Galvin's Conjecture and Weakly Precipitous Ideals

TL;DR

This paper explores the relationship between large cardinal assumptions and properties of ideals on , demonstrating that weaker assumptions suffice for certain combinatorial and set-theoretic results related to Galvin's conjecture.

Contribution

It shows that weakly precipitous ideals on  can be obtained under milder large cardinal assumptions, extending the applicability of previous results.

Findings

Weakly precipitous ideals exist under mild large cardinal assumptions.

The proof of Galvin's conjecture can be adapted to weaker assumptions.

Connections between ideals on  and large cardinal hypotheses are clarified.

Abstract

We investigate a combinatorial game on $\omega_1$ and show that mild large cardinal assumptions imply that every normal ideal on $\omega_1$ satisfies a weak version of precipitousness. As an application, we show that that the Raghavan-Todor\v{c}evi\'{c} proof of a longstanding conjecture of Galvin (done assuming the existence of a Woodin cardinal) can be pushed through under much weaker large cardinal assumptions.