THE STRENGTH OF THE TREE THEOREM FOR PAIRS IN REVERSE MATHEMATICS
Ludovic Patey · Journal of Symbolic Logic · 2016
Abstract No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA 0 ) 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 0 and $RT_2^2$ over RCA 0 , but its exact strength remains open. In this paper, we prove that $RT_2^2$ together with weak König’s lemma (WKL 0 ) does not imply $TT_2^2$ , thereby answering a question of Montálban. 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.