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MEASURABLE PERFECT MATCHINGS FOR ACYCLIC LOCALLY COUNTABLE BOREL GRAPHS

Published online by Cambridge University Press:  21 March 2017

CLINTON T. CONLEY  and
BENJAMIN D. MILLER
CLINTON T. CONLEY
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USA E-mail: clintonc@andrew.cmu.edu URL: http://www.math.cmu.edu/math/faculty/conley
BENJAMIN D. MILLER
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITÄT WIEN WÄHRINGER STRASSE 25 1090 WIEN, AUSTRIA E-mail: benjamin.miller@univie.ac.at URL: http://www.logic.univie.ac.at/benjamin.miller
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Abstract

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating μ-hyperfinite equivalence relations admit μ-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a counterexample in the locally countable case.

Keywords

Primary 03E1528A05Secondary 05C15Marriagematchingmeasurable

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 258 - 271
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

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