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

Abstract

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.