Small embedding of an into an

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.