A survey on resolvable quadruple systems

A Steiner system (X, ß), denoted S_λ(t, k, v), is a set X of points, of cardinality v, and a collection β of k-subsets of X called blocks, with the property that every t-subset of X is contained in precisely λ blocks. A quadruple system is a Steiner system S₁(3,4, v). A triple (X, β, γ) is called an (s, μ)-resolvable system if, for some s<t, it is a partition of an S_λ(t, k, v) system (X, β) into subsystems (X, γ₁), each of which is an S_μ (s, k, v) system, such that γ=γ₁|γ₂|…|γ_c is a partition of β. A system is doubly resolvable if it is resolvable and each (X, γ₁) is also resolvable. This article surveys the work done on the existence of (s, μ)-resolvable and doubly-resolvable quadruple systems for (s, μ)=(2, 1), (2, 3) and (1, 1).