On the Weihrauch Degree of the Additive Ramsey Theorem over the Rationals

We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-like theorems concerning colourings of the rationals. The theorems we are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of rationals satisfying propertie...

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Bibliographic Details
Published inRevolutions and Revelations in Computability Vol. 13359; pp. 259 - 271
Main Authors Pradic, Cécilia, Soldà, Giovanni
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2022
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Additive ramsey
induction
Reverse mathematics
Weihrauch reducibility
Online AccessGet full text
ISBN3031087399
9783031087394
ISSN0302-9743
1611-3349
DOI10.1007/978-3-031-08740-0_22

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Summary:We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-like theorems concerning colourings of the rationals. The theorems we are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of rationals satisfying properties determined by some additional algebraic structure on the set of colours. In the context of reverse mathematics, most of the principles we study are equivalent to Σ20 $$\varSigma ^0_2$$ -induction over RCA0 $$\textsf{RCA}_0$$ . The associated problems in the Weihrauch lattice are related to TCN∗ $$\textsf{TC}_{\mathbb {N}}^*$$ , (LPO′)∗ $$(\textsf{LPO}')^*$$ or their product, depending on their precise formalizations.
Bibliography:Original Abstract: We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-like theorems concerning colourings of the rationals. The theorems we are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of rationals satisfying properties determined by some additional algebraic structure on the set of colours. In the context of reverse mathematics, most of the principles we study are equivalent to Σ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varSigma ^0_2$$\end{document}-induction over RCA0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{RCA}_0$$\end{document}. The associated problems in the Weihrauch lattice are related to TCN∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{TC}_{\mathbb {N}}^*$$\end{document}, (LPO′)∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\textsf{LPO}')^*$$\end{document} or their product, depending on their precise formalizations.
The second author was supported by an LMS Early Career Fellowship.
ISBN:3031087399
9783031087394
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-031-08740-0_22