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A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension

Author(s): Dominique Lecomte
Journal: Trans. Amer. Math. Soc. 361 (2009), 4181-4193.
MSC (2000): Primary 03E15; Secondary 54H05
Posted: February 10, 2009
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Abstract: We study the extension of the Kechris-Solecki-Todorčević dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism.

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B. Bollobás, Modern graph theory, Graduate Texts in Math., vol. 184, Springer-Verlag, New York, 1998. MR 1633290 (99h:05001)

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A. S. Kechris, S. Solecki and S. Todorčević, Borel chromatic numbers, Adv. Math. 141 (1999), 1-44. MR 1667145 (2000e:03132)

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Additional Information:

Dominique Lecomte
Affiliation: Equipe d'Analyse Fonctionnelle, Université Paris 6, tour 46-0, boîte 186, 4, place Jussieu, 75 252 Paris Cedex 05, France
Address at time of publication: Université de Picardie, I.U.T. de l'Oise, site de Creil,13, allée de la faïencerie, 60 107 Creil, France
Email: dominique.lecomte@upmc.fr

DOI: 10.1090/S0002-9947-09-04792-8
PII: S 0002-9947(09)04792-8
Keywords: Borel chromatic number, dimension
Received by editor(s): July 23, 2007
Posted: February 10, 2009
Additional Notes: The author would like to thank the anonymous referee for the simplification of some proofs in this paper.
Copyright of article: Copyright 2009, American Mathematical Society

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