Definable maximal cofinitary groups of intermediate size

Using almost disjoint coding, we show that for each $1<M<N<\omega$ consistently $\mathfrak{d}=\mathfrak{a}_g=\aleph_M<\mathfrak{c}=\aleph_N$, where $\mathfrak{a}_g=\aleph_M$ is witnessed by a $\Pi^1_2$ maximal cofinitary group.Comment: 22 page

Definability of maximal cofinitary groups

We present a proof of a result, previously announced by the second author, that there is a closed (even $\Pi^0_1$) set generating an $F_\sigma$ (even $\Sigma^0_2$) maximal cofinitary group (short, mcg) which is isomorphic to a free group. In this isomorphism class, this is the lowest possible definitional complexity of an mcg.Comment: This work is part of the first authors thesi

Universally Sacks-indestructible combinatorial families of reals

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 $\text{CH}$.Comment: 33 pages, submitte