ℬ 1 and uνt , xyw ∈ ℬ 2 implies s ≠ t . Orthogonal triple systems are associated with a class of Room squares, with the skew orthogonal triple systems corresponding to skew Room squares. Also, SOSTS are related to separable weakly union‐free TTS . SOSTS are much rarer than OSTS ; for example SOSTS(ν) do not exist for ν=3,9,15. Furthermore, a fundamental construction for the earlier OSTS proofs when ν ≡ 3 (mod 6) cannot exist. In the case ν≡ 1 ( mod 6) we are able to show existence except possibly for 22 values, the largest of which is 1315. There are at least two non‐isomorphic OSTS (19)s one of which is SOSTS (19) and the other not. A SOSTS (27) was found, implying the existence of SOSTS (ν) for ν ≡ 3 (mod 6) with finitely many possible exceptions.