高维Galvin问题
Galvin's problem in higher dimensions
摘要：
证明了对每个自然数$n$，如果$\left| \mathbb{R} \right| = {\aleph}_{n}$，则存在一个将${\left[ \mathbb{R} \right]}^{n+2}$染色成${\aleph}_{0}$色的染色，该染色取${\left[ X \right]}^{n+2}$上的所有色，只要$X$是与$\mathbb{Q}$同胚的实数集.这推广了鲍姆加特纳的一个定理，并进一步阐明了20世纪70年代高尔文的一个问题。我们的结果也补充和对比了我们以前的结果，即当存在大基数时，任何将${\left[ \mathbb{R} \right]}^{2}$染色成多个颜色的着色在同胚于$\mathbb{Q}$的实数集的对上最多可约化为$2$颜色。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
