Errata for “on the Strength of Ramsey’s Theorem for Pairs”

Several proofs given in [2] contain significant errors or gaps, although to our knowledge all results claimed there are provable. The needed corrections are described below. All references are to [2] unless otherwise stated, and we adopt the notation and terminology of that paper. 1. Lemma 7.10 asserts that the principles D2 and SRT 2 2 are equivalent over RCA0. However, the proof that D 2 2 implies SRT 2 2 has a hidden application of BΣ2 and thus is actually carried out in RCA0 + BΣ 0 2. The problem is that, in the construction of H by adding one element at a time, each element c added to H must form a pair of the appropriate color with all previously chosen elements. To get the existence of such a c one seems to need BΣ2. This gap was recently closed by Chong, Lempp, and Yang, who showed in [3], Theorem 1.4, that, in RCA0, D 2 2 implies BΣ2, and hence D 2 2 implies SRT 2 2. 2. Lemma 7.11 asserts that RT2 is equivalent to SRT 2 2 & COH over RCA0. However, the proof given there that RT 2 2 implies COH in RCA0 is seriously flawed. This was pointed out by Joseph Mileti and later by Jeffrey Hirst. A proof that RT2 implies COH in RCA0 + IΣ2 can easily be extracted from the proof of Theorem 12.5. Mileti, and simultaneously Lempp and Jockusch, observed that it is possible to eliminate the use of IΣ2 by effectively bounding in terms of k the number of changes changes in the characteristic function of A when it is restricted to Ak, so that proving that this number is finite requires only Σ1-induction. Thus, it is provable in RCA0 that RT 2 2 implies COH, and hence that RT2 is equivalent to SRT 2 2 & COH. 3. Joseph Mileti pointed out a gap in the proof of the claim at the bottom of page 50 that a certain computable 2-coloring of pairs C is “jump universal” in the sense that for every C-homogeneous set A and every computable coloring C̃, there exists an infinite C̃-homogeneous set B with B′ ≤T A′ . The proof provided works only when C̃ is stable. However, this assumption can be eliminated by using the density of the Turing degrees under << (see [6], Theorem 6.5) to stabilize C̃ . Namely,