Parametrizing topological Ramsey spaces

We prove a general theorem indicating that essentially all infinite-dimensional Ramsey-type theorems proven using topological Ramsey space theory can be parametrized by products of infinitely many perfect sets. This theorem has applications in several known spaces, showing that certain ultrafilters are preserved under both side-by-side and iterated Sacks forcing. In particular, the well-known result of 'selective ultrafilters on the natural numbers are preserved under Sacks forcing' is extended to the corresponding ultrafilters on richer structures. We also characterize ultrafilters in topological Ramsey spaces in an abstract setting. The technique of combinatorial forcing is crucial in the proof of the general parametrization theorem, and ultra-Ramsey theory plays an important role in the applications.Ph.D