Dominating the Erdos-Moser theorem in reverse mathematics

TL;DR

This paper investigates the computational strength of the Erdos-Moser theorem within reverse mathematics, constructing models to demonstrate its limitations and separation from other principles like ADS and the atomic model theorem.

Contribution

It constructs a standard model of EM, weak K"onig's lemma, and cohesiveness that does not satisfy the atomic model theorem, clarifying EM's computational weakness.

Findings

EM is weaker than previously thought in reverse mathematics.

A model satisfies EM, WKL, and COH but not the atomic model theorem.

This separation answers an open question in the field.

Abstract

The Erdos-Moser theorem (EM) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs (RT^2_2) by providing an alternate proof of RT^2_2 in terms of EM and the ascending descending sequence principle (ADS). In this paper, we study the computational weakness of EM and construct a standard model (omega-model) of simultaneously EM, weak K\"onig's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdos-Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner.