k vertices of a regular n-gon P are coloured. A colouring is called almost uniform if for every positive integer m the following condition is satisfied:
If M1 is a set of m consecutive vertices of P and M2 is another such set then the number of coloured vertices of M1 differs from the number of coloured vertices of M2 at most by 1.
Prove that for all positive integers k and n (k≤n) an almost uniform colouring exists and that it is unique within a rotation.(M Kontsevich, Moscow) Coloringcombinatoricscombinatorial geometryregular polygon