Galvin’s problem in higher dimensions

Abstract

It is proved that for each natural number n n , if | R | = ℵ n \left |\mathbb {R}\right |= {\aleph }_{n} , then there is a coloring of [ R ] n + 2 {\left [\mathbb {R}\right ]}^{n+2} into ℵ 0 {\aleph }_{0} colors that takes all colors on [ X ] n + 2 {\left [X\right ]}^{n+2} whenever X X is any set of reals which is homeomorphic to Q \mathbb {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 {\left [\mathbb {R}\right ]}^{2} into finitely many colors can be reduced to at most 2 2 colors on the pairs of some set of reals which is homeomorphic to Q \mathbb {Q} when large cardinals exist.