Ramsey's theorem for trees: the polarized tree theorem and notions of stability
type: Item
instance of: scholarly article
title: Ramsey's theorem for trees: the polarized tree theorem and notions of stability
zbMATH Open document ID: 1215.03018
author: Damir D. Dzhafarov
author: Jeffry L. Hirst
author: Tamara J. Lakins
published in: Archive for Mathematical Logic
publication date: 2010-04-15T00:00:00Z
review text: 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 \(H_1\),\dots, \(H_n\) 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.
reviewed by: Antonio Montalbán
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03D80
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 05D10
zbMATH DE Number: 5693394
zbMATH Keywords: polarized version of Ramsey's theorem for trees
MaRDI profile type: Publication
full work available at URL: s00153-010-0179-6
OpenAlex ID: W2000565016
cites work: On the strength of Ramsey's theorem for pairs
cites work: Corrigendum to: “On the strength of Ramsey's Theorem for pairs”
cites work: On the role of the collection principle for Σ⁰₂-formulas in second-order reverse mathematics
cites work: Reverse mathematics, computability, and partitions of trees
cites work: The polarized Ramsey's theorem
cites work: Combinatorial principles weaker than Ramsey's Theorem for pairs
cites work: Generalized cohesiveness
cites work: Ramsey's theorem and recursion theory
cites work: Q3819052
cites work: Q4220572
DOI: 10.1007/S00153-010-0179-6
description: scientific article; zbMATH DE number 5693394