Borel chromatic number of closed graphs

TL;DR

This paper constructs specific closed graphs with a Borel chromatic number of two and varying Baire class chromatic numbers, demonstrating the diversity of chromatic properties within closed graphs.

Contribution

It introduces a method to construct closed graphs with prescribed Baire class chromatic numbers, expanding understanding of graph coloring complexities.

Findings

Existence of closed graphs with Borel chromatic number two

Construction of graphs with Baire class $\xi$ chromatic number $\xi ext{-}\aleph_0$

Demonstration of the diversity in chromatic properties of closed graphs

Abstract

We construct, for each countable ordinal $\xi$, a closed graph with Borel chromatic number two and Baire class $\xi$ chromatic number $\aleph\_0$.