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Transactions of the American Mathematical Society

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


Authors: Dominique Lecomte and Miroslav Zeleny
Journal: Trans. Amer. Math. Soc. 366 (2014), 2345-2373
MSC (2010): Primary 03E15; Secondary 54H05
DOI: https://doi.org/10.1090/S0002-9947-2014-05876-5
Published electronically: January 28, 2014
MathSciNet review: 3165641
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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 $ {\bf\Sigma }^{0}_{\xi }$-measurable countable colorings when $ \xi \leq 3$. A $ {\bf\Sigma }^{0}_{\xi }$-measurable countable coloring gives a covering of the diagonal consisting of countably many $ {\bf\Sigma }^{0}_{\xi }$ squares. This leads to the study of countable unions of $ {\bf\Sigma }^{0}_{\xi }$ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when $ \xi \leq 2$.


References [Enhancements On Off] (What's this?)

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Additional Information

Dominique Lecomte
Affiliation: Université de Picardie, I.U.T. de l’Oise, site de Creil, 13, allée de la faïencerie, 60 107 Creil, France
Address at time of publication: Institut de Mathématiques de Jussieu, Université Paris 6, Projet Analyse Fonctionnelle, Couloir 16-26, 4ème étage, Case 247, 4, place Jussieu, 75 252 Paris Cedex 05, France
Email: dominique.lecomte@upmc.fr

Miroslav Zeleny
Affiliation: Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic
Email: zeleny@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/S0002-9947-2014-05876-5
Keywords: Borel chromatic number, Borel class, coloring, dichotomy, Hurewicz, partition, product
Received by editor(s): April 16, 2011
Received by editor(s) in revised form: May 11, 2012
Published electronically: January 28, 2014
Additional Notes: The work was part of the research project MSM 0021620839 financed by MSMT and partly supported by the grant GAČR 201/09/0067.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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