Borel partitions of infinite subtrees of a perfect tree

Louveau, A., S. Shelah and B. Veličković, Borel partitions of infinite subtrees of a perfect tree, Annals of Pure and Applied Logic 63 (1993) 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.