  We prove that for every $m$ there is a finite point set $\mathcal{P}$ in the
plane such that no matter how $\mathcal{P}$ is three-colored, there is always a
disk containing exactly $m$ points, all of the same color. This improves a
result of Pach, Tardos and T\'oth who proved the same for two colors. The main
ingredient of the construction is a subconstruction whose points are in convex
position. Namely, we show that for every $m$ there is a finite point set
$\mathcal{P}$ in the plane in convex position such that no matter how
$\mathcal{P}$ is two-colored, there is always a disk containing exactly $m$
points, all of the same color. We also prove that for unit disks no similar
construction can work, and several other results.
