GALVIN’S PROBLEM IN HIGHER DIMENSIONS

Abstract

It is proved that for each natural number n, if |R| = ℵn, then there is a coloring of [R]n+2 into ℵ0 colors that takes all colors on [X]n+2 whenever X is any set of reals which is homeomorphic to Q. This generalizes a theorem of Baumgartner and sheds further light on a problem of Galvin from the 1970s. Our result also complements and contrasts with our earlier result saying that any coloring of [R]2 into finitely many colors can be reduced to at most 2 colors on the pairs of some set of reals which is homeomorphic to Q when large cardinals exist.