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Pré-Publication, Document De Travail Année : 2023

The weakness of the Erd\H{o}s-Moser theorem under arithmetic reductions

Résumé

The Erd\H{o}s-Moser theorem $(\mathsf{EM})$ says that every infinite tournament admits an infinite transitive subtournament. We study the computational behavior of the Erd\H{o}s-Moser theorem with respect to the arithmetic hierarchy, and prove that $\Delta^0_n$ 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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Dates et versions

hal-04265681 , version 1 (31-10-2023)

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Ludovic Levy Patey, Ahmed Mimouni. The weakness of the Erd\H{o}s-Moser theorem under arithmetic reductions. 2023. ⟨hal-04265681⟩
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