Families of trees decompose the random graph in any arbitrary way

Let F ={H1, . . . ,Hk} be a family of graphs. A graph G with m edges is called totallyF -decomposable if for every linear combination of the form α1e(H1)+ · · ·+αke(Hk) = m whereeach αi is a nonnegative integer, there is a coloring of the edges of G with α1 + · · ·+ αk colorssuch that exactly αi color classes induce each a copy of Hi, for i = 1, . . . , k. We prove that ifF is any fixed family of trees then logn/n is a sharp threshold function for the property thatthe random graph G(n, p) is totally F -decomposable. In particular, if H is a tree, then log n/nis a sharp threshold function for the property that G(n, p) contains ⌊e(G)/e(H)⌋ edge-disjointcopies of H .