Dense ideals

Abstract

In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a $\ddot{I}ƒ$-complete, ${\aleph }_1$-dense ideal on ${\aleph }_{n+1}$ for every $n<\ddot{I}‰$, answering a question of Foreman. Using this construction we show the consistency of the existence of various irregular ultrafilters on $\ddot{I}{‰}_n$, the consistency of the Foreman-Laver reflection property for the chromatic number of graphs for all possible pairs of cardinals below ${\aleph }_{\ddot{I}}‰$, and the simultaneous consistency of the partition hypotheses ${\mathrm{PH}}_n(\ddot{I}{‰}_m)$ for $n<m$.