On notions of computability-theoretic reduction between Π21 principles

Journal of Mathematical Logic 16 (1):1650002 (2016)
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Abstract

Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and RT[Formula: see text] under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey-theoretic properties, we show that for each [Formula: see text], there is an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] such that every [Formula: see text]-coloring of [Formula: see text] has an infinite homogeneous set that does not compute any infinite homogeneous set for [Formula: see text], and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to [Formula: see text]-models of RCA0, and related notions that allow us to count how many applications of a principle [Formula: see text] are needed to reduce another principle to [Formula: see text]. Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman.

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Citations of this work

Ramsey-like theorems and moduli of computation.Ludovic Patey - 2022 - Journal of Symbolic Logic 87 (1):72-108.
Using Ramsey’s theorem once.Jeffry L. Hirst & Carl Mummert - 2019 - Archive for Mathematical Logic 58 (7-8):857-866.
Regressive versions of Hindman’s theorem.Lorenzo Carlucci & Leonardo Mainardi - 2024 - Archive for Mathematical Logic 63 (3):447-472.

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References found in this work

On the Strength of Ramsey's Theorem.David Seetapun & Theodore A. Slaman - 1995 - Notre Dame Journal of Formal Logic 36 (4):570-582.
Open questions in reverse mathematics.Antonio Montalbán - 2011 - Bulletin of Symbolic Logic 17 (3):431-454.
Ramsey's theorem and recursion theory.Carl G. Jockusch - 1972 - Journal of Symbolic Logic 37 (2):268-280.

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