Almost universal graphs

We study the question as to when a random graph with n vertices and m edges contains a copy of almost all graphs with n vertices and cn/2 edges, c constant. We identify a ”phase transition” at c = 1. For c < 1, m needs to grow slightly faster than n, and we prove that m = O(n log log n/ log log log n) is sufficient. When c > 1, m needs to grow at a rate m = n1+a, where a = a(c) > 0 for every c > 1, and a(c) is between 1− 2 (1+o(1))c and 1− 1c for large enough c.