The reverse mathematics of the pigeonhole hierarchy

Quentin Le Houérou, Ludovic Levy Patey, Ahmed Mimouni

Published 2024-07-01Version 1

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.