Homogeneous embeddings of disjoint Mendelsohn triple systems (Q1918190)

Let \({\mathcal S}_X= \{(X, {\mathcal A}_i)\mid 1\leq i\leq n\}\) and \({\mathcal S}_Y= \{(Y, {\mathcal B}_i)\mid 1\leq i\leq n\}\) be two sets of pairwise disjoint Mendelsohn triple systems of order \(v\) and \(u\), respectively. The set \({\mathcal S}_X\) is homogeneously embeddable in \({\mathcal S}_Y\) if for each \(i\) with \(1\leq i\leq n\), the system \((X, {\mathcal A}_i)\) is embeddable in \((Y, {\mathcal B}_i)\). It is proved that if \(u- v\equiv 1\) or \(2\pmod 3\), then \(n\) disjoint Mendelsohn triple systems of order \(v\) can be homogeneously embedded in \(n\) disjoint Mendelsohn triple systems of order \(u\) if and only if \(u, v\equiv 0\) or \(1\pmod 3\), \(u\geq 2v+ 1\), \(v\geq 3\), \(v\neq 6\), for any \(n\) such that \(1\leq n\leq v- 2\).