BOREL PARTITIONS OF INFINITE SUBTREES OF A PERFECT TREE

We define a notion of type of a perfect tree and show that, for any gi ven type tau, if the set of all subtrees of a given perfect tree T whi ch have type tau is partitioned into two Borel classes then there is a perfect subtree S of T such that all subtrees of S of type tau belong to the same class. This result simultaneously generalizes the partiti on theorems of Galvin-Prikry and Galvin-Blass. The key ingredient of t he proof is the theorem of Halpern-Lauchli on partitions of products o f perfect trees.