## Journal of Symbolic Logic

### On the Strength of Ramsey's Theorem for Pairs

#### Abstract

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT$^n_k$ denote Ramsey's theorem for k-colorings of n-element sets, and let RT$^n_{<\infty}$ denote ($\forall k)RT^n_k$. Our main result on computability is: For any n $\geq$ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' $\leq_T 0^{(n)}$. Let $I\Sigma_n$ and $B\Sigma_n$ denote the $\Sigma_n$ induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low$_2$) to models of arithmetic enables us to show that $RCA_0 + I\Sigma_2 + RT^2_2$ is conservative over $RCA_0 + I\Sigma_2$ for $\Pi^1_1$ statements and that $RCA_0 + I\Sigma_3 + RT^2_{<\infty}$, is $\Pi^1_1$-conservative over $RCA_0 + I\Sigma_3$. It follows that $RCA_0 + RT^2_2$ does not imply $B\Sigma_3$. In contrast, J. Hirst showed that $RCA_0 + RT^2_{<\infty}$ does imply $B\Sigma_3$, and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{<\infty}$ is strictly stronger than $RT^2_2$ over $RCA_0$.

#### Article information

Source
J. Symbolic Logic, Volume 66, Issue 1 (2001), 1-55.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183746359

Mathematical Reviews number (MathSciNet)
MR1825173

Zentralblatt MATH identifier
0977.03033

JSTOR