Coloring closed Noetherian graphs

TL;DR

This paper investigates the coloring properties of closed Noetherian graphs within a set-theoretic framework, showing consistency results related to countable chromatic number and the non-existence of Vitali sets.

Contribution

It establishes the consistency of certain coloring properties of closed Noetherian graphs in ZF+DC set theory without assuming the Axiom of Choice.

Findings

Closed Noetherian graphs can be countably chromatic without an infinite clique.

Under ZF+DC, Vitali sets may not exist for such graphs.

The results depend on set-theoretic assumptions and are consistent with choiceless set theory.

Abstract

If G is a closed Noetherian graph on a sigma-compact Polish space without an infinite clique, it is consistent with the choiceless set theory ZF+DC that G is countably chromatic and there is no Vitali set.