Dense ideals

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 $\sigma$-complete, ${\mathrm{\aleph }}_1$-dense ideal on ${\mathrm{\aleph }}_{n+1}$ for every $n<\omega$, answering a question of Foreman. Using this construction we show the consistency of the existence of various irregular ultrafilters on ${\omega }_n$, the consistency of the Foreman-Laver reflection property for the chromatic number of graphs for all possible pairs of cardinals below ${\mathrm{\aleph }}_{\omega }$, and the simultaneous consistency of the partition hypotheses ${\mathrm{P}\mathrm{H}}_n({\omega }_m)$ for $n<m$.