Families of trees decompose the random graph in an arbitrary way

Let F= {H₁, . . . ,H_k} be a family of graphs. A graph G is called totally F-decomposable if for every linear combination of the form α₁ e(H₁) + ⋯ + α_ke(H _k) = e(G) where each α_i is a nonnegative integer, there is a colouring of the edges of G with α₁ + ⋯ + α_k colours such that exactly α_i colour classes induce each a copy of H_i, for i = 1, . . . ,k. We prove that if F is any fixed nontrivial family of trees then log n/n is a sharp threshold function for the property that the random graph G(n,p) is totally F-decomposable. In particular, if H is a tree with more than one edge, then log n/n is a sharp threshold function for the property that G(n,p) contains ⌊e(G)/e(H) ⌋ edge-disjoint copies of H.