Abstract
The infinite pigeonhole principle for 2-partitions asserts the existence, for every set A, of an infinite subset of A or of its complement. In this paper, we develop a new notion of forcing enabling a fine analysis of the computability-theoretic features of the pigeonhole principle. We deduce various consequences, such as the existence, for every set A, of an infinite subset of it or its complement of non-high degree. We also prove that every Δ03 set has an infinite low3 solution and give a simpler proof of Liu's theorem that every set has an infinite subset in it or its complement of non-PA degree.
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