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PARTITION GENERICITY AND PIGEONHOLE BASIS THEOREMS

Part of: General logic Computability and recursion theory

Published online by Cambridge University Press:  03 October 2022

BENOIT MONIN  and
LUDOVIC PATEY Open the ORCID record for LUDOVIC PATEY [Opens in a new window]
BENOIT MONIN
Affiliation:
LACL UNIVERSITÉ DE CRÉTEIL CRÉTEIL, FRANCE E-mail: benoit.monin@computability.fr
LUDOVIC PATEY*
Affiliation:
CNRS, IMJ-PRG, PARIS, FRANCE
*
E-mail: ludovic.patey@computability.fr
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Abstract

There exist two main notions of typicality in computability theory, namely, Cohen genericity and randomness. In this article, we introduce a new notion of genericity, called partition genericity, which is at the intersection of these two notions of typicality, and show that many basis theorems apply to partition genericity. More precisely, we prove that every co-hyperimmune set and every Kurtz random is partition generic, and that every partition generic set admits weak infinite subsets, for various notions of weakness. In particular, we answer a question of Kjos-Hanssen and Liu by showing that every Kurtz random admits an infinite subset which does not compute any set of positive effective Hausdorff dimension. Partition genericity is a partition regular notion, so these results imply many existing pigeonhole basis theorems.

Keywords

partition regularitypartition genericitypigeonhole principle

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics) 03D80: Applications of computability and recursion theory
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 29
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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

Dorais, F. G., A variant of Mathias forcing that preserves ACA 0 . Archive for Mathematical Logic , vol. 51 (2012), nos. 7–8, pp. 751–780.CrossRefGoogle Scholar
Downey, R., Hirschfeldt, D. R., Lempp, S., and Solomon, R., A ${\varDelta}_2^0$ set with no infinite low subset in either it or its complement, this Journal, vol. 66 (2001), no. 3, pp. 1371–1381.Google Scholar
Dzhafarov, D. D. and Jockusch, C. G., Ramsey’s theorem and cone avoidance, this Journal, vol. 74 (2009), no. 2, pp. 557–578.Google Scholar
Greenberg, N. and Miller, J. S., Lowness for Kurtz randomness, this Journal, vol. 74 (2009), no. 2, pp. 665–678.Google Scholar
Hindman, N. and Strauss, D., Algebra in the Stone–Čech Compactification , De Gruyter Textbook, Walter de Gruyter, Berlin, 2012.Google Scholar
Hirschfeldt, D. R., Jockusch, C. G., Kjos-Hanssen, B., Lempp, S., and Slaman, T. A., The strength of some combinatorial principles related to Ramsey’s theorem for pairs , Computational Prospects of Infinity, Part II: Presented Talks , World Scientific, Singapore, 2008, pp. 143–161. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=MR2449463.CrossRefGoogle Scholar
Jockusch, C. G. and Soare, R. I.,  ${\varPi}_1^0$ classes and degrees of theories . Transactions of the American Mathematical Society , vol. 173 (1972), pp. 33–56.Google Scholar
Kjos-Hanssen, B., Infinite subsets of random sets of integers . Mathematics Research Letters , vol. 16 (2009), pp. 103–110.CrossRefGoogle Scholar
Kjos-Hanssen, B., A strong law of computationally weak subsets . Journal of Mathematical Logic , vol. 11 (2011), no. 1, pp. 1–10.CrossRefGoogle Scholar
Kjos-Hanssen, B. and Liu, L., Extracting randomness within a subset is hard . European Journal of Mathematics , vol. 6 (2020), no. 4, pp. 1438–1451.CrossRefGoogle Scholar
Kjos-Hanssen, B., Merkle, W., and Stephan, F., Kolmogorov complexity and the recursion theorem . Transactions of the American Mathematical Society , vol. 363 (2011), no. 10, pp. 5465–5480.CrossRefGoogle Scholar
Liu, L., RT ${}_2^2$ does not imply WKL 0, this Journal, vol. 77 (2012), no. 2, pp. 609–620.Google Scholar
Liu, L., Cone avoiding closed sets . Transactions of the American Mathematical Society , vol. 367 (2015), no. 3, pp. 1609–1630.CrossRefGoogle Scholar
Miller, J. S., Pi-0-1 Classes in Computable Analysis and Topology , Cornell University, 2002. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=2703773.Google Scholar
Monin, B. and Patey, L., The weakness of the pigeonhole principle under hyperarithmetical reductions . Journal of Mathematical Logic , vol. 21 (2021), no. 3, Article no. 2150013, 41 pp.CrossRefGoogle Scholar
Patey, L., Iterative forcing and hyperimmunity in reverse mathematics , Evolving Computability (Beckmann, A., Mitrana, V., and Soskova, M., editors), Lecture Notes in Computer Science, vol. 9136, Springer, 2015, pp. 291–301. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=3382369.CrossRefGoogle Scholar
Patey, L., Iterative forcing and hyperimmunity in reverse mathematics , Computability , vol. 6 (2017), no. 3, pp. 209–221.CrossRefGoogle Scholar
Rosenstein, J. G., Linear Orderings , Pure and Applied Mathematics, vol. 98, Academic Press, New York–London, 1982.Google Scholar
Stephan, F. and Yu, L., Lowness for weakly 1-generic and Kurtz-random , Theory and Applications of Models of Computation , Lecture Notes in Computer Science, vol. 3959, Springer, Berlin, 2006, pp. 756–764.CrossRefGoogle Scholar
Terwijn, S. A and Zambella, D., Algorithmic randomness and lowness, Mathematical Logic and Foundations (ML), 1997. Available at https://eprints.illc.uva.nl/id/eprint/1388/.Google Scholar
Wang, W., The definability strength of combinatorial principles, this Journal, vol. 81 (2016), no. 4, pp. 1531–1554.Google Scholar