  A clique colouring of a graph is a colouring of the vertices such that no
maximal clique is monochromatic (ignoring isolated vertices). The least number
of colours in such a colouring is the clique chromatic number. Given $n$ points
$x_1, \ldots,x_n$ in the plane, and a threshold $r>0$, the corresponding
geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and
$v_j$ are adjacent when the Euclidean distance between $x_i$ and $x_j$ is at
most $r$. We investigate the clique chromatic number of such graphs.
  We first show that the clique chromatic number is at most 9 for any geometric
graph in the plane, and briefly consider geometric graphs in higher dimensions.
Then we study the asymptotic behaviour of the clique chromatic number for the
random geometric graph $RG$ in the plane, where $n$ random points are
independently and uniformly distributed in a suitable square. We see that as
$r$ increases from 0, with high probability the clique chromatic number is 1
for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and
finally drops back to 2.
