Erdős–Moser and IΣ2

The first-order part of Ramsey’s theorem for pairs with an arbitrary number of colors is known to be precisely B Σ 3 0 . 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 I Σ 2 0 is entirely due to the arbitrary color analog of ADS. Specifically, we show that ADS for an arbitrary number of colors implies B Σ 3 0 while EM for an arbitrary number of colors is Π 1 1 -conservative over I Σ 2 0 and it does not imply I Σ 2 0 .