SCHWEITZER 2018 Problem 1

Let $S\subset \mathbb{R}$ be a closed set and $f:\mathbb{R}^{2n}\to \mathbb{R}$ be a continuous function. Define a graph $G$ as follows: Let $x$ be a vertex of $G$ iff $x\in \mathbb{R}^{n}$ and $f(x,x)\not\in S$ , then connect the vertices $x$ and $y$ by an edge in $G$ iff $f(x,y)\in S$ or $f(y,x)\in S$ . Show that the chromatic number of $G$ is countable.