Journal of Symbolic Logic

Effective Versions of Ramsey's Theorem: Avoiding the Cone Above $\mathbf{0}$'

Tamara Lakins Hummel

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Ramsey's Theorem states that if $P$ is a partition of $\lbrack\omega\rbrack^\kappa$ into finitely many partition classes, then there exists an infinite set of natural numbers which is homogeneous for $P$. We consider the degrees of unsolvability and arithmetical definability properties of infinite homogeneous sets for recursive partitions. We give Jockusch's proof of Seetapun's recent theorem that for all recursive partitions of $\lbrack\omega\rbrack^2$ into finitely many pieces, there exists an infinite homogeneous set $A$ such that $\emptyset' \nleq_T A$. Two technical extensions of this result are given, establishing arithmetical bounds for such a set $A$. Applications to reverse mathematics and introreducible sets are discussed.

Article information

J. Symbolic Logic, Volume 59, Issue 4 (1994), 1301-1325.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03D30: Other degrees and reducibilities
Secondary: 03D55: Hierarchies 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]


Hummel, Tamara Lakins. Effective Versions of Ramsey's Theorem: Avoiding the Cone Above $\mathbf{0}$'. J. Symbolic Logic 59 (1994), no. 4, 1301--1325.

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