Borel Asymptotic Dimension and AsymptoticSeparation Index

In this exposition, we explore the notion of Borel asymptotic dimension, and its recent applications to the study of countable Borel equivalence relations and Borel combinatorics. Originally introduced as a Borel analogue of Gromov's asymptotic dimension, Borel asymptotic dimension proves to be a key step toward understanding hyperfinite countable Borel equivalence relations, and provides a framework for analyzing Borel graphs. In Chapter 2, we review the standard asymptotic dimension as an invariant in metric geometry. In Chapter 3 we introduce Borel asymptotic dimension and asymptotic separation index, and prove several important properties and techniques. In Chapter 4, we prove how asymptotic separation index extends results in Borel combinatorics. In Chapter 5, we discuss the connections among local algorithms, Borel asymptotic dimension and asymptotic separation index, and growth rate of Borel graphs. In Chapter 6, we witness finite Borel asymptotic dimension as a key tool in rendering hyperfiniteness of countable Borel equivalence relations.