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Generalized r-cohesiveness and the arithmetical hierarchy: A correction to “Generalized cohesiveness”. (English) Zbl 1011.03033
The paper under review is connected with the authors’ paper “Generalized cohesiveness” [J. Symb. Log. 64, 489-516 (1999; Zbl 0935.03050)]. The main result of the present paper for $$n = 2$$ refutes a result previously claimed by the authors, and for $$n \geq 3$$ answers a question raised in the mentioned paper. To explain the result, let, for $$X \subseteq \omega$$, $$[X^n]$$ denote the class of all subsets of $$X$$. An infinite set $$A \subseteq \omega$$ is called $$n$$-r-cohesive iff for each computable function $$f: [\omega]^{n} \rightarrow \{ 0,1 \}$$ there is a finite set $$F$$ such that $$f$$ is constant on $$[A- f^n]$$. It is shown that for each $$n \geq 2$$ there is no $$\Pi_{n}^{0}$$ set $$A \subseteq \omega$$ which is $$n$$-r-cohesive.
##### MSC:
 03D55 Hierarchies of computability and definability 03D28 Other Turing degree structures
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##### References:
 [1] Proceedings of the London Mathematical Society 30 pp 264– (1930) [2] Ramsey’s theorem and recursion theory 37 pp 268– (1972) [3] Recursively enumerable sets and degrees (1987) [4] Generalized cohesiveness 64 pp 489– (1999) · Zbl 0935.03050 [5] Ramsey’s theorem for computably enumerable colorings 66 pp 873– (2001)
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