Locally dense graph thinning lemma

For every $\epsilon >0$ there is $\delta >0$ such that a graph satisfying $e(A,B)\ge p|A||B|$ whenever $|A|,|B|\ge \delta n$, with $p\ge \delta$, has a spanning subgraph $G^{\prime }$ satisfyingfor all disjoint $A,B$. Partition into a fixed large number of nearly equal parts, independently thin each cross-pair to expected density $p$, and apply a concentration inequality and a union bound over all pairs $A,B$.