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A Dichotomy Characterizing Analytic Digraphs of Uncountable Borel Chromatic Number in Any Dimension

Dominique Lecomte
Transactions of the American Mathematical Society
Vol. 361, No. 8 (Aug., 2009), pp. 4181-4193
Stable URL: http://www.jstor.org/stable/40302694
Page Count: 13
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Dichotomy Characterizing Analytic Digraphs of Uncountable Borel Chromatic Number in Any Dimension
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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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