  We study a model of random graph where vertices are $n$ i.i.d. uniform random
points on the unit sphere $S^d$ in $\mathbb{R}^{d+1}$, and a pair of vertices
is connected if the Euclidean distance between them is at least $2- \epsilon$.
We are interested in the chromatic number of this graph as $n$ tends to
infinity.
  It is not too hard to see that if $\epsilon > 0$ is small and fixed, then the
chromatic number is $d+2$ with high probability. We show that this holds even
if $\epsilon \to 0$ slowly enough. We quantify the rate at which $\epsilon$ can
tend to zero and still have the same chromatic number. The proof depends on
combining topological methods (namely the Lyusternik--Schnirelman--Borsuk
theorem) with geometric probability arguments. The rate we obtain is best
possible, up to a constant factor --- if $\epsilon \to 0$ faster than this, we
show that the graph is $(d+1)$-colorable with high probability.
