On Homomorphism Graphs

TL;DR

This paper explores the combinatorial properties of acyclic bounded degree Borel graphs using a generalized determinacy method, revealing complexity results and limitations of classical graph coloring theorems in the Borel setting.

Contribution

It introduces new examples of acyclic Borel graphs, applies a generalized determinacy approach, and demonstrates the complexity of classifying such graphs with respect to Borel chromatic number.

Findings

Acyclic Δ-regular Borel graphs with Borel chromatic number ≤ Δ form a Σ^1_2-complete set for Δ>2

Classical Brooks'-like theorems fail in the Borel context

Unified framework for previous and new results in descriptive combinatorics

Abstract

We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for $\Delta>2$ it is impossible to give a simple characterization of acyclic $\Delta$-regular Borel graphs with Borel chromatic number at most $\Delta$: such graphs form a $\mathbf{\Sigma}^1_2$-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.