Supersaturation Theorem

Statement

Consider an $h$-uniform hypergraph $H$ with $v(H)$ vertices. The supersaturation theorem states that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $G$ is an $h$-uniform hypergraph on $n$ vertices and at least $(\pi(H) + \epsilon) \binom{n}{h}$ edges for $n$ sufficiently large, then there are at least $\delta n^{v(H)}$ copies of $H$.

Equivalently, we can restate this theorem as the following: If a graph $G$ with $n$ vertices has $o(n^{v(H)})$ copies of $H$, then there are at most $\pi(H) \binom{n}{2} + o(n^2)$ edges in $G$.