We continue our discussion of coloring graphs subject to measurability constraints. This time we focus on the issue of constructing acyclic Borel graphs on standard probability spaces which are unexpectedly hard to color. To do so, we examine free measure-preserving actions of free groups and analyze restrictions on the measures of independent sets for the associated graph. The corresponding parameter, the independence number, turns out to have surprisingly deep connections with the ergodic-theoretic properties of the action we started with. We show how this implies that graphs associated with Bernoulli shift actions have large chromatic number, and (time permitting) how to use these techniques to generate infinitely many weakly inequivalent mixing actions. The talk includes joint work with Alexander Kechris, Robin Tucker-Drob, and Brandon Seward.