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August 2021 Effectiveness for the Dual Ramsey Theorem
Damir Dzhafarov, Stephen Flood, Reed Solomon, Linda Westrick
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Notre Dame J. Formal Logic 62(3): 455-490 (August 2021). DOI: 10.1215/00294527-2021-0024

Abstract

We analyze the dual Ramsey theorem for k partitions and colors (DRTk) in the context of reverse math, effective analysis, and strong reductions. Over RCA0, the dual Ramsey theorem stated for Baire colorings Baire-DRTk is equivalent to the statement for clopen colorings ODRTk and to a purely combinatorial theorem CDRTk. When the theorem is stated for Borel colorings and k3, the resulting principles are essentially relativizations of CDRTk. For each α, there is a computable Borel code for a Δα0-coloring such that any partition homogeneous for it computes (α) or (α1) depending on whether α is infinite or finite. For k=2, we present partial results giving bounds on the effective content of the principle. A weaker version for Δn0-reduced colorings is equivalent to D2n over RCA0+IΣn10 and in the sense of strong Weihrauch reductions.

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Damir Dzhafarov. Stephen Flood. Reed Solomon. Linda Westrick. "Effectiveness for the Dual Ramsey Theorem." Notre Dame J. Formal Logic 62 (3) 455 - 490, August 2021. https://doi.org/10.1215/00294527-2021-0024

Information

Received: 31 August 2018; Accepted: 25 February 2021; Published: August 2021
First available in Project Euclid: 6 October 2021

Digital Object Identifier: 10.1215/00294527-2021-0024

Subjects:
Primary: 03D80
Secondary: 03B30, 03F35, 03F60, 05C55

Rights: Copyright © 2021 University of Notre Dame

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Vol.62 • No. 3 • August 2021
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