The strength of the tree theorem for pairs in reverse mathematics

TL;DR

This paper investigates the relative strength of the tree theorem for pairs (TT^2_2) in reverse mathematics, showing it cannot be derived from RT^2_2 combined with WKL, thereby clarifying its position between ACA and RT^2_2.

Contribution

It proves that RT^2_2 plus WKL does not imply TT^2_2, providing a key separation result and a tutorial on the method used for such distinctions in reverse mathematics.

Findings

RT^2_2 + WKL does not imply TT^2_2

The separation uses a computability-theoretic property

Clarifies the position of TT^2_2 in reverse mathematics hierarchy

Abstract

No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA) and Ramsey's theorem for pairs (RT^2_2) in reverse mathematics. The tree theorem for pairs (TT^2_2) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle TT^2_2 is known to lie between ACA and RT^2_2 over RCA, but its exact strength remains open. In this paper, we prove that RT^2_2 together with weak K\"onig's lemma (WKL) does not imply TT^2_2, thereby answering a question of Montalban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics.