Adding highly generic subsets of $\omega_2$

TL;DR

This paper constructs a model of set theory where a large subset of 02 has the property that all its countably infinite subsets are Cohen generic over the ground model, preserving GCH.

Contribution

It introduces a method to create a GCH-preserving extension with a subset of 02 having all countably infinite subsets Cohen generic.

Findings

Existence of a 02 subset with all countably infinite subsets Cohen generic

Preservation of GCH in the constructed extension

Cardinal 02 remains unchanged

Abstract

Starting from the $\rm{GCH},$ we build a cardinal and $\rm{GCH}$ 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.