  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
$\Delta^0_3$ set has an infinite low${}_3$ 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.
