The infinite pigeonhole principle for 2-partitions (RT21) 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 RT21 admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every Δn0 set, of an infinite lown 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 et al.

中文翻译：

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超算术归约下鸽巢原理的弱点

2 分区的无限鸽巢原理 (R吨21) 断言存在，对于每个集合一种, 的无限子集一种或其补语。在本文中，我们从可计算性理论的角度研究了无限鸽笼原理。我们特别证明R吨21承认算术和超算术减少的强锥避免。我们也证明存在，对于每一个Δn0集合，无限低n它的子集或其补集。这回答了王的一个问题。为此，我们设计了一个新的强制概念，它概括了 Cholak 的第一次和第二次跳跃控制等。