Threshold functions for

Let H be a graph on h vertices, and let G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n=h vertex disjoint copies of H. The fractional arboricity of H is a(H) = maxf jE 0 j jV 0 j?1 g, where the maximum is taken over all subgraphs (V 0 ; E 0) of H with jV 0 j > 1. Let (H) denote the minimum degree of a vertex of H. It is shown that if (H) < a(H) then n ?1=a(H) is a sharp threshold function for the property that the random graph G(n; p) contains an H-factor. I.e., there are two positive constants c and C so that for p(n) = cn ?1=a(H) , almost surely G(n; p(n)) does not have an H-factor, whereas for p(n) = Cn ?1=a(H) , almost surely G(n; p(n)) contains an H-factor (provided h divides n). A special case of this answers a problem of Erdd os.