Annals of Pure and Applied Logic 63 (3):271-281 (1993)

Abstract
Louveau, A., S. Shelah and B. Velikovi, Borel partitions of infinite subtrees of a perfect tree, Annals of Pure and Applied Logic 63 271–281. We define a notion of type of a perfect tree and show that, for any given type τ, if the set of all subtrees of a given perfect tree T which have type τ is partitioned into two Borel classes then there is a perfect subtree S of T such that all subtrees of S of type τ belong to the same class. This result simultaneously generalizes the partition theorems of Galvin-Prikry and Galvin-Blass. The key ingredient of the proof is the theorem of Halpern-Laüchli on partitions of products of perfect trees
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DOI 10.1016/0168-0072(93)90151-3
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References found in this work BETA

Borel Sets and Ramsey's Theorem.Fred Galvin & Karel Prikry - 1973 - Journal of Symbolic Logic 38 (2):193-198.
A Partition Theorem.J. D. Halpern - 1974 - Journal of Symbolic Logic 39 (1):181-182.

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