The investigation of Borel graphs and Borel chromatic numbers has been a very active area in the past couple of years. Recently, thanks to the work of Bernshteyn, even a connection between Borel combinatorics and distributed computing has been found.
One question of particular interest is the understanding of Borel chromatic numbers of acyclic Borel graphs with bounded degrees. In this talk I will discuss two methods for constructing interesting examples of such graphs. The first one is inspired by distributed computing and it is a generalization of the Borel determinacy method of Marks, while the second is based on the Galvin-Prikry theorem and graphs with large expansion.