  The infinite pigeonhole principle for $k$ colors ($\mathsf{RT}_k$) states,
for every $k$-partition $A_0 \sqcup \dots \sqcup A_{k-1} = \mathbb{N}$, the
existence of an infinite subset~$H \subseteq A_i$ for some~$i < k$. This
seemingly trivial combinatorial principle constitutes the basis of Ramsey's
theory, and plays a very important role in computability and proof theory. In
this article, we study the infinite pigeonhole principle at various levels of
the arithmetical hierarchy from both a computability-theoretic and reverse
mathematical viewpoint. We prove that this hierarchy is strict
over~$\mathsf{RCA}_0$ using an elaborate iterated jump control construction,
and study its first-order consequences. This is part of a large
meta-mathematical program studying the computational content of combinatorial
theorems.
