Coloring trees in reverse mathematics
Damir Dzhafarov, Ludovic Patey

TL;DR
This paper investigates the reverse mathematical strength of the tree theorem for pairs ($ ext{TT}^2_2$), showing it is strictly between $ ext{RCA}_0$ and $ ext{ACA}_0$, and introduces new forcing techniques for tree-based combinatorial principles.
Contribution
It proves that $ ext{TT}^2_2$ does not imply $ ext{ACA}_0$, and provides a new proof that $ ext{TT}^2_2$ is stronger than $ ext{RT}^2_2$, establishing a new natural intermediate principle.
Findings
$ ext{TT}^2_2$ does not imply $ ext{ACA}_0$ over $ ext{RCA}_0$
$ ext{TT}^2_2$ is strictly stronger than $ ext{RT}^2_2$
Introduces an extension of bushy tree forcing for combinatorial statements on trees
Abstract
The tree theorem for pairs (), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree , there is a set of nodes isomorphic to which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs (), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of , by showing that this principle does not imply the arithmetic comprehension axiom () over the base system, recursive comprehension axiom (), of second-order arithmetic. In addition, we give a new and self-contained proof of a recent result of Patey that is strictly stronger than .…
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