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Pigeons do not jump high. (English) Zbl 1441.03013
The authors prove new computability theoretic results related to the pigeonhole principle for two colors. They show that for every set $$A$$ there is an infinite non-high set $$H$$ with $$H \subseteq A$$ or $$H \subseteq \bar A$$. Also, they show that for every $$\Delta^0_3$$ set $$A$$ there is an infinite $$\text{low}_3$$ set $$H$$ with $$H \subseteq A$$ or $$H \subseteq \bar A$$. The proofs use a new notion of forcing particularly adapted to the fine analysis of computability theoretic aspects of the pigeonhole principle. This forcing is used to succinctly prove that for every set $$A$$ there is an infinite non-PA set $$H$$ with $$H \subseteq A$$ or $$H \subseteq \bar A$$, the main combinatorial lemma of J. Liu [J. Symb. Log. 77, No. 2, 609–620 (2012; Zbl 1245.03095)]. The paper explains the connections between computability theoretic results and a number of open questions in reverse mathematics. Subsequently, the authors announced an answer to Question 1.1 in [“$$\mathsf{SRT}^2_2$$ does not imply $$\mathsf{RT}^2_2$$ in $$\omega$$-models”, Preprint, arxiv:1905.08427].
##### MSC:
 03B30 Foundations of classical theories (including reverse mathematics) 03F35 Second- and higher-order arithmetic and fragments 03D80 Applications of computability and recursion theory
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