Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Galvin's problem in higher dimensions   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.

14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin