A note on measure-theoretic domatic partitions

TL;DR

This paper proves that in standard probability spaces, certain measure-preserving graphs can be colored measurably so that each vertex's neighborhood contains all colors, extending understanding of graph colorings in measure theory.

Contribution

It establishes the existence of a measurable vertex coloring with all colors appearing in each neighborhood for measure-preserving, regular Borel graphs.

Findings

Existence of measurable colorings in measure-preserving graphs.

Every vertex's neighborhood contains all colors in the coloring.

Applicable to $ u$-preserving $eth_0$-regular Borel graphs.

Abstract

We show that if $(X,\mu)$ is a standard probability space, then every $\mu$-preserving $\aleph_0$-regular Borel graph on $X$ admits a $\mu$-measurable vertex $\aleph_0$-coloring in which every vertex sees every color in its neighborhood.