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We prove that any proof of a ∀ Σ 2 0 sentence in the theory WKL 0 + RT 2 2 can be translated into a proof in RCA 0 at the cost of a polynomial increase in size. In fact, the proof in RCA 0 can be obtained by a polynomial-time algorithm. On the other hand, RT 2 2 has nonelementary speedup over the weaker base theory RCA 0 * for proofs of Σ 1 sentences. We also show that for n ≥ 0 , proofs of Π n + 2 sentences in B Σ n + 1 + exp can be translated into proofs in I Σ n + exp at a polynomial cost in size. Moreover, the Π n + 2 -conservativity of B Σ n + 1 + exp over I Σ n + exp can be proved in PV , a fragment of bounded arithmetic corresponding to polynomial-time computation. For n ≥ 1 , this answers a question of Clote, Hájek and Paris. [ABSTRACT FROM AUTHOR]

For A ⊆ ω , the coarse similarity class of A, denoted by [ A ] , is the set of all B ⊆ ω such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric δ on the space of coarse similarity classes defined by letting δ ([ A ] , [ B ]) be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form { [ A ] : A ∈ } where is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of . We then define a distance between Turing degrees based on Hausdorff distance in the metric space (, δ). We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly 0, 1 / 2 , and 1, and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree a to be attractive if the class of all degrees at distance 1 / 2 from a has measure 1, and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability. Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic. Finally, we study the completeness of (, δ) from the perspectives of computability theory and reverse mathematics. [ABSTRACT FROM AUTHOR]

The Ginsburg–Sands theorem from topology states that every infinite topological space has an infinite subspace homeomorphic to exactly one of the following five topologies on ω : indiscrete, discrete, initial segment, final segment, and cofinite. The original proof is nonconstructive, and features an interesting application of Ramsey's theorem for pairs (RT 2 2). We analyze this principle in computability theory and reverse mathematics, using Dorais's formalization of CSC spaces. Among our results are that the Ginsburg–Sands theorem for CSC spaces is equivalent to ACA 0 , while for Hausdorff spaces it is provable in RCA 0. Furthermore, if we enrich a CSC space by adding the closure operator on points, then the Ginsburg–Sands theorem turns out to be equivalent to the chain/antichain principle (CAC). The most surprising case is that of the Ginsburg–Sands theorem restricted to T 1 spaces. Here, we show that the principle lies strictly between ACA 0 and RT 2 2 , yielding arguably the first natural theorem from outside logic to occupy this interval. As part of our analysis of the T 1 case we introduce a new class of purely combinatorial principles below ACA 0 and not implied by RT 2 2 which form a hierarchy generalizing the stable Ramsey's theorem for pairs (SRT 2 2). We show that one of these, the Σ 2 0 subset principle (Σ 2 0 - Subset), has the property that it, together with the cohesive principle (COH), is equivalent over RCA 0 to the Ginsburg–Sands theorem for T 1 CSC spaces. [ABSTRACT FROM AUTHOR]