| Abstract: | Two Steiner triple systems, S1=(V,ℬ︁1) and S2=(V,ℬ︁2), are orthogonal (S1 ⟂ S2) if ℬ︁1 ∩ ℬ︁2=∅︁ and if {u,ν} ≠ {x,y}, uνw,xyw ∈ ℬ︁1, uνs, xyt ∈ ℬ︁2 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 S1∼S2) if, in addition, we require uνw, xys ∈ ℬ︁1 and uνt, xyw∈ ℬ︁2 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. |
