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THE OPEN AND CLOPEN RAMSEY THEOREMS IN THE WEIHRAUCH LATTICE

Published online by Cambridge University Press:  01 February 2021

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

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 $\mathrm {ATR}_0$ .

Type
Article
Copyright
© The Association for Symbolic Logic 2021

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