The collection of \(n\)-element subsets of a Hausdorff space carries a natural topology. A continuous \(n\)-coloring on a Polish space \(X\) is a continuous map that assigns to each \(n\)-element subset of \(X\) one of two colors.

An \(n\)-coloring is uncountably homogeneous if the underlying space \(X\) is not the union of countably many sets on which the coloring is constant.

Generalizing a previous result about \(2\)-colorings (i.e., graphs) and answering a question of Ben Miller, it is shown that the class of uncountably homogeneous, continuous \(n\)-colorings on Polish spaces has a finite basis.

I.e., there is a finite collection of uncountably homogeneous, continuous \(n\)-colorings on the Cantor space such that every uncountably homogeneous, continuous \(n\)-coloring on any Polish space contains a copy of one of the finitely many colorings.

This complements some recent results of Lecomte and Miller on the nonexistence of small bases for uncountably chromatic analytic graphs.