Self-orthogonal Mendelsohn triple systems (Q1906135)

A Mendelsohn triple system of order \(v\), briefly \(\text{MTS}(v)\), is a pair \((X, B)\) where \(X\) is a \(v\)-set of points and \(B\) is a collection of triples \((a, b, c)\) that contain ordered pairs \((a, b)\), \((b, c)\), and \((c, a)\) such that every ordered pair of distinct points occurs in exactly one tripe of \(B\). One can construct a latin square of order \(v\) from \(\text{MTS}(v)\) and if the latin square is orthogonal to its transpose, then the \(\text{MTS}(v)\) is said to be self-orthogonal, denoted by \(\text{SOMTS}(v)\). In this paper it is shown that a \(\text{SOMTS}(v)\) exists for all \(v\equiv 0\pmod 3\), except possibly for \(v= 18\).