Let $X$ be a non-empty set. Consider $\mathcal{P}(X)$, the power-set of $X$. We say that $a,b \in \mathcal{P}(X)$ form an edge if and only if their symmetric difference is a singleton, i.e. $\textrm{card}((a\setminus b) \cup (b\setminus a)) = 1$.

It is clear that for a finite set $X$ the resulting graph has chromatic number 2: color those subsets of $X$ with an even number of members with color 1, and the rest with color 2.

For $X$ infinite, does the resulting graph still have chromatic number 2?