Familiar graph-theoretic problems (for example, vertex coloring) exhibit a stark change of character when measurability constraints are placed on the structures and functions involved. While discussing some ramifications in descriptive set theory, we also pay special attention to interactions with probability (concerning random colorings of Cayley graphs) and ergodic theory (characterizing various dynamical properties of groups).

The talk will include joint work with Alexander Kechris, Andrew Marks, Benjamin Miller, and Robin Tucker-Drob.