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RAMSEY-LIKE THEOREMS FOR THE SCHREIER BARRIER

Part of: General logic Computability and recursion theory

Published online by Cambridge University Press:  17 July 2025

LORENZO CARLUCCI Open the ORCID record for LORENZO CARLUCCI [Opens in a new window] ,
ORIOLA GJETAJ ,
QUENTIN LE HOUÉROU Open the ORCID record for QUENTIN LE HOUÉROU [Opens in a new window]  and
LUDOVIC LEVY PATEY Open the ORCID record for LUDOVIC LEVY PATEY [Opens in a new window]
LORENZO CARLUCCI
Affiliation:
DEPARTMENT OF MATHEMATICS ‘GUIDO CASTELNUOVO’ https://ror.org/02be6w209 SAPIENZA UNIVERSITY ROME ITALY E-mail: lorenzo.carlucci@uniroma1.it
ORIOLA GJETAJ
Affiliation:
DEPARTMENT OF MATHEMATICS: ANALYSIS, LOGIC AND DISCRETE MATHEMATICS https://ror.org/00cv9y106 GHENT UNIVERSITY GHENT BELGIUM E-mail: oriola.gjetaj@ugent.be
QUENTIN LE HOUÉROU
Affiliation:
LABORATOIRE D’ALGORITHMIQUE COMPLEXITÉ ET LOGIQUE https://ror.org/05ggc9x40 UNIVERSITÉ PARIS-EST-CRÉTEIL-VAL-DE-MARNE CRÉTEIL FRANCE E-mail: quentin.le-houerou@computability.fr
LUDOVIC LEVY PATEY*
Affiliation:
CNRS, ÉQUIPE DE LOGIQUE https://ror.org/03fk87k11 INSTITUT DE MATHÉMATIQUES DE JUSSIEU-PARIS RIVE GAUCHE UNIVERSITÉ PARIS CITÉ - CAMPUS DES GRANDS MOULINS PARIS, FRANCE
*
E-mail: ludovic.patey@computability.fr
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Abstract

The family of finite subsets s of the natural numbers such that $|s|=1+\min s$ is known as the Schreier barrier in combinatorics and Banach Space theory, and as the family of exactly $\omega $-large sets in Logic. We formulate and prove the generalizations of Friedman’s Free Set and Thin Set theorems and of Rainbow Ramsey’s theorem to colorings of the Schreier barrier. We analyze the strength of these theorems from the point of view of Computability Theory and Reverse Mathematics. Surprisingly, the exactly $\omega $-large counterparts of the Thin Set and Free Set theorems can code $\emptyset ^{(\omega )}$, while the exactly $\omega $-large Rainbow Ramsey theorem does not code the halting set.

Keywords

Ramsey’s theoremSchreier barrierrainbow Ramsey theoremthin setfree set

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics) 03D80: Applications of computability and recursion theory

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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