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.