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

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