Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

Cone avoiding closed sets


Author: Lu Liu
Journal: Trans. Amer. Math. Soc. 367 (2015), 1609-1630
MSC (2010): Primary 03B30; Secondary 03F35, 03C62, 68Q30, 03D32, 03D80, 28A78
DOI: https://doi.org/10.1090/S0002-9947-2014-06049-2
Published electronically: November 4, 2014
MathSciNet review: 3286494
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for an arbitrary subtree $ T$ of $ 2^{<\omega }$ with each element extendable to a path, a given countable class $ \mathcal {M}$ closed under disjoint union, and any set $ A$, if none of the members of $ \mathcal {M}$ strongly $ k$-enumerate $ T$ for any $ k$, then there exists an infinite set contained in either $ A$ or $ \bar {A}$ such that for every $ C\in \mathcal {M}$, $ C\oplus G$ also does not strongly $ k$-enumerate $ T$. We give applications of this result, which include: (1) $ \mathsf {RT_2^2}$ doesn't imply $ \mathsf {WWKL_0}$; (2) [Ambos-Spies et al., 2004] $ \mathsf {DNR}$ is strictly weaker than $ \mathsf {WWKL_0}$; (3) [Kjos-Hanssen, 2009] for any Martin-Löf random set $ A$, either $ A$ or $ \bar {A}$ contains an infinite subset that does not compute any Martin-Löf random set; etc. We also discuss further generalizations of this result.


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


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 03B30, 03F35, 03C62, 68Q30, 03D32, 03D80, 28A78

Retrieve articles in all journals with MSC (2010): 03B30, 03F35, 03C62, 68Q30, 03D32, 03D80, 28A78


Additional Information

Lu Liu
Affiliation: Department of Mathematics, Central South University, ChangSha 410083, People’s Republic of China
Email: g.jiayi.liu@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-06049-2
Keywords: Ramsey's theorem, weak weak K\"onig's lemma, Martin-L\"of randomness, randomness extraction
Received by editor(s): November 20, 2010
Received by editor(s) in revised form: December 12, 2012
Published electronically: November 4, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society