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A theory of stationary trees and the balanced Baumgartner-Hajnal-Todorcevic theorem for trees. (English) Zbl 1324.03013

This paper is devoted to a proof of its main theorem, which generalizes Theorem 3.1 of J. E. Baumgartner et al. [NATO ASI Ser., Ser. C, Math. Phys. Sci. 411, 1–17 (1993; Zbl 0846.03021)], by replacing the cardinal \((2^{<\kappa})^+\) by suitable trees: if \(\kappa\) is regular and infinite, \(\xi\) an ordinal with \(2^{| \xi|}<\kappa\) and \(k\in\omega\) then for every tree \(T\) that is not \(2^{<\kappa}\)-special one has \(T\to (\kappa+\xi)^2_k\), that is, for every colouring \(\chi:[T]^2\to k\) there is a \(\chi\)-homogeneous subset of \(T\) of order type \(\kappa+\xi\).
The paper is enriched by a careful exposition of earlier results and necessary background material, and would make a good basis for a seminar on infinitary combinatorics.
Reviewer: K. P. Hart (Delft)

MSC:

03E02 Partition relations
03C62 Models of arithmetic and set theory
05C05 Trees
05D10 Ramsey theory
06A07 Combinatorics of partially ordered sets

Citations:

Zbl 0846.03021
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References:

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