Dense ideals

TL;DR

This paper proves the consistency of dense ideals on successor cardinals and explores their implications for ultrafilters, graph coloring, and partition hypotheses, advancing understanding of large cardinal assumptions in set theory.

Contribution

It establishes the simultaneous existence of dense ideals on all successors of regular cardinals and derives several combinatorial and ultrafilter results from this construction.

Findings

Existence of a $\sigma$-complete, $\aleph_1$-dense ideal on $\aleph_{n+1}$ for all $n<\omega$

Consistency of irregular ultrafilters on $\omega_n$

Validation of the Foreman-Laver reflection property for graph chromatic numbers

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 $\sigma$-complete, $\aleph_1$-dense ideal on $\aleph_{n+1}$ 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 $\aleph_\omega$, and the simultaneous consistency of the partition hypotheses $\mathrm{PH}_n(\omega_m)$ for $n < m$.