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THE REVERSE MATHEMATICS OF THE THIN SET AND ERDŐS–MOSER THEOREMS

Part of: Computability and recursion theory Proof theory and constructive mathematics General logic

Published online by Cambridge University Press:  24 November 2021

LU LIU Open the ORCID record for LU LIU [Opens in a new window]  and
LUDOVIC PATEY Open the ORCID record for LUDOVIC PATEY [Opens in a new window]
LU LIU
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS HNP-LAMA, CENTRAL SOUTH UNIVERSITY CHANGSHA410083, PEOPLE’S REPUBLIC OF CHINA E-mail: g.jiayi.liu@gmail.com
LUDOVIC PATEY
Affiliation:
CNRS, INSTITUT CAMILLE JORDAN UNIVERSITÉ CLAUDE BERNARD LYON 1 43 BOULEVARD DU 11 NOVEMBRE 1918 F-69622VILLEURBANNE CEDEX, FRANCEE-mail:ludovic.patey@computability.fr
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Abstract

The thin set theorem for n-tuples and k colors ( $\operatorname {\mathrm {\sf {TS}}}^n_k$ ) states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$ , nor the free set theorem ( $\operatorname {\mathrm {\sf {FS}}}^n$ ) imply the Erdős–Moser theorem ( $\operatorname {\mathrm {\sf {EM}}}$ ) whenever k is sufficiently large (answering a question of Patey and giving a partial result towards a question of Cholak Giusto, Hirst and Jockusch). Given a problem $\mathsf {P}$ , a computable instance of $\mathsf {P}$ is universal iff its solution computes a solution of any other computable $\mathsf {P}$ -instance. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erdős–Moser theorem remained open so far. We prove that Erdős–Moser theorem does not admit a universal instance (answering a question of Patey).

Keywords

computability theoryreverse mathematicsErdős–Moser theoremthin set theoremfree set theorem

MSC classification

Primary: 03D80: Applications of computability and recursion theory
Secondary: 03F35: Second- and higher-order arithmetic and fragments 03B30: Foundations of classical theories (including reverse mathematics)
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 313 - 346
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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