Let ω∗ = βω \ ω, where βω denotes the Stone-Cˇech compactification of the natural numbers. This space, called the corona or the remainder of ω, has been extensively studied in the fields of set theory and topology. We investigate minimal actions of G = Homeo(ω∗) on various hyperspaces of ω∗. Using the dual Ramsey Theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system (Exp(Exp(ω∗)),G), where Exp(Z) stands for the hyperspace comprising the closed subsets of the compact space Z equipped with the Vietoris topology. The dynamical importance of this dynamical system stems from Uspenskij’s characterization of the universal ambit of G. These results apply as well to the Polish group Homeo(C), where C is the Cantor set.