Universally Sacks-indestructible combinatorial families of reals

Abstract

We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory. We then prove that every combinatorial family of reals of arithmetical type, which is indestructible by the product of Sacks forcing $\mathbb{S}^{\aleph_0}$, is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under $\text{CH}$ we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under $\text{CH}$.