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On Baire measurable colorings of group actions

Part of: Set theory Topological dynamics Graph theory

Published online by Cambridge University Press:  10 January 2020

ANTON BERNSHTEYN Open the ORCID record for ANTON BERNSHTEYN [Opens in a new window]
ANTON BERNSHTEYN*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, IL, USA Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA email abernsht@math.cmu.edu
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Abstract

The field of descriptive combinatorics investigates to what extent classical combinatorial results and techniques can be made topologically or measure-theoretically well behaved. This paper examines a class of coloring problems induced by actions of countable groups on Polish spaces, with the requirement that the desired coloring be Baire measurable. We show that the set of all such coloring problems that admit a Baire measurable solution for a particular free action $\unicode[STIX]{x1D6FC}$ is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by $\unicode[STIX]{x1D6FC}$ is smooth on a comeager set); this result confirms the ‘hardness’ of finding a topologically well-behaved coloring. When $\unicode[STIX]{x1D6FC}$ is the shift action, we characterize the class of problems for which $\unicode[STIX]{x1D6FC}$ has a Baire measurable coloring in purely combinatorial terms; it turns out that closely related concepts have already been studied in graph theory with no relation to descriptive set theory. We remark that our framework permits a wholly dynamical interpretation (with colorings corresponding to equivariant maps to a given subshift), so this article can also be viewed as a contribution to generic dynamics.

Keywords

group actionssymbolic dynamicstopological dynamics

MSC classification

Primary: 03E15: Descriptive set theory 05C15: Coloring of graphs and hypergraphs 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 3 , March 2021 , pp. 818 - 845
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Copyright
© Cambridge University Press, 2020

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