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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Infinite dimensional perfect set theorems


Author: Tamás Mátrai
Journal: Trans. Amer. Math. Soc. 365 (2013), 23-58
MSC (2010): Primary 03E15
Published electronically: June 8, 2012
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Abstract: What largeness and structural assumptions on $ A \subseteq [\mathbb{R}]^{\omega }$ can guarantee the existence of a non-empty perfect set $ P \subseteq \mathbb{R}$ such that $ [P]^{\omega } \subseteq A$? Such a set $ P$ is called $ A$-homogeneous. We show that even if $ A$ is open, in general it is independent of ZFC whether for a cardinal $ \kappa $, the existence of an $ A$-homogeneous set $ H \in [\mathbb{R}]^{\kappa }$ implies the existence of a non-empty perfect $ A$-homogeneous set.

On the other hand, we prove an infinite dimensional analogue of Mycielski's Theorem: if $ A$ is large in the sense of a suitable Baire category-like notion, then there exists a non-empty perfect $ A$-homogeneous set. We introduce fusion games to prove this and other infinite dimensional perfect set theorems.

Finally we apply this theory to show that it is independent of ZFC whether Tukey reductions of the maximal analytic cofinal type can be witnessed by definable Tukey maps.


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Additional Information

Tamás Mátrai
Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada
Address at time of publication: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 13-15 Reáltanoda Street, Budapest H-1053 Hungary
Email: matrait@renyi.hu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05468-7
PII: S 0002-9947(2012)05468-7
Received by editor(s): October 8, 2009
Received by editor(s) in revised form: September 20, 2010
Published electronically: June 8, 2012
Additional Notes: This research was partially supported by the OTKA grants K 61600, K 49786 and K 72655 and by the NSERC grants 129977 and A-7354.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.