 Dense bipartite common-neighbour lemma

In a bipartite graph with parts of sizes $m,n$ and density $\delta >0$, a neighbourhood $V^{\prime }$ of size at least $\delta m/2$ can be chosen so that at least a proportion $1-2ϵ/{\delta }^2$ of its ordered pairs have at least $ϵn$ common neighbours. Choose a random vertex on the other side, use the second moment of its degree, and penalize pairs with few common neighbours. This is a simple dependent random choice tool for controlling restricted sumsets.