We consider continuous colorings of the n-element subsets of a Polish space, which we call n-colorings for short, and their so-called homogeneity numbers. It turns out that there is a finite list of n-colorings on 2ω such that an n-coloring on a Polish space X has uncountable homogeneity number iff it contains a coloring from the list. The proof is based on a generalization of a Ramsey-style theorem of Blass. If time permits, I will also say something about the existence of universal 2-colorings.