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We discuss the minimal number of vertices in a graph with a large chromaticnumber such that each ball of a fixed radius in it has a small chromaticnumber. 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$\chi(G)\leq n$.

Number of vertices in graphs with locally small chromatic number and large chromatic number