Galvin’s problem in higher dimensions

Abstract:

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