Maximum embedding of an H(v-w, 3, 1) into a TS(v, λ)

Let H be a subgraph of a graph G, and let V ⊆ X. We say that an H-design (V, C) of order u and index μ is embedded into a G-design (X,B) of order v and index λ, μ ≤ λ, if 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 (V, C) into (X,B). For every pair of positive integers v, λ, we determine the minimum value of w such that there exists a triple system TS(v, λ) which embeds a handcuffed path design H(v − w, 3, 1).