Numbers of common blocks in Mendelsohn and directed triple systems with repeated elements (Q1323536)

An ordered triple system with repeats, of order \(v\), is a pair \((S,T)\) where \(S\) is a \(v\)-set and \(T\) is a collection of ordered triples of elements of \(S\) of type \((a,b,c)\) where \(a\), \(b\), \(c\) need not all be distinct, and so that every ordered pair of not necessarily distinct elements of \(S\) belongs to exactly one ordered triple in \(T\). If each triple \((a,b,c)\) is said to contain the pairs \((a,b)\), \((b,c)\), \((c,a)\), then the system is a Mendelsohn triple system with repeats, while if it contains the pairs \((a,b)\), \((b,c)\), \((a,c)\), it is a directed triple system with repeats. The authors show for each type that if \(v\equiv 0\pmod 3\) then two such directed triple systems can be constructed on \(v\) points with any prescribed number of common ordered triples, apart from obvious exceptions.