Realizing an m-Uniform Four-Chromatic Hypergraph with Disks

We prove that for every positive integer m there is a finite point set $$\cal{P}$$ in the plane such no matter how three-colored, always disk containing exactly points, all of same color. This improves result Pach, Tardos and Tóth who proved two colors. The main ingredient construction subconstruction whose points are convex position. Namely, we show position two-colored, also unit disks similar can work, several other results.