Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: March 5, 2014
MathSciNet review: 3182030
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Lewis Bowen, Weak isomorphisms between Bernoulli shifts, Israel J. Math. 183 (2011), 93-102. MR 2811154 (2012k:37008),
  • [2] Clinton Conley and Alexander Kechris, Measurable chromatic and independence numbers for ergodic graphs and group actions, Groups Geom. Dyn. 7 (2013), no. 1, 127-180. MR 3019078,
  • [3] Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289-324. MR 0578656 (58 #28261a)
  • [4] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • [5] Alexander S. Kechris, Global aspects of ergodic group actions, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010. MR 2583950 (2011b:37003)
  • [6] Alexander S. Kechris and Benjamin D. Miller, Topics in orbit equivalence, Lecture Notes in Mathematics, vol. 1852, Springer-Verlag, Berlin, 2004. MR 2095154 (2005f:37010)
  • [7] A. S. Kechris, S. Solecki, and S. Todorcevic, Borel chromatic numbers, Adv. Math. 141 (1999), no. 1, 1-44. MR 1667145 (2000e:03132),
  • [8] Russell Lyons and Fedor Nazarov, Perfect matchings as IID factors on non-amenable groups, European J. Combin. 32 (2011), no. 7, 1115-1125. MR 2825538 (2012m:05423),
  • [9] Benjamin D. Miller, Measurable chromatic numbers, J. Symbolic Logic 73 (2008), no. 4, 1139-1157. MR 2467208 (2009k:03079),
  • [10] Saharon Shelah, Can you take Solovay's inaccessible away?, Israel J. Math. 48 (1984), no. 1, 1-47. MR 768264 (86g:03082a),
  • [11] Asger Törnquist, Orbit equivalence and actions of $ \mathbb{F}_n$, J. Symbolic Logic 71 (2006), no. 1, 265-282. MR 2210067 (2007a:37005),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E15, 28A05, 37A20

Retrieve articles in all journals with MSC (2010): 03E15, 28A05, 37A20

Additional Information

Clinton T. Conley
Affiliation: Department of Mathematics, 584 Malott Hall, Cornell University, Ithaca, New York 14853

Benjamin D. Miller
Affiliation: Institut für mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany

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.

American Mathematical Society