Pan-orientable block designs

A balanced incomplete block design BIBD(v, k, λ) is a pair (V,B) where V is a v-set (points) and B is a collection of k-subsets of V (blocks) such that each pair of elements of V occurs in exactly λ blocks. A k-tournament is a directed graph on k vertices in which there is exactly one arc between any two distinct vertices. Given a k-tournament T , we call a BIBD(v, k, 2) T -orientable if it is possible to replace each block B by a copy of T on the set B such that every ordered pair of distinct points appears in exactly one of the tournaments. We call a BIBD(v, k, 2) pan-orientable if it is T -orientable for every possible k-tournament T . There is an extensive literature on oriented triple systems. In this paper, we investigate the case k = 4. We prove that pan-orientable BIBD(v, 4, 2)s exist for any admissible order v with a finite number of possible exceptions and show for each admissible order v except v = 7 the existence of a BIBD(v, 4, 2) which is not pan-orientable. Moreover, we discuss the asymptotic existence of pan-orientable designs for general k, and study the repeated block problem.