Krull dimension in set theory

Abstract

For every number $n\ge 2$, let ${\Gamma }_n$ be the hypergraph on $R^n$ of arity four consisting of all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that the chromatic number of ${\Gamma }_n$ is countable while that of ${\Gamma }_{n+1}$ is not.