On the Existence of \((v,n,4,\lambda)\)-\(IPMD\) for Even \(\lambda\)

Let \(v\), \(k\),\(\lambda\) and \(n\) be positive integers. \((x_1, x_2, \ldots, x_k)\) is defined to be \(\{(x_1, x_2), (x_2, x_3), \ldots, (x_k-1, x_k), (x_k, x_1)\}\), and is called a cyclically ordered \(k\)-subset of \(\{x_1, x_2, \ldots, x_1\}\). An incomplete perfect Mendelsohn design, denoted by \((v, n, 4, \lambda)\)-IPMD, is a triple \((X, Y, \mathcal{B})\), where \(X\) is a \(v\)-set (of points), \(Y\) is an \(n\)-subset of \(X\), and \(\mathcal{B}\) 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 \(\lambda\) blocks of \(\mathcal{B}\) and no ordered pair \((x, y) \in Y \times Y\) appears in any block of \(\mathcal{B}\) for any \(t\), where \(1 \leq t \leq (k – 1)\). In this paper, the necessary condition for the existence of a \((v, n, 4, \lambda)\)-IPMD for even \(\lambda\), namely \(v \geq (3n + 1)\), is shown to be sufficient.