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THE OPEN AND CLOPEN RAMSEY THEOREMS IN THE WEIHRAUCH LATTICE

Part of: General logic Graph theory Computability and recursion theory

Published online by Cambridge University Press:  01 February 2021

ALBERTO MARCONE  and
MANLIO VALENTI
ALBERTO MARCONE
Affiliation:
DEPARTMENT OF MATHEMATICS, COMPUTER SCIENCE AND PHYSICS UNIVERSITY OF UDINEUDINE, 33100, ITALYE-mail: alberto.marcone@uniud.itE-mail: manliovalenti@gmail.com
MANLIO VALENTI
Affiliation:
DEPARTMENT OF MATHEMATICS, COMPUTER SCIENCE AND PHYSICS UNIVERSITY OF UDINEUDINE, 33100, ITALYE-mail: alberto.marcone@uniud.itE-mail: manliovalenti@gmail.com
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Abstract

We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm {ATR_0}$ from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight different multivalued functions (five corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be strictly stronger than any previously studied multivalued functions arising from statements around $\mathrm {ATR}_0$ .

Keywords

computable analysisWeihrauch degreeshomogeneous setsopen and clopen Ramsey theorem

MSC classification

Primary: 03D78: Computation over the reals
Secondary: 03D30: Other degrees and reducibilities 03B30: Foundations of classical theories (including reverse mathematics) 05C55: Generalized Ramsey theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021 , pp. 316 - 351
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Copyright
© The Association for Symbolic Logic 2021

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