Borel chromatic numbers of closed graphs and forcing with uniform trees

TL;DR

This paper investigates the uncountable Borel chromatic numbers of closed graphs, demonstrating their potential to differ or equal the continuum, using Axiom A forcing techniques.

Contribution

It introduces methods to distinguish or equate uncountable Borel chromatic numbers of closed graphs within set theory.

Findings

Uncountable Borel chromatic numbers can be consistently different.

Uncountable Borel chromatic numbers can be consistently equal to the continuum.

Uses Axiom A forcing notions to establish results.

Abstract

In this work, we continue the tradition initiated by Geschke, 2011 of viewing the uncountable Borel chromatic number of analytic graphs as cardinal invariants of the continuum. We show that various uncountable Borel chromatic numbers of closed graphs can be consistently different, as well as consistently equal to the continuum. This is done using arguments that are typical to Axiom A forcing notions.