\(\mathsf{RT}_{2}^{2}\) does not imply \(\mathsf{WKL}_{0}\)
In this article, the author settles a long-standing question in reverse mathematics by showing that \(\text{RCA}_0\) plus \(\text{RT}^2_\infty\) (Ramsey's theorem for pairs for an arbitrary number of colors) does not prove \( \text{WKL}_0\) (weak König's lemma). He does this by constructing an \(\omega\)-model of \(\text{RCA}_0 + \text{RT}^2_2\) in which the set universe avoids all cones above PA degrees. The construction relies on the following central computability-theoretic result: If \(C\) is not of PA degree and \(A\) is any set then there is an infinite set \(G\) such that \(G \oplus C\) is not of PA degree and \(G\) is either a subset of \(A\) or disjoint from \(A\). The proof of this theorem uses a Mathias forcing argument based on techniques previously used by \textit{D. Seetapun} and \textit{T. A. Slaman} [``On the strength of Ramsey's theorem'', Notre Dame J. Formal Logic 36, No. 4, 570--582 (1995; Zbl 0843.03034)], \textit{P. A. Cholak}, \textit{C. G. Jockusch} and \textit{T. A. Slaman} [``On the strength of Ramsey's theorem for pairs'', J. Symb. Log. 66, No. 1, 1--55 (2001; Zbl 0977.03033)] and \textit{D. D. Dzhafarov} and \textit{C. G. Jockusch} [``Ramsey's theorem and cone avoidance'', J. Symb. Log. 74, No. 2, 557--578 (2009; Zbl 1166.03021)]. The article assumes familiarity with this previous work.
scientific article; zbMATH DE number 6047779