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**1 - 3**of**3**### 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 asse ..."

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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 2 and SRT2 2 are equivalent over RCA0. However, the proof that D2 2 implies SRT2 2 has a hidden application of BΣ0 2 and thus is actually carried out in RCA0 + BΣ0 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 BΣ0 2. This gap was recently closed by Chong, Lempp, and Yang, who showed in [3], Theorem 1.4, that, in RCA0, D2 2 implies BΣ0 2, and hence D2 2 implies SRT2 2. 2. Lemma 7.11 asserts that RT2 2 is equivalent to SRT2 2 & COH over RCA0. However, the proof given there that RT2 2 implies COH in RCA0 is seriously flawed. This was pointed out by Joseph Mileti and later by Jeffrey Hirst. A proof that RT2 2 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 RT2 2 implies COH, and hence that RT2 2 is equivalent to SRT2 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 ˜ C, there exists an infinite ˜ C-homogeneous set B with B ′ ≤T A ′. The proof provided works only when ˜ C is stable. However, this assumption can be eliminated by using the density of the Turing degrees under << (see [6], Theorem 6.5) to stabilize ˜ C. Namely,

### GENERICS FOR COMPUTABLE MATHIAS FORCING

"... Abstract. We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak n-generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, ..."

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Abstract. We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak n-generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 2 then it satisfies the jump property G (n−1) ≡T G ′ ⊕ ∅ (n). We prove that every such G has generalized high Turing degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real. 1.