Compactness versus hugeness at successor cardinals

Abstract

If $\kappa$ is regular and $2^{<\kappa }\le {\kappa }^{+}$, then the existence of a weakly presaturated ideal on ${\kappa }^{+}$ implies ${\square }_{\kappa }^{\ast }$. 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.