Abstrakt
The G(0)-dichotomy due to Kechris, Solecki and Todorcevic characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the G(0)-dichotomy for Sigma(0)(xi)-measurable countable colorings when xi {= 3.
A Sigma(0)(xi)-measurable countable coloring gives a covering of the diagonal consisting of countably many Sigma(0)(xi) squares. This leads to the study of countable unions of Sigma(0)(xi) rectangles.
We also give a Hurewicz-like dichotomy for such countable unions when xi {= 2.