  We improve results of Kahn, Kalai, and Linial from the late 80s on the
existence of influential large coalitions for Boolean functions, and we give
counterexamples to conjectures (of Benny Chor and others) also from the late
80s, by exhibiting functions for which the influences of large coalitions are
unexpectedly small relative to the expectations of the functions. The large
gaps between the new upper and lower bounds leave a lot of room for further
study.
