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No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA0) 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 ACA0 and $RT_2^2$ over RCA0, but its exact strength remains open. In this paper, we prove that $RT_2^2$ together with weak König’s lemma (WKL0) 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.



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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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