MathDB

Some

k{}

vertices of a regular

n{}

-gon are colored red. We will call a coloring uniform if for any

m

the number of red vertices in any two sets of

m

consecutive vertices of the

n{}

-gon coincide or differ by 1. Prove that a uniform coloring exists for any

k<n

and is unique, up to rotations of the

n{}

-gon.
Proposed by M. Kontsevich

Problem Statement