KöMaL 2015 A657

Let $\{x_n\}$ be the van der Korput sequence, that is, if the binary representation of the positive integer $n$ is $n = \sum_i a_i2^i$ ($a_i\in\{0,1\}$), then $x_n = \sum_i a_i2^{-i-1}$. Let $V$ be the set of points $(n,x_n)$ in the plane where $n$ runs over the positive integers. Let $G$ be the graph with vertex set $V$ that is connecting any two distinct points $p$ and $q$ if and only if there is a rectangle $R$ which lies in a parallel position to the axes and $R\cap V = \{p,q\}$. Prove that the chromatic number of $G$ is finite.