Large Sets Of Resolvable Oriented Triple Systems

Let X be a finite set of v elements. In what follows an ordered pair of X is always a pair (x,y) with x,y∈X and x≠y. A cyclic triple on X is a set of three ordered pairs (x, y),(y, z) and (z, x) of X, which is denoted by (x, y, z)(or<y, z, x), or<z, x, y)). A transitive triple on X is a set of three ordered pairs (x, y),(y, z) and (x, z) of X, which is denoted by (x, y, z). A Mendelsohn (or directed) triple system of order v, briefly MTS(v)(or DTS(v)), is a pair (X, B) where B is a collection of cyclic (or transitive) triples on X, called blocks, such that every ordered pair of X belongs to exactly one block of B. Two MTS(v)s (or DTS(v)s) on the same set X are said to be disjoint if they have no common blocks. A large set of Mendelsohn (or directed) triple systems of order v is a partition of all cyclic (or transitive) triples on a v-element set into pairwise disjoint Mendelsohn (or directed) triple systems of order v.A set P of some cyclic (or transitive) triples on X is said to be a. parallel class if P forms a partition of X. An MTS(v)(or DTS(v))(X, B) is called resolvable if its block set B can be partitioned into parallel classes. A resolvable MTS(v)(or DTS(v)) is denoted by RMTS(v)(or RDTS(v)). An LRMTS(v)(or LRDTS(v)) denotes an LMTS(v)(or LDTS(v)) in which each member MTS(v)(or DTS(v)) is resolvable.This paper addresses questions related to the construction of large sets of resolv-able Mendelsohn triple systems and large sets of resolvable directed triple systems. An improved recursive method is established and a number of new infinite series large sets of prime power sizes are settled by recursion, in combination with already known small cases. To be specific, for all prime powers q<400and q≡1(mod3), large sets of resolvable Mendelsohn and directed triple systems of order qn+2are proved to exist for any positive integer n, possibly except for q=379,397.