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, Jockusch and Slaman.Comment: 29 page