The weakness of the Erdős-Moser theorem under arithmetic reductions

Ludovic Levy Patey; Ahmed Mimouni

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arXiv · 2310.17968

The weakness of the Erdős-Moser theorem under arithmetic reductions

Ludovic Levy Patey·Ahmed Mimouni

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Abstract

The Erdős-Moser theorem $(\mathsf{EM})$ says that every infinite tournament admits an infinite transitive subtournament. We study the computational behavior of the Erdős-Moser theorem with respect to the arithmetic hierarchy, and prove that $Δ^0_n$ instances of $\mathsf{EM}$ admit low${}_{n+1}$ solutions for every $n \geq 1$, and that if a set $B$ is not arithmetical, then every instance of $\mathsf{EM}$ admits a solution relative to which $B$ is still not arithmetical. We also provide a level-wise refinement of this theorem. These results are part of a larger program of computational study of combinatorial theorems in Reverse Mathematics.

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BibTeX RIS

Ludovic Levy Patey, Ahmed Mimouni. 2023-10-27. The weakness of the Erdős-Moser theorem under arithmetic reductions. https://arxiv.org/abs/2310.17968

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