The lowest non-trivial complexity class in the theory of Countable Borel Equivalence Relations (CBERs) is the class of hyperfinite CBERs. One difficulty that arises in studying this class is determining which CBERs are hyperfinite. Measure theory can be used to answer this question, but not many techniques can. For instance, a Baire category approach cannot distinguish hyperfinite CBERS: a result of Hjorth and Kechris states that every CBER on a Polish space is hyperfinite when restricted to some comeager set. We will discuss a classical proof of Mathias's theorem that every CBER on the Ellentuck Ramsey space is hyperfinite when restricted to some pure Ellentuck cube. Mathias's theorem implies that a Ramsey-theoretic approach also cannot distinguish hyperfinite CBERs. This is joint work with A. Panagiotopoulos.