Incomplete Perfect Mendelsohn Designs

Let \(v, k\), and \(n\) be positive integers. An incomplete perfect Mendelsohn design, denoted by \(k\)-IMPD\((v, n)\), is a triple \((X, Y, B)\) where \(S\) is a \(v\)-set (of points), \(Y\) is an \(n\)-subset of \(X\), and \(B\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks) such that every ordered pair \((a, b) \in (X \times X) \setminus (Y \times Y)\) appears \(t\)-apart in exactly one block of \(B\) and no ordered pair \((a, b) \in Y \times Y\) appears in any block of \(B\) for any \(t\), where \(1 \leq t \leq k – 1\). In this paper, some basic necessary conditions for the existence of a \(k\)-IMPD\((v, n)\) are easily obtained, namely,
\((v – n)(v – (k – 1)n – 1) \equiv 0 \pmod{k} \quad {and} \quad v > (k – 1)n + 1.\) It is shown that these basic necessary conditions are also sufficient for the case \(k = 3\), with the one exception of \(v = 6\) and \(n = 1\). Some problems relating to embeddings of perfect Mendelsohn designs and associated quasigroups are mentioned.