Erdos-Moser and ISigma_2

TL;DR

This paper investigates the logical strength of principles related to Ramsey's Theorem for pairs with multiple colors, showing that the Erdős-Moser principle is weaker than the ascending-descending sequence principle in this context.

Contribution

It establishes that the strength of the ascending-descending sequence principle for multiple colors exceeds that of the Erdős-Moser principle, clarifying their roles in reverse mathematics.

Findings

ADS for multiple colors implies BSigma03.

EM for multiple colors is Pi11-conservative over ISigma02.

EM does not imply ISigma02.

Abstract

The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the Erd\H{o}s-Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond ISigma02 is entirely due to the arbitrary color analog of ADS. Specifically, we show that ADS for an arbitrary number of colors implies BSigma03 while EM for an arbitrary number of colors is Pi11-conservative over ISigma02 and it does not imply ISigma02.