Antipodal triple systems.
Peter B. Gibbons, Eric Mendelsohn, Hao Shen · 1994
An antipodal triple system of order v is a triple (V, B, 1), where 1 V 1 = v, B is a set of cyclically oriented 3-subsets of V, and f: V-+ V is an involution with one fixed point such that: (i) (V, B U f(B)) is a Mendelsohn triple system. Oi) B n f(B) = 0. (iii) f is an isomorphism between the Steiner triple system (ST S) (V, B') and the STS (V,f(B')), where B ' is the same as B without orientation. (iv) f preserves orientation. An ST S (V, B) is hemispheric if there exists a cyclic orientation B * of its block set B and an involution f such that (V, B*, 1) is an antipodal system. We prove that for all admissible v> 3, there exists an antipodal system. This is the first step in establishing the conjecture that every ST S(V, B) of order