The reverse mathematics of the Thin set and Erd\H{o}s-Moser theorems

TL;DR

This paper investigates the logical strength of the thin set and Erd ext{o}s-Moser theorems within reverse mathematics, showing that certain combinatorial principles do not imply each other and that the Erd ext{o}s-Moser theorem lacks a universal instance.

Contribution

It proves that the thin set theorem and the free set theorem do not imply the Erd ext{o}s-Moser theorem for large k, and establishes that Erd ext{o}s-Moser has no universal instance in computability theory.

Findings

Neither $ ext{TS}^n_k$ nor $ ext{FS}^n$ imply $ ext{EM}$ for large $k$

Erd ext{o}s-Moser theorem does not admit a universal instance

Answers open questions in reverse mathematics and computability theory

Abstract

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 (answering a question of Patey and giving a partial result towards a question of Cholak Giusto, Hirst and Jockusch). 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. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erd\H{o}s-Moser theorem remained open so far. We prove that Erd\H{o}s-Moser theorem does not admit a universal instance (answering a question of Patey).