Local Coloring Problems on Smooth Graphs

TL;DR

The paper constructs a smooth locally finite Borel graph with a local coloring problem that admits a solution but cannot be realized as a Borel coloring, highlighting limitations of Borel methods.

Contribution

It demonstrates the existence of a smooth locally finite Borel graph with a local coloring problem solvable in theory but not via Borel coloring, revealing new limitations.

Findings

Existence of a smooth locally finite Borel graph with a solvable local coloring problem.

Such a graph admits a coloring solving the problem but no Borel coloring exists.

Highlights limitations of Borel methods in graph coloring problems.

Abstract

We construct a smooth locally finite Borel graph $G$ and a local coloring problem $\Pi$ such that $G$ has a coloring $V(G) \to \mathbb{N}$ that solves $\Pi$, but no such coloring can be Borel.