论文标题
加尔文在更高维度的问题
Galvin's problem in higher dimensions
论文作者
论文摘要
事实证明，如果$ \ left | \ mathbb {r} \ right | = {\ aleph} _ {n} $，然后有$ {\ left [\ mathbb {r} \ right]}^{n+2} $ in $ {\ aleph} _ {0同型至$ \ Mathbb {q} $。这概括了鲍姆加特纳的定理，并进一步阐明了1970年代的加尔文问题。我们的结果还与我们的早期结果相称，并说$ {\ left [\ mathbb {r} \ right]}^{2} $的任何颜色都可以降低到有限的许多颜色中，最多可将$ 2 $的颜色降低到一组真实的成对上，当时是同理型的$ \ \ \ mathbb {q {q {q {q {
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.
