Basis theorems for continuous n-colorings

This article is devoted to the study of continuous colorings of the n-element subsets of a Polish space. The homogeneity number hm(c) of an n-coloring c : [X]n → 2 is the least size of a family of c-homogeneous sets that covers X. An n-coloring is uncountably homogeneous if hm(c) > א0. Answering a question of B. Miller, we show that for every n > 1 there is a finite family B of continuous n-colorings on 2ω such that every uncountably homogeneous, continuous n-coloring on a Polish space contains a copy of one of the colorings from B. We also give upper and lower bounds for the minimal size of such a basis B.