I'm unsure of how to solve the following question. Intuitively, I am considering using an FKG inequality by creating some increasing and decreasing sequences, but I'm having trouble formalising it.
Let $G$ be a graph with maximum degree $d$, and let $V = V_1 \cup V_2 \cup \ldots \cup V_n$ be a partition of the vertices such that $|V_i| \geq 2ed$ for every $i$. Show that there is an independent set $\{v_1, \ldots, v_n\}$ such that $v_i \in V_i$ for every $i$.