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Ramsey’s theorem for trees: the polarized tree theorem and notions of stability. (English) Zbl 1215.03018
The authors study the reverse mathematics of a polarized version of Ramsey’s theorem for trees. Ramsey’s theorem for trees (studied by Chubb, Hirst and McNicholl) extends the usual infinite Ramsey’s theorem from \(\mathbb N\) to \(2^{<\mathbb N}\): It says that for every coloring of the increasing \(n\)-tuples of binary strings with \(k\) colors, there is a subset of \(2^{<\mathbb N}\), order-isomorphic to \(2^{<\mathbb N}\), which is homogeneous. The polarized Ramsey’s theorem (studied by Dzhafarov and Hirst) extends the usual infinite Ramsey’s theorem in a different way: It says that for every coloring of \([\mathbb N]^n\) with \(k\) colors, there exist sets \(H_1\),…, \(H_n\) of natural numbers such that \(H_1\times\cdots \times H_n \cap [\mathbb N]^n\) is monochromatic. In this paper, a version of Ramsey’s theorem which mixes both of these ideas is introduced.
For those exponents greater than 2, both the reverse mathematics and the computability theory associated with this theorem parallel that of its linear analog. For pairs the situation is more complex. In particular, there are many notions of stability in the tree setting, complicating the analysis of the related results.

03B30 Foundations of classical theories (including reverse mathematics)
03D80 Applications of computability and recursion theory
03F35 Second- and higher-order arithmetic and fragments
05D10 Ramsey theory
Full Text: DOI
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