Let $G$ be a graph with $a\choose 2$ edges (and an arbitrary number of vertices). Is it true that it has at most $a\choose 3$ triangles?
Context: this continues the question Number of triangles in a graph based on number of edges. As a matter of fact, someone referred to this fact in his/her answers, but without a proof.
Let $G(V,E)$ be a graph and let $T(G)$ be the number of triangles in $G$. The maximum number of triangles in $G$ would be when every choice of 3 vertices forms a triangle in $G$, so that would be when $G$ is complete and if $|V(G)|=n$ then $T(G)\le\binom{n}{3}$. On the other hand, the clique number of $G$, denoted $\omega(G)$, would give you a lower bound on the number of triangles $G$ could have because $\omega(G)$ is the order of the largest complete sub graph in $G$. Thus $\binom{\omega(G)}{3}\le T(G)\le\binom{n}{3}$.