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FORCING CONSTRUCTIONS AND COUNTABLE BOREL EQUIVALENCE RELATIONS

Part of: Topological dynamics Connections with other structures, applications Set theory

Published online by Cambridge University Press:  14 March 2022

SU GAO ,
STEVE JACKSON ,
EDWARD KROHNE  and
BRANDON SEWARD
SU GAO
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITYTIANJIN300071P.R. CHINAE-mail:[email protected]
STEVE JACKSON
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS 1155 UNION CIRCLE #311430 DENTON, TX76203, USAE-mail:[email protected]:[email protected]
EDWARD KROHNE
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS 1155 UNION CIRCLE #311430 DENTON, TX76203, USAE-mail:[email protected]:[email protected]
BRANDON SEWARD
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA SAN DIEGO 9500 GILMAN DRIVE #0112 LA JOLLA, CA92903-0112, USAE-mail:[email protected]
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Abstract

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.

Keywords

Borel equivalence relationcomplete sectionorbit forcing

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 03E40: Other aspects of forcing and Boolean-valued models 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 37B20: Notions of recurrence
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 873 - 893
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Conley, C. and Marks, A., Distance from marker sequences in locally finite Borel graphs, Trends in Set Theory, Contemporary Mathematics 752, American Mathematical Society, Providence, 2020, pp. 89–92.CrossRefGoogle Scholar
Conley, C. and Miller, B., A bound on measurable chromatic numbers of locally finite Borel graphs. Mathematical Research Letters, vol. 23 (2016), no. 6, pp. 1633–1644.CrossRefGoogle Scholar
Gao, S., Invariant Descriptive Set Theory, Pure and Applied Mathematics, A Series of Monographs and Textbooks, vol. 293, Taylor & Francis Group, Boca Raton, 2009.Google Scholar
Gao, S. and Jackson, S., Countable abelian group actions and hyperfinite equivalence relations. Inventiones Mathematicae, vol. 201 (2015), no. 1, pp. 309–383.CrossRefGoogle Scholar
Gao, S., Jackson, S., Krohne, E., and Seward, B., Continuous combinatorics of abelian group actions, preprint, 2018, arXiv:1803.03872.Google Scholar
Gao, S., Jackson, S., Krohne, E., and Seward, B., Borel combinatorics of abelian group actions, unpublished manuscript, 2019.Google Scholar
Gao, S., Jackson, S., and Seward, B., A coloring property for countable groups. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 147 (2009), no. 3, pp. 579–592.CrossRefGoogle Scholar
Gao, S., Jackson, S., and Seward, B., Group colorings and Bernoulli subflows. Memoirs of the American Mathematical Society, vol. 241 (2016), no. 1141.CrossRefGoogle Scholar
Kechris, A. and Miller, B., Topics in Orbit Equivalence, Lecture Notes in Mathematics, vol. 1852, Springer-Verlag, Berlin, 2004.CrossRefGoogle Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
Marks, A., A determinacy approach to Borel combinatorics. Journal of the American Mathematical Society, vol. 29 (2016), no. 2, pp. 579–600.CrossRefGoogle Scholar
Marks, A., Uniformity, universality, and recursion theory, Journal of Mathematical Logic, vol. 17 (2017), no. 1, Article no. 1750003, 50 pp.CrossRefGoogle Scholar
Marks, A., Structure in complete sections of the shift action of a residually finite group. www.math.ucla.edu/~marks/.Google Scholar
Schneider, S. and Seward, B., Locally nilpotent groups and hyperfinite equivalence relations, preprint, 2013, arXiv:1308.5853.Google Scholar
Seward, B. and Tucker-Drob, R., Borel structurability on the 2-shift of a countable group. Annals of Pure and Applied Logic, vol. 167 (2016), no. 1, pp. 1–21.CrossRefGoogle Scholar
Slaman, T. and Steel, J., Definable functions on degrees , Cabal Seminar 81–85, Lecture Notes in Mathematics, vol. 1333, Springer-Verlag, Berlin, 1988, pp. 37–55.CrossRefGoogle Scholar
Thomas, S., Martin’s conjecture and strong ergodicity, Archive for Mathematical Logic, vol. 48 (2009), pp. 749–759.CrossRefGoogle Scholar