## Abstract

Ramsey's theorem for n-tuples and k-colors (RT_{k} ^{n}) asserts that every k-coloring of [N]^{n} admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two colors, namely, the set of its Π_{1} ^{0} consequences, and show that RT_{2} ^{2} is Π_{3} ^{0} conservative over IΣ_{1} ^{0}. This strengthens the proof of Chong, Slaman and Yang that RT^{2} _{2} does not imply IΣ_{2} ^{0}, and shows that RT_{2} ^{2} is finitistically reducible, in the sense of Simpson's partial realization of Hilbert's Program. Moreover, we develop general tools to simplify the proofs of Π_{3} ^{0}-conservation theorems.

Original language | English |
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Pages (from-to) | 1034-1070 |

Number of pages | 37 |

Journal | Advances in Mathematics |

Volume | 330 |

DOIs | |

Publication status | Published - 2018 May 25 |

## Keywords

- Proof-theoretic strength
- Ramsey's theorem
- Reverse mathematics