Resolvable Group Divisible Designs with Block Size 3

Let v be a non negative integer, let λ be a positive integer, and let K and M be sets of positive integers. Agroup divisible design, denoted by GD[K, λ, M. ν] is a triple (α γ β) where X is a set of points, γ = {G₁, G₂…} is a partition of α, and β is a class of subsets of X with the following properties. (Members of Γ are called groups and members of β are called blocks 1. The cardinality of X is ν. 2. The cardinality of each group is a member of M. 3. The cardinality of each block is a member of K. 4. Every 2–subset {x, y} of X such that x and y belong to distinct groups is contained in precisely λ blocks. 5. Every 2–subset {x, y) of X such that x and y belong to the same group is contained in no block. A group divisible design is resolvable if there exists a partition II= P₁, P₂, of β such that each part P₁, is itself a partition of X. In this paper we investigate the existence of resolvable group divisible designs with K = {3}, M a singleton set, and all λ The case where M = {1} has been solved by Ray-Chaudhuri and Wilson for λ = 1, and by Hanani for all λ > The case where M is a singleton set, and λ= 1 has recently been investigated by Rees and Stinson. We give some small improvements to Rees and Stinson's results, and give new results for the cases where λ> We also investigate a class of designs, introduced by Hanani, which we call frame resolvable group divisible designs and prove necessary and sufficient conditions for their existence.