Adding highly generic subsets of ${\omega }_2$

Abstract

Starting from the $\mathrm{GCH},$ we build a cardinal and $\mathrm{GCH}$ preserving generic extension of the universe, in which there exists a set $A\subseteq {\omega }_2$ of size ${\aleph }_2$ so that every countably infinite subset of $A$ or ${\omega }_2\setminus A$ is Cohen generic over the ground model.