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# The weakness of the pigeonhole principle under hyperarithmetical reductions

Abstract : The infinite pigeonhole principle for 2-partitions ($\mathsf{RT}^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\mathsf{RT}^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.
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https://hal.archives-ouvertes.fr/hal-02991292
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Submitted on : Thursday, November 5, 2020 - 10:48:44 PM
Last modification on : Monday, December 14, 2020 - 6:11:01 PM
Long-term archiving on: : Saturday, February 6, 2021 - 8:20:21 PM

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• HAL Id : hal-02991292, version 1
• ARXIV : 1905.08425

### Citation

Benoit Monin, Ludovic Patey. The weakness of the pigeonhole principle under hyperarithmetical reductions. Journal of Mathematical Logic, World Scientific Publishing, In press. ⟨hal-02991292⟩

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