Definable expansions on countable groups and countable Borel equivalence relations

We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure and category settings, and establish some basic correspondences between the two notions. We also prove some general structure theorems for measure and category. We then explore in detail many examples, including finding spanning trees in graphs, finding monochromatic sets in Ramsey’s Theorem, and linearizing partial orders.