Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2009 / A / 3 / c / Solution

Let $S\subseteq X$ be the vertices with fewer than $dn/2$ neighbours in $Y$. If $|S|>ϵn$, its density to all of $Y$ is below $d/2$. But epsilon-uniformity applies to $(S,Y)$ and requiresa contradiction. Therefore $|S|\le ϵn$.