Consider a fixed graph with vertices and edges respectively. Assume that is balanced which means that the maximum edge density among all possible subgraphs of is achieved from $H$. Take two copies of , call them and create a new graph . In case the two copies are edge disjoint then the edge density of is the same as the the edge density of . If however the intersection then prove that the edge density of is greater than the edge density of . Recall that the edge density of a graph is defined to be the number of edges divided by the number of vertices. If you solve the first part, use induction to generalize the statement when the number of copies is greater than 2.

It is a simple observation but turns out to be useful. The reason why I mention *fixed *graph in the title is because it is very useful when using the method of moments to find the number of copies of a fixed graph in . Use the observation to prove the following statement:

**Theorem:** If is a non empty strictly balanced graph and then the number of copies of is is asymptotically Poisson() where .