The reverse mathematics of the pigeonhole hierarchy - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2024

The reverse mathematics of the pigeonhole hierarchy

Résumé

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 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.
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Dates et versions

hal-04633199 , version 1 (03-07-2024)

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Quentin Le Houérou, Ludovic Levy Patey, Ahmed Mimouni. The reverse mathematics of the pigeonhole hierarchy. 2024. ⟨hal-04633199⟩
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