Saturated ideals obtained via restricted iterated collapse of huge cardinals

A uniform method to define a (restricted iterated) forcing notion to collapse a huge cardinal to a small one to obtain models with various types of highly saturated ideals over small cardinals is presented. The method is discussed in great technical details in the first chapter, while in the second chapter the application of the method is shown on three different models: Model I with an ℵ1-complete ℵ2-saturated ideal over ℵ1 that satisfies Chang’s conjecture, Model II with an ℵ1-complete ℵ3-saturated ideal over ℵ3, and Model III with an ℵ1-complete (ℵ2, ℵ2, ℵ0)-saturated ideal over ℵ1.