Galvin’s problem in higher dimensions

It is proved that for each natural number n n , if alttext="StartAbsoluteValue double-struck upper R EndAbsoluteValue equals normal alef Subscript n"> | R = mathvariant="normal">ℵ encoding="application/x-tex">\left |\mathbb {R}\right |= {\aleph }_{n} then there a coloring of alttext="left-bracket right-bracket Superscript n plus 2"> [ ] + 2 encoding="application/x-tex">{\left [\mathbb ]}^{n+2} into alttext="normal 0"> 0 encoding="application/x-tex">{\aleph }_{0} colors takes all on X X [X\right whenever alttext="upper X"> encoding="application/x-tex">X any set reals which homeomorphic to alttext="double-struck Q"> mathvariant="double-struck">Q encoding="application/x-tex">\mathbb {Q} . This generalizes theorem Baumgartner and sheds further light problem Galvin from the 1970s. Our result also complements contrasts with our earlier saying squared"> ]}^{2} finitely many can be reduced at most alttext="2"> encoding="application/x-tex">2 pairs some when large cardinals exist.