Factors in random graphs

Let H be a fixed graph on v vertices. For an n-vertex graph G with n divisible by v, an H-factor of G is a collection of n/v copies of H whose vertex sets partition V (G). In this paper we consider the threshold th H (n) of the property that an Erd˝ os-Rényi random graph (on n points) contains an H-factor. Our results determine th H (n) for all strictly balanced H. The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random k-uniform hyper-graph, solving the well-known " Shamir's problem. "