Edge density of a bipartite graph ($d(A,B)$)

For disjoint nonempty vertex sets $A,B$, their edge density is

A pair $(A,B)$ is $\epsilon$-regular whenwhenever $X\subseteq A$, $Y\subseteq B$, $|X|\ge \epsilon |A|$, and $|Y|\ge \epsilon |B|$.

For every $\epsilon >0$ and $m_0$ there are $M,n_0$ such that every graph on at least $n_0$ vertices has a partitionwhere $m_0\le m\le M$, $|V_0|\le \epsilon |V|$, the other parts have equal size, and all but at most $\epsilon m^2$ pairs $(V_i,V_j)$ are $\epsilon$-regular.