Generalized R-Cohesiveness and The Arithmetical Hierarchy: A Correction to "Generalized Cohesiveness"

ForX ⊆ ω, let [X] denote the class of all n-element subsets of X. An infinite set A ⊆ ω is called n-r-cohesive if for each computable function f : [ω] → {0, 1} there is a finite set F such that f is constant on [A−F ]. We show that for each n ≥ 2 there is no Πn set A ⊆ ω which is n-r-cohesive. For n = 2 this refutes a result previously claimed by the authors, and for n ≥ 3 it answers a question raised by the authors.