Galvin's problem in higher dimensions

TL;DR

This paper generalizes Galvin's problem to higher dimensions, showing that for any natural number n, there exists a coloring of n+2 subsets of reals with countably many colors that is fully realized on any set homeomorphic to the rationals, extending previous results and addressing a longstanding problem.

Contribution

It proves a new theorem for higher-dimensional subsets of reals, extending Galvin's problem and providing insights into colorings on sets homeomorphic to the rationals.

Findings

Existence of a coloring with countably many colors for (n+2)-subsets of reals

Any such coloring takes all colors on sets homeomorphic to the rationals

Generalization of Baumgartner's theorem to 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.