For a positive integer $n$, let $\mathcal{P}$ be the power set of $[n]$. Consider the graph $G$ with $\mathcal{P}$ as its vertex set, and, for $S_1,S_2 \in \mathcal{P}$, the edge $(S_1,S_2)$ exists iff $S_1 \subset S_2$. (That is, $G$ is the transitive closure of the Hasse diagram of the "$\subset$" relation.)

**Question:** Suppose we can color the vertices of $G$ using $k$ colors. How many vertices $v$ are there such that $v$ has at least one neighbor that has the same color as $v$?

Remarks:

- Note that the question asks for a lower bound on the number of such vertices that holds no matter how $G$ is colored with $k$ colors.
- Obviously, the bound will depend on $k$. I'm mostly interested in large $k$, e.g., $k \approx \epsilon|\mathcal{P}|$, for some small constant $\epsilon>0$.