Baire class one colorings and a dichotomy for countable unions of $F_\sigma$ rectangles

TL;DR

This paper investigates Baire class one colorings and establishes a dichotomy for countable unions of $F_\sigma$ rectangles, advancing understanding of their structure and partition properties.

Contribution

It introduces a Hurewicz-like dichotomy for countable unions of $F_\sigma$ rectangles, providing new insights into their partitioning and coloring.

Findings

Established a dichotomy for countable unions of $F_\sigma$ rectangles.

Connected Baire class one colorings to partitions into $F_\sigma$ sets.

Enhanced understanding of the structure of $F_\sigma$ rectangles in descriptive set theory.

Abstract

We study the Baire class one countable colorings, i.e., the countable partitions into $F_\sigma$ sets. Such a partition gives a covering of the diagonal into countably many $F_\sigma$ squares. This leads to the study of countable unions of $F_\sigma$ rectangles. We give a Hurewicz-like dichotomy for such countable unions.