The proof-theoretic strength of Ramsey's theorem for pairs and two colors

Ramsey's theorem for $n$-tuples and $k$-colors ($\mathsf{RT}^n_k$) asserts that every k-coloring of $[\mathbb{N}]^n$ 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 $\Pi^0_1$ consequences, and show that $\mathsf{RT}^2_2$ is $\Pi^0_3$ conservative over $\mathsf{I}\Sigma^0_1$. This strengthens the proof of Chong, Slaman and Yang that $\mathsf{RT}^2_2$ does not imply $\mathsf{I}\Sigma^0_2$, and shows that $\mathsf{RT}^2_2$ 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 $\Pi^0_3$-conservation theorems.Comment: 32 page

Degrees bounding principles and universal instances in reverse mathematics

A Turing degree d bounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove that the stable version of the ascending descending sequence principle (SADS) as well as the stable version of the thin set theorem for pairs (STS(2)) do not admit a bound of low_2 degree. Therefore no principle between Ramsey's theorem for pairs RT22 and SADS or STS(2) admit a universal instance. We construct a low_2 degree bounding the Erd\H{o}s-Moser theorem (EM), thereby showing that previous argument does not hold for EM. Finally, we prove that the only Delta^0_2 degree bounding a stable version of the rainbow Ramsey theorem for pairs (SRRT22) is 0'. Hence no principle between the stable Ramsey theorem for pairs SRT22 and SRRT22 admit a universal instance. In particular the stable version of the Erd\H{o}s-Moser theorem does not admit one. It remains unknown whether EM admits a universal instance.Comment: 23 page

Where Pigeonhole Principles meet K\"onig Lemmas

We study the pigeonhole principle for $\Sigma_2$-definable injections with domain twice as large as the codomain, and the weak K\"onig lemma for $\Delta^0_2$-definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of $2$-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for $\Sigma_2$-definable injections.Comment: 33 page