Triangles and Vitali sets

TL;DR

This paper explores the consistency of certain set-theoretic properties related to triangles and Vitali sets within Euclidean plane under specific axioms, revealing independence results in set theory.

Contribution

It demonstrates the relative consistency of the hypergraph of equilateral triangles having countable chromatic number without the existence of Vitali sets under ZF+DC.

Findings

Hypergraph of equilateral triangles can have countable chromatic number

Vitali sets may not exist in certain models of set theory

Consistency results depend on an inaccessible cardinal

Abstract

It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles in Euclidean plane has countable chromatic number, while there is no Vitali set.