Weak saturation properties and side conditions

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.

Theorem 28 of the first version was false. There were some gaps in the proof of Lemma 32 and in the argument for the tree property in the last paragraph of page 24