Compactness versus hugeness at successor cardinals

TL;DR

This paper explores the relationship between certain ideal properties and combinatorial principles at successor cardinals, revealing limitations of forcing methods in achieving specific set-theoretic configurations.

Contribution

It establishes new implications between weakly presaturated ideals and the square principle, and identifies barriers to combining the tree property with saturated ideals.

Findings

Weakly presaturated ideals imply irc_____ in certain contexts

Presaturated ideals on _2 with semiproper quotient imply CH

Barriers exist to obtaining the tree property and saturated ideals simultaneously via forcing

Abstract

If $\kappa$ is regular and $2^{<\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on $\omega_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $\omega_2$ such that $\mathcal{P}(\omega_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.