# Unfriendly colorings of graphs with finite average degree

## TL;DR

This paper proves that certain classes of graphs with finite average degree or subexponential growth admit Borel unfriendly colorings, advancing understanding of graph colorings in measure-theoretic and descriptive set theory contexts.

## Contribution

It establishes the existence of Borel unfriendly colorings for graphs with finite average degree and subexponential growth, extending previous results to broader graph classes.

## Key findings

- Graphs with finite average degree admit Borel unfriendly colorings.
- Graphs of subexponential growth admit Borel unfriendly colorings.
- Results apply to probability measure preserving Borel graphs.

## Abstract

In an unfriendly coloring of a graph the color of every node mismatches that of the majority of its neighbors. We show that every probability measure preserving Borel graph with finite average degree admits a Borel unfriendly coloring almost everywhere. We also show that every bounded degree Borel graph of subexponential growth admits a Borel unfriendly coloring.