Weak saturation properties and side conditions

TL;DR

This paper explores the use of side-conditions forcing to combine compactness and hugeness properties at 02, reducing the consistency strength needed for weak Chang's Conjecture but identifying limitations of current methods.

Contribution

It demonstrates a reduction in the consistency strength for weak Chang's Conjecture at 02 using Neeman's forcing and provides a counterexample to Neeman's claim about iterated forcing effects.

Findings

Reduced the upper bound on the consistency strength of weak Chang's Conjecture at 02.

Identified limitations and barriers in applying side-conditions forcing methods.

Constructed a counterexample to Neeman's claim regarding iterated forcing effects.

Abstract

Towards combining "compactness" and "hugeness" properties at $\omega_2$, we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at $\omega_2$ using Neeman's forcing. But we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating such forcing.