Galvin's Conjecture and Weakly Precipitous Ideals

TL;DR

The paper advances Galvin's Conjecture by showing that weak precipitousness of normal ideals on $\omega_1$ can be obtained under weaker large-cardinal hypotheses (notably Ramsey cardinals or suitable generic embeddings), via a two-player game that operates on small-rank objects. It then integrates generic embeddings, saturation notions for colorings, and topology to produce an everywhere weakly precipitous, $\langle i,j\rangle$-winning framework for appropriate colorings. Using these ingredients, the Raghavan–Todorčević construction is adapted to yield, for an uncountable $X\subseteq\mathbb{R}$ and a finite coloring $c:[X]^2\to l$, a subset $Y$ homeomorphic to $\mathbb{Q}$ on which $c$ takes at most two colors. The result broadens the known consistency strength needed for Galvin's Conjecture and clarifies the role of embeddings, forcing, and topological methods in Ramsey-type partition relations on uncountable sets.

Abstract

We investigate a combinatorial game on $ω_1$ and show that mild large cardinal assumptions imply that every normal ideal on $ω_1$ satisfies a weak version of precipitousness. As an application, we show that that the Raghavan-Todorčević 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.

Theorems & Definitions (34)

• Theorem 1: Raghavan-Todorčević dilipstevo
• Definition 2.1
• Proposition 2.2
• proof
• Definition 3.1
• Definition 3.2
• Theorem 2
• proof
• Theorem 3
• proof