Borel Partitions of Infinite Subtrees of a Perfect Tree

Abstract Louveau, A., S. Shelah and B. Velickovic, 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-Lauchli on partitions of products of perfect trees.