Weak saturation properties and side conditions

Abstract

Towards combining “compactness” and “hugeness” properties at ω2, we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at ω2 using Neeman's forcing. On the other hand, 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.