On the Existence of Nearly Kirkman Triple Systems with Subsystems

Let υ be a positive integer, and K and M be two sets of positive integers. A group divisible design GD(K,M;υ) is an ordered triple (X,G,B) where X is a set with cardinality υ, G is a set of subsets (called groups) of X such that G partitions X and ∣G∣ ∈ M for each G ∈ G, and B is a set of subsets (called blocks) of X such that ∣B∣ ∈ K for each B ∈ B, with the property that each pair of distinct elements of X is contained either in a unique group or in a unique block, but not both. The number υ is called the order of the group divisible design.