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

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.