Almost disjoint refinements and mixing reals

Abstract

We investigate families of subsets of $\omega$ with almost disjoint refinements in the classical case as well as with respect to given ideals on $\omega$. More precisely, we study the following topics and questions: 1) Examples of projective ideals. 2) We prove the following generalization of a result due to J. Brendle: If $V\subseteq W$ are transitive models, ${\omega }_1^W\subseteq V$, $\mathcal{P}(\omega )\cap V\ne \mathcal{P}(\omega )\cap W$, and $\mathcal{I}$ is an analytic or coanalytic ideal coded in $V$, then there is an $\mathcal{I}$-almost disjoint refinement ($\mathcal{I}$-ADR) of ${\mathcal{I}}^{+}\cap V$ in $W$, that is, a family $\{A_X:X\in {\mathcal{I}}^{+}\cap V\}\in W$ such that (i) $A_X\subseteq X$, $A_X\in {\mathcal{I}}^{+}$ for every $X$ and (ii) $A_X\cap A_Y\in \mathcal{I}$ for every distinct $X$ and $Y$. 3) The existence of perfect $\mathcal{I}$-almost disjoint ($\mathcal{I}$-AD) families, and the existence of a "nice" ideal $\mathcal{I}$ on $\omega$ with the property: Every $\mathcal{I}$-AD family is countable but $\mathcal{I}$ is nowhere maximal. 4) The existence of $(\mathcal{I},\text{Fin})$-almost disjoint refinements of families of $\mathcal{I}$-positive sets in the case of everywhere meager (e.g. analytic or coanalytic) ideals. We prove a positive result under Martin's Axiom. 5) Connections between classical properties of forcing notions and adding mixing reals (and mixing injections), that is, a (one-to-one) function $f:\omega \to \omega$ such that $|f[X]\cap Y|=\omega$ for every $X,Y\in [\omega {]}^{\omega }\cap V$.