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We prove that any proof of a $forall Sigma^0_2$ sentence in the theory $mathrm{WKL}_0 + mathrm{RT}^2_2$ can be translated into a proof in $mathrm{RCA}_0$ at the cost of a polynomial increase in size. In fact, the proof in $mathrm{RCA}_0$ can be found by a polynomial-time algorithm. On the other hand, $mathrm{RT}^2_2$ has non-elementary speedup over the weaker base theory $mathrm{RCA}^*_0$ for proofs of $Sigma_1$ sentences. We also show that for $n ge 0$, proofs of $Pi_{n+2}$ sentences in $mathrm{B}Sigma_{n+1}+exp$ can be translated into proofs in $mathrm{I}Sigma_{n} + exp$ at polynomial cost. Moreover, the $Pi_{n+2}$-conservativity of $mathrm{B}Sigma_{n+1} + exp$ over $mathrm{I}Sigma_{n} + exp$ can be proved in $mathrm{PV}$, a fragment of bounded arithmetic corresponding to polynomial-time computation. For $n ge 1$, this answers a question of Clote, Hajek, and Paris.
In this paper, we show that $mathrm{RT}^{2}+mathsf{WKL}_0$ is a $Pi^{1}_{1}$-conservative extension of $mathrm{B}Sigma^0_3$.
We study the reverse mathematics and computability-the-o-re-tic strength of (stable) Ramseys Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a question of Mileti b
The well-known Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey: Given any Borel subset $mathcal{X}subseteq [omega]^{omega}$, where $[omega]^{omega}$ is endowed with the metric topology, each infinite subset $Xsubseteq om
We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our m
A definable set in a pair (K, k) of algebraically closed fields is co-analyzable relative to the subfield k of the pair if and only if it is almost internal to k. To prove this and some related results for tame pairs of real closed fields we introduc