BG-MO 1998 Problem 6

The sides and diagonals of a regular $n$ -gon $R$ are colored in $k$ colors so that: (i) For each color $a$ and any two vertices $A$ , $B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$ . (ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors. Prove that $k\leq 2$ .