Hyperfiniteness on Topological Ramsey Spaces

We investigate the behavior of countable Borel equivalence relations (CBERs) on topological Ramsey spaces. First, we give a simple proof of the fact that every CBER on \([\mathbb{N}]^{\mathbb{N}}\) is hyperfinite on some set of the form \([A]^{\mathbb{N}}\). Using the idea behind the proof, we show the analogous result for every topological Ramsey space.