Krull dimension in set theory

TL;DR

This paper explores the set-theoretic properties of hypergraphs formed by rectangles in Euclidean spaces, demonstrating that certain chromatic number behaviors are consistent with ZF+DC.

Contribution

It shows the consistency of having a countable chromatic number in one hypergraph while its higher-dimensional counterpart has an uncountable chromatic number.

Findings

Chromatic number of Gn can be countable under ZF+DC.

Chromatic number of Gn+1 can be uncountable under the same assumptions.

Set-theoretic consistency results for hypergraph colorings in Euclidean spaces.

Abstract

Let n>1 be a number. Let Gn be the hypergraph of all rectangles in an n-dimensional Euclidean space. It is consistent that ZF+DC holds, the chromatic number of Gn is countable, yet the chromatic number of Gn+1 is uncountable.