Galvin's Problem In Higher Dimensions

In 1970's Galvin conjectured that for all colorings of the reals there is a subset H of the reals homeomorphic to the rational numbers that gets at most two colors. i.e. the 2-dimensional Ramsey degree of the rational with respect to the reals is 2. Baumgartner proved that if we consider any infinite countable Hausdorff space X then there is a coloring with \omega many colors, which takes all its values on subsets homeomorphic to \mathbb{Q}. In paper from 2018 Raghavan and Todorcevic showed that assuming existance of Woodin cardinal the 2-dimensional Ramsey degree of \mathbb{Q} within respect to the class of all uncountable sets is 2. In this talk we give a result of Raghavan and Todorcevic in higher dimensions which is that there is no generalization of their previous theorem to dimension 3 and higher.