Ramsey property for Boolean algebras with ideals and $\mathcal{P}(\omega_1)/\rm{fin}$

Abstract

We apply the Dual Ramsey Theorem of Graham and Rothschild to prove the Ramsey property for classes of finite Boolean algebras with distinguished ideals. This allows us to compute the universal minimal flow of the group of automorphisms of $\mathcal{P}(\omega_1)/\rm{fin},$ should it be isomorphic to $\mathcal{P}(\omega)/\rm{fin}$ or not, and of other quotients of power set algebras. Taking Fra\"iss\'e limits of these classes, we can compute universal minimal flows of groups of homeomorphisms of the Cantor set fixing some closed subsets.