Coloring triangles and rectangles

TL;DR

This paper explores the chromatic properties of geometric hypergraphs, showing that under certain set-theoretic assumptions, rectangles can be colored with countably many colors while equilateral triangles cannot.

Contribution

It demonstrates the consistency of differing chromatic numbers for hypergraphs of rectangles and equilateral triangles in Euclidean space under ZF+DC.

Findings

Rectangles' hypergraph can have countable chromatic number.

Equilateral triangles' hypergraph can require uncountably many colors.

Set-theoretic assumptions influence geometric hypergraph colorings.

Abstract

It is consistent that ZF+DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles in two-dimensional Euclidean space does not.