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

Dominique Lecomte (1) , Miroslav Zeleny (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 Department of Mathematical Analysis [Prague]

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 $\xi\!\leq\! 2$.

Mots clés: Borel chromatic number; Borel class; coloring; dichotomy; Hurewicz; partition; product

Domaines: Logique [math.LO]; Topologie générale [math.GN]

Dominique Lecomte, Miroslav Zeleny. Baire-class $\xi$ colorings: the first three levels. 2011. ⟨hal-00586487⟩