  We consider a planar Poisson process and its associated Voronoi map. We show
that there is a proper coloring with 6 colors of the map which is a
deterministic isometry-equivariant function of the Poisson process. As part of
the proof we show that the 6-core of the corresponding Delaunay triangulation
is empty.
  Generalizations, extensions and some open questions are discussed.
