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

Abstract:

The $\mathbb {G}_0$-dichotomy due to Kechris, Solecki and Todorčević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the $\mathbb {G}_0$-dichotomy for $\mathbf {\Sigma }^0_\xi$-measurable countable colorings when $\xi \leq 3$. A $\mathbf {\Sigma }^0_\xi$-measurable countable coloring gives a covering of the diagonal consisting of countably many $\mathbf {\Sigma }^0_\xi$ squares. This leads to the study of countable unions of $\mathbf {\Sigma }^0_\xi$ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when $\xi \leq 2$.