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 $i$.
Alon proved this in https://link.springer.com/content/pdf/10.1007/BF02783300.pdf .
For simplicity assume that $|V_i|=\lceil 2ed\rceil=:k$ for all $i$. Let $I=\{x_1,\ldots,x_t\}$ where $x_i$ is chosen uniformly at random from $V_i$. Otherwise, arbitrarily choose subsets of size $k$ from each $V_i$ and restrict our view to those.
For every edge $uv\in E(H)$ crossing the partition, define the 'bad' event $B_{uv}=\{uv\subseteq I\}$. Suppose $u\in V_i$ and $v\in V_j$. Then $\mathbb P[B_{uv}]=\frac{1}{|V_i||V_j|}=k^{-2}$. Now since each $B_{uv}$ is determined by variables $x_i$ and $x_j$, we can say $B_{uv}\sim B_{u'v'}$ if $uv\neq u'v'$ and one of $u',v'\in V_i\cup V_j$. The total number of choices for $u'$ or $v'$ is at most $2kd-1$, since $|V_i\cup V_j|=2k$ and each has $\leq d$ neighbors. By the Lovasz Local Lemma, all bad events can be avoided as long as $e\cdot k^{-2}\cdot 2kd\leq 1$. But we have assumed that $\frac{2ed}{k}\leq 1$, so we are done.
