Cohen preservation and independence

Abstract

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $i$ is strictly below $c$, including iterations of Sacks forcing, Miller partition forcing, h-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $i={\aleph }_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of “Cohen preservation”, which we discuss in detail.