Borel chromatic numbers of locally countable $F_\sigma$ graphs and forcing with superperfect trees

TL;DR

This paper investigates the Borel chromatic numbers of low complexity graphs, demonstrating their boundedness in certain models after adding Laver reals, and addresses open questions about regularity properties of subsets of the real line.

Contribution

It establishes bounds on Borel chromatic numbers for specific classes of graphs in models with added Laver reals, answering open questions in the field.

Findings

Borel chromatic number bounded by continuum in models with Laver reals

Answers to questions on regularity properties of subsets of the real line

Extension of results to locally countable $F_\sigma$ graphs

Abstract

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 a strong form of locally countable graphs with compact totally disconnected set of vertices have Borel chromatic number bounded by the continuum of the ground model in the model obtained by adding $\aleph_2$ Laver reals. 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.