Baire-class $\xi$ colorings: the first three levels

TL;DR

This paper extends the $G_0$-dichotomy to $oraxi$-measurable countable colorings for $\xi extless= 3$, exploring their structure and providing new dichotomies for analytic relations.

Contribution

It develops a $oraxi$-measurable version of the $G_0$-dichotomy for $\xi extless= 3$, and introduces a Hurewicz-like dichotomy for unions of $oraxi$ rectangles when $\xi extless= 2.

Findings

Extended $G_0$-dichotomy to $oraxi$-measurable colorings for $\xi extless= 3$

Characterized coverings of the diagonal via countably many $oraxi$ squares

Established a Hurewicz-like dichotomy for unions of $oraxi$ rectangles when $\xi extless= 2

Abstract

The $\mathbb{G}_0$-dichotomy due to Kechris, Solecki and Todor\vcevi\'c characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the $\mathbb{G}_0$-dichotomy for $\boraxi$-measurable countable colorings when $\xi\leq 3$. A $\boraxi$-measurable countable coloring gives a covering of the diagonal consisting of countably many $\boraxi$ squares. This leads to the study of countable unions of $\boraxi$ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when $\xi\leq 2$.