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Minimal definable graphs of definable chromatic number at least three

Part of: Set theory Classical measure theory

Published online by Cambridge University Press:  28 January 2021

Raphaël Carroy ,
Benjamin D. Miller ,
David Schrittesser  and
Zoltán Vidnyánszky
Raphaël Carroy
Affiliation:
Raphaël Carroy, Dipartimento di Matematica “Giuseppe Peano”, Università degli studi di Torino Palazzo Campana, Via Carlo Alberto 10, 10123Torino, Italy, E-mail: raphael.carroy@univie.ac.at; http://www.logique.jussieu.fr/~carroy/indexeng.html
Benjamin D. Miller
Affiliation:
University of Vienna, Department of Mathematics, Oskar Morgenstern Platz 1, Wien1090, Austria, E-mail: benjamin.miller@univie.ac.at; https://homepage.univie.ac.at/benjamin.miller/
David Schrittesser
Affiliation:
University of Vienna, Department of Mathematics, Oskar Morgenstern Platz 1, Wien1090, Austria, E-mail: david.schrittesser@univie.ac.at; http://homepage.univie.ac.at/david.schrittesser/
Zoltán Vidnyánszky
Affiliation:
University of Vienna, Department of Mathematics, Oskar Morgenstern Platz 1, Wien1090, Austria; and California Institute of Technology, Department of Mathematics, Pasadena, CA91125, E-mail: zoltan.vidnyanszky@univie.ac.at; http://www.logic.univie.ac.at/~vidnyanszz77/

Abstract

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We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property and show that the analogous result for digraphs fails.

Keywords

Borel bipartiteBorel coloring

MSC classification

Primary: 03E15: Descriptive set theory 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Type
Foundations
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e7
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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