Let $V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\}$ and consider the metric that is defined for $f,g\in V$ by $$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$
We set $E = \{\{f,g\}: f,g \in V\text{ and } d(f,g) = 1\}$. Setting $G:=(V,E)$ it is easy to see that $G$ has a countable clique, but do we also have $\chi(G) = \aleph\_0$?



Graph of bounded continous functions with distance 1
It is a separable space and so may be covered by countably many sets of diameter less then 1.


