Almost disjoint refinements and mixing reals
Volume 242 / 2018
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 $.
We prove the following generalization of a result due to J. Brendle: If $V\subseteq W$ are transitive models, $\omega _1^W\subseteq V$, $\mathcal {P}(\omega )\cap V\not =\mathcal {P}(\omega )\cap W$, and $\mathcal {I}$ is an analytic or coanalytic ideal coded in $V$, then there is an $\mathcal {I}$-almost disjoint refinement of $\mathcal {I}^+\cap V$ in $W$.
We study the existence of perfect $\mathcal {I}$-almost disjoint families, and the existence of $\mathcal {I}$-almost disjoint refinements in which any two distinct sets have finite intersection.
We introduce the notion of mixing real (motivated by the construction of an almost disjoint refinement of $[\omega ]^\omega \cap V$ after adding a Cohen real to $V$) and discuss logical implications between the existence of mixing reals in forcing extensions and classical properties of forcing notions.
