# Relation between number of edges and vertices in a DAG

I conjecture that, in a Directed Acyclic Graph, $O(|V|) = O(|E|)$. Is this statement correct, can it be refined? This is probably standard material; is there a simple reference about this?

The maximum number of edges in a DAG with n vertices is $\Theta(n^2)$.
Take the complete bipartite graph $K_{n,n}$ and direct all edges from left to right. This is a DAG with $2n$ vertices and $n^2$ edges.