Balanced Transitive Orientations

A transitive orientation of a partial triple system \((S,T)\) of index \(2\lambda\) is a partial transitive triple system formed by replacing each triple \(t \in T\) with a transitive triple defined on the same vertex set as \(t\), such that each ordered pair occurs in at most \(\lambda\) of the resulting transitive triples. A transitive orientation \((S_1,T_1)\) of \((S,T)\) is said to be balanced if for all \(\{u,v\} \subseteq S\), if \(\{u,v\}\) occurs in \(\ell\) triples in \(T\) then \(\left\lfloor{\ell}/{2}\right\rfloor\)
and \(\left\lceil{\ell}/{2}\right\rceil\) transitive triples in \(T_1\) contain the arcs \((u,v)\) and \((v,u)\) respectively. In this paper, it is shown that every partial triple
system has a balanced transitive orientation. This result is then used to prove the existence of certain transitive group divisible designs.