  The Hadwiger--Nelson problem is about determining the chromatic number of the
plane (CNP), defined as the minimum number of colours needed to colour the
plane so that no two points of distance 1 have the same colour. In this paper
we investigate a related problem for spheres and we use a few natural
restrictions on the colouring. Thomassen showed that with these restrictions,
the chromatic number of all manifolds satisfying certain properties (including
the plane and all spheres with a large enough radius) is at least 7. We prove
that with these restrictions, the chromatic number of any sphere with a large
enough radius is at least 8. This also gives a new lower bound for the minimum
colours needed for colouring the 3-dimensional space with the same
restrictions.
