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

Abstract

The ${\mathbb{G}}_0$-dichotomy due to Kechris, Solecki and Todor\vcević 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 $\xi \le 3$. 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 $\xi \le 2$.