Ramsey's theorem for pairs, collection, and proof size

Leszek Aleksander Kołodziejczyk, Tin Lok Wong, Keita Yokoyama

Published 2020-05-14Version 1

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, H\'ajek, and Paris.