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An antibasis result for graphs of infinite Borel chromatic number


Authors: Clinton T. Conley and Benjamin D. Miller
Journal: Proc. Amer. Math. Soc. 142 (2014), 2123-2133
MSC (2010): Primary 03E15; Secondary 28A05, 37A20
DOI: https://doi.org/10.1090/S0002-9939-2014-11918-6
Published electronically: March 5, 2014
MathSciNet review: 3182030
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Abstract: We answer in the negative a question posed by Kechris-Solecki-Todorcevic as to whether the shift graph on Baire space is minimal among graphs of indecomposably infinite Borel chromatic number. To do so, we use ergodic-theoretic techniques to construct a new graph amalgamating various properties of the shift actions of free groups. The resulting graph is incomparable with any graph induced by a function. We then generalize this construction and collect some of its useful properties.


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Additional Information

Clinton T. Conley
Affiliation: Department of Mathematics, 584 Malott Hall, Cornell University, Ithaca, New York 14853
Email: clintonc@math.cornell.edu

Benjamin D. Miller
Affiliation: Institut für mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
Email: bdm@uni-muenster.de

DOI: https://doi.org/10.1090/S0002-9939-2014-11918-6
Received by editor(s): September 4, 2011
Received by editor(s) in revised form: June 26, 2012, and June 28, 2012
Published electronically: March 5, 2014
Additional Notes: The authors were supported in part by SFB Grant 878.
Communicated by: Julia Knight
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.