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 numberhm(c) of an n-coloring c:nX]→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 ω² 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 basisB.