Small embeddings of an S_3(2,4,u) into an S_lambda(2,4,u+w)

Let H be a subgraph of a graph G. An H-design (U, C) of order u and index μ is embedded into a G-design (V,B) of order v and index λ if μ ≤ λ, U ⊆ V and there is an injective mapping f : C → B such that B is a subgraph of f (B) for every B ∈ C. The mapping f is called the embedding of (U, C) into (V,B). We determine, for every admissible value of u and λ, the minimum value of w (except 12 values of (u, λ)) such that every S_3(2, 4, u) can be embedded into an S_λ(2, 4, u + w). This result implies that we determine also the minimum value of w such that there exists an S_λ(2, 4, u + w) which embeds an E_2(u, 1), where E_2 is the graph with two parallel edges and without isolated vertices.