Authors: Raghavan, Dilip
Todorčević, Stevo 
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: GALVIN’S PROBLEM IN HIGHER DIMENSIONS
Journal: Proceedings of the American Mathematical Society
Volume: 151
Issue: 7
First page: 3103
Last page: 3110
Issue Date: 2023
Rank: ~M22
ISSN: 0002-9939
DOI: 10.1090/proc/16386
Abstract: 
It is proved that for each natural number n, if |R| = ℵn, then there is a coloring of [R]n+2 into ℵ0 colors that takes all colors on [X]n+2 whenever X is any set of reals which is homeomorphic to 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 [R]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 Q when large cardinals exist.
Keywords: Partition calculus | Ramsey degree | rationals | strong coloring
Publisher: American Mathematical Society

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