Minimal definable graphs of definable chromatic number at least three

TL;DR

This paper constructs a Borel graph with chromatic number three that maps onto all such graphs and characterizes low-degree graphs with this property, highlighting differences with directed graphs.

Contribution

It introduces a universal Borel graph for chromatic number three and characterizes low-degree graphs with this property, revealing distinctions between graphs and digraphs.

Findings

Existence of a universal Borel graph with chromatic number three

Characterization of Borel graphs with degree at most two

Failure of the analogous property for digraphs

Abstract

We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property, and show that the analogous result for digraphs fails.