  We discuss the minimal number of vertices in a graph with a large chromatic
number such that each ball of a fixed radius in it has a small chromatic
number. It is shown that for every graph $G$ on $\sim((n+rc)/(c+rc))^{r+1}$
vertices such that each ball of radius $r$ is properly $c$-colorable, we have
$\chi(G)\leq n$.
