Large sets of three types of oriented triple systems (Q1361669)

An ordered triple \((x,y,z)\) is said to be cyclic if it contains the ordered pairs \((x,y)\), \((y,z)\) and \((z,x)\); it is said to be transitive (or directed) if it contains the ordered pairs \((x,y)\), \((y,z)\) and \((x,z)\). If \(X\) is a \(v\)-set, a Mendelsohn triple system \(\text{MTS} (v,\lambda)\) is a pair \((X,B)\) where \(B\) is a collection of cyclic triples which include each ordered pair of distinct elements of \(X\) exactly \(\lambda\) times. Similarly, a directed triple system, DTS\((v,\lambda)\), is a pair \((X,B)\) where \(B\) is a collection of directed triples which include each ordered pair of distinct elements of \(X\) exactly \(\lambda\) times. A hybrid triple system \(\text{HTS}(v,\lambda)\) is a pair \((X,B)\) where \(B\) contains both cyclic and directed triples (but still includes each ordered pair of distinct elements of \(X\) exactly \(\lambda\) times). A large set of \(\text{MTS}(v,\lambda)\) (\(\text{DTS}(v,\lambda)\), \(\text{HTS}(v,\lambda)\)), denoted \(\text{LMTS}(v,\lambda)\) (\(\text{LDTS}(v,\lambda)\), \(\text{LHTS}(v,\lambda)\)) is a partition of the collection of all cyclic (directed, cyclic and directed) triples from a \(v\)-set into a set of \(\text{MTS}(v,\lambda)\) (\(\text{DTS}(v,\lambda)\), \(\text{HTS}(v,\lambda)\)). This paper summarises all results on existence of these large sets; it includes a comprehensive reference list, and it completes the necessary and sufficient conditions for existence of all \(\text{LMTS}(v,\lambda)\), \(\text{LDTS}(v,\lambda)\) and \(\text{LHTS}(v,\lambda)\).