The spectrum of Meta(K-4-e > K-3+e,lambda) with any lambda

Let (X, Β) be a (λK_v, G_1)-design and G_2 a subgraph of G_1. Define sets Β(G_2) and D(G_2\G_2) as follows: for each block B ε Β, partition B into copies of G_2and G_2 \ G 2 and place the copy of G_2 in Β(G_2) and the edges belonging to the copy of G_2\ G_2 in D(G_2 \ G_2). If the edges belonging to D(G1\ G_2) can be assembled into a collection D(G_2) of copies of G_2, then (X, Β(G_2) ∪ D(G_2)) is a (λK_v, G_2)-design, called a metamorphosis of the (λKv, G_1)design (X, Β). For brevity we denote such (λK_v, G_1)-design (X, Β) with a metamorphosis of (λK_v,G_2)-design (λK_v, (G_2) ∪ D(G_2)) by (λKv, G1 > G2)-design. Let Meta(G1 > G2, λ) denote the set of all integers v such that there exists a (λK_ v, G_1 > G_2)-design. In this paper we completely determine the set M eta(K4 - e > K_3 + e, λ) for any λ.