A llison Y. W ang . Complexity in Countable Borel Equivalence Relations and Topological Ramsey Spaces . Carnegie Mellon University. 2026. Supervised by Clinton T. Conley. MSC: 03E15, 54H05, 28A05.

This dissertation investigates several questions concerning complexity in the context of countable Borel equivalence relations (CBERs) and topological Ramsey spaces. Chapter 2 presents relevant background material about topological Ramsey spaces. Also discussed are several examples of such spaces, including the Milliken space of strong subtrees and the Carlson–Simpson space of infinite partitions of ω ω omega . Chapter 3, which is joint work with Aristotelis Panagiotopoulos, studies CBERs on the Carlson–Simpson space. Ramsey-theoretic methods cannot distinguish CBERs beyond hyperfiniteness. More specifically, Mathias showed that every CBER on [ ω ] ω [ω ]^ω left bracket omega right bracket Superscript omega is hyperfinite on a non-empty Ellentuck cube, and Bursics and Vidnyánszky recently extended this result to all topological Ramsey spaces. An even stronger result holds in some topological Ramsey spaces. For example, Kanovei, Sabok, and Zapletal showed that every CBER on the Milliken space of strong subtrees of 2 < ω 2<ω 2 Superscript less than omega is smooth on a non-empty Ellentuck cube; in fact, the same holds true for any analytic equivalence relation on this space. The main result of this chapter is that every CBER on the Carlson–Simpson space is smooth on a non-empty Ellentuck cube, but that this statement fails for arbitrary analytic equivalence relations. Chapter 4 examines the complexity of a set arising naturally from notions in topological Ramsey theory. Sabok proved that the set of codes for the G δ G_δ upper G Subscript delta Ramsey positive sets in [ ω ] ω [ω ]^ω left bracket omega right bracket Superscript omega is Σ 2 1 Σ ¹₂ normal upper Sigma 2 Superscript 1 -complete, and that the same holds for analytic Ramsey positive sets. A natural question is whether the same results hold in other topological Ramsey spaces. In this chapter, it is shown that the analogous sets of codes in other topological Ramsey spaces are Σ 2 1