Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2011 / A / 3 / b

Let $B\subseteq X$ be the vertices with fewer than $(d/2)|Y|$ neighbours in $Y$. If $|B|\ge \epsilon |X|$, regularity applies to the pair $(B,Y)$ and givesBut averaging the defining strict degree inequality over $B$ gives $d(B,Y)<d/2$, a contradiction. ThereforeRegularity controls the neighbourhood of almost every vertex because a large set of exceptional low-degree vertices would itself witness an irregular density.