Some applications of combinatorics in descriptive set theory

We investigate the relationship between classical combinatorics and descriptive set theory. Establishing new results in the areas of intersecting families and partitions of the complete binary tree, we subsequently analyze ramifications in the study of Borel equivalence relations and canonization theorems on Cantor space. We also examine metric space valued random variables as well as finite measure spaces. In all of these inquiries we employ the same basic strategy: approximate the object in question by combinatorially simple structures, and analyze how the combinatorics of the approximations yield desired information.