The $\mathbb{G}_0$-dichotomy due to Kechris, Solecki and Todor\v cević 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 $\xi\!\leq\! 3$. 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 $\xi\!\leq\! 2$.