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ON UNIFORM RELATIONSHIPS BETWEEN COMBINATORIAL PROBLEMS
"... Abstract. The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a we ..."
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Abstract. The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to nonuniform decisions about how to proceed in a given construction. In practice, however, if a theorem Q implies a theorem P, it is usually because there is a direct uniform translation of the problems represented by P into the problems represented by Q, in a precise sense formalized by Weihrauch reducibility. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all n, j, k ≥ 1, if j < k then Ramsey’s theorem for ntuples and k many colors is not uniformly, or Weihrauch, reducible to Ramsey’s theorem for ntuples and j many colors. The two theorems are classically equivalent, so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions. We also study Weak König’s Lemma, the Thin Set Theorem, and the Rainbow Ramsey’s Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve infinitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related. 1.
GENERICS FOR COMPUTABLE MATHIAS FORCING
"... Abstract. We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The ngenerics and weak ngenerics 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 ngenerics and weak ngenerics 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 ngeneric 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 1generic degree. On the other hand, we show that every Mathias ngeneric real computes a Cohen ngeneric real. 1.
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 Σ1induction. 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 2coloring of pairs C is “jump universal ” in the sense that for every Chomogeneous set A and every computable coloring ˜ C, there exists an infinite ˜ Chomogeneous 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,