Borel chromatic numbers of locally countable $F_σ$ graphs and forcing with superperfect trees

Raiean Banerjee, Michel Gaspar

Published 2022-08-16Version 1

In this work we study the uncountable Borel chromatic numbers, defined by Geschke (2011) as cardinal characteristics of the continuum, of low complexity graphs. We show that locally countable graphs with compact totally disconnected set of vertices have Borel chromatic number bounded by the continuum of the ground model. From this, we answer a question from Geschke and the second author (2022), and another question from Fisher, Friedman and Khomskii (2014) concerning regularity properties of subsets of the real line.