The reverse mathematics of the pigeonhole hierarchy

Abstract

The infinite pigeonhole principle for $k$ colors ($\mathsf{RT}^1_k$) 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 the hierarchy of pigeonhole principles induced by restricting instances to levels of the arithmetic 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.