The weakness of the Erdős–Moser theorem under arithmetic reductions

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The Erdős–Moser ([Formula: see text]) theorem says that every infinite tournament admits an infinite transitive subtournament. We study the computational behavior of the [Formula: see text] theorem with respect to the arithmetic hierarchy, and prove that [Formula: see text] instances of [Formula: see text] admit low[Formula: see text] solutions for every [Formula: see text], and that if a set [Formula: see text] is not arithmetical, then every instance of [Formula: see text] admits a solution relative to which [Formula: see text] 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.