Erdős–Moser and IΣ2

Abstract

The first-order part of Ramsey’s theorem for pairs with an arbitrary number of colors is known to be precisely BΣ ⁰₃ . We compare this to the known division of Ramsey’s theorem for pairs into the weaker principles, EM (the Erdős–Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond IΣ ⁰₂ is entirely due to the arbitrary color analog of ADS.

Specifically, we show that ADS for an arbitrary number of colors implies BΣ ⁰₃ while EM for an arbitrary number of colors is Π ¹₁ -conservative over IΣ ⁰₂ and it does not imply IΣ ⁰₂ .