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Article Dans Une Revue The Journal of Symbolic Logic Année : 2022

The reverse mathematics of the Thin set and the Erdos-Moser theorems

Résumé

The thin set theorem for $n$-tuples and $k$ colors ($\mathsf{TS}^n_k$) states that every $k$-coloring of $[\mathbb{N}]^n$ admits an infinite set of integers $H$ such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\mathsf{TS}^n_k$, nor the free set theorem ($\mathsf{FS}^n$) imply the Erd\H{o}s-Moser theorem ($\mathsf{EM}$) whenever $k$ is sufficiently large. Given a problem $\mathsf{P}$, a computable instance of $\mathsf{P}$ is universal iff its solution computes a solution of any other computable $\mathsf{P}$-instance. We prove that Erd\H{o}s-Moser theorem does not admit a universal instance.
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Dates et versions

hal-03362591 , version 1 (01-10-2021)

Identifiants

Citer

Lu Liu, Ludovic Patey. The reverse mathematics of the Thin set and the Erdos-Moser theorems. The Journal of Symbolic Logic, 2022, 87 (1), pp.313--346. ⟨10.1017/jsl.2021.98⟩. ⟨hal-03362591⟩
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