[Paper Review] Galvin's property at large cardinals and an application to partition calculus

Tom Benhamou, Shimon Garti+1 more|arXiv (Cornell University)|Jul 15, 2022

Advanced Topology and Set Theory5 citations

TL;DR

This paper investigates the consistency of Galvin’s property for ultrafilters on large cardinals, particularly supercompact and measurable cardinals, using forcing and iteration techniques. It shows that it is consistent for a supercompact cardinal to carry both non-Galvin and Galvin ultrafilters, and applies this to obtain new instances of the partition relation $ rightarrow( heta, heta+1)^2$ under weak cardinal arithmetic assumptions, especially in models of determinacy.

ABSTRACT

In the first part of this paper, we explore the possibility for a very large cardinal $κ$ to carry a $κ$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $κ$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $κ$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. Finally, we apply this property to obtain consistently new instances of the classical problem in partition calculus $λ ightarrow(λ,ω+1)^2$.

Motivation & Objective

• To investigate whether very large cardinals, such as supercompact or $C^{(n)}$-extendible cardinals, can carry $ heta$-complete ultrafilters that fail Galvin’s property.
• To determine the consistency strength of the statement that every ground model $ heta$-complete ultrafilter extends to a non-Galvin one.
• To explore the opposite scenario: whether every $ heta$-complete ultrafilter can extend to a $P$-point, hence a Galvin ultrafilter.
• To apply these forcing constructions to partition calculus, specifically to obtain new instances of $ heta ightarrow( heta, heta+1)^2$ without assuming strong cardinal arithmetic.
• To analyze the role of Galvin’s property in models of determinacy, particularly $V=L(bR)$ and under $f{AD}$, to derive new positive partition relations.

Proposed method

• Using a generic extension via forcing to construct a model where a supercompact cardinal $ heta$ carries a $ heta$-complete ultrafilter that fails Galvin’s property.
• Applying iterations of Generalized Mathias forcing to extend any $ heta$-complete filter to a $ heta$-complete ultrafilter satisfying Galvin’s property.
• Adapting a sophisticated iteration technique from Gitik and Shelah to force every ground model $ heta$-complete ultrafilter to extend to a $P$-point ultrafilter.
• Constructing a 2-coloring $c:[ heta]^2 o 2$ to witness the negative partition relation $ heta rightarrow( heta, heta+1)^2$ under specific assumptions on Galvin’s property failure.
• Leveraging determinacy axioms and the structure of $f{L}(bR)$ to show that under $f{AD}+V=L(bR)$, certain singular cardinals $ heta$ satisfy $ heta ightarrow( heta, heta+1)^2$.
• Using the club filter and normal measures on measurable limits of cardinals to verify the assumptions of the main partition calculus theorem.

Experimental results

Research questions

• RQ1 Is it consistent that a supercompact cardinal $ heta$ carries a $ heta$-complete ultrafilter that fails Galvin’s property?
• RQ2 Can every $ heta$-complete ultrafilter on a supercompact $ heta$ extend to a $P$-point ultrafilter, hence satisfying Galvin’s property?
• RQ3 What is the consistency strength of the partition relation $ heta ightarrow( heta, heta+1)^2$ when $2^ heta eq heta^+$?
• RQ4 Under $f{AD}$, does $ heta ightarrow( heta, heta+1)^2$ hold for singular $ heta$ of uncountable cofinality?
• RQ5 What conditions on Galvin’s property are necessary to force $ heta rightarrow( heta, heta+1)^2$?

Key findings

• It is consistent that a supercompact cardinal $ heta$ carries a $ heta$-complete ultrafilter that fails Galvin’s property, using a generic extension with appropriate forcing.
• It is also consistent that every $ heta$-complete ultrafilter on a supercompact $ heta$ extends to a $P$-point ultrafilter, hence satisfies Galvin’s property.
• The iteration of Generalized Mathias forcing ensures that every $ heta$-complete filter extends to a $ heta$-complete ultrafilter satisfying Galvin’s property.
• A new instance of the partition relation $ heta ightarrow( heta, heta+1)^2$ is obtained by replacing the cardinal arithmetic assumption $2^ heta < heta^+$ with an instance of Galvin’s property.
• Under $f{AD}+V=L(bR)$, if $ heta$ is a limit of regular cardinals with $ heta = ext{cf}( heta) > heta$, then $ heta ightarrow( heta, heta+1)^2$ holds.
• In particular, $ heta = eth_{ heta}$ and $ heta = eth_{ heta+1}$ are consistent with $ heta ightarrow( heta, heta+1)^2$ under $f{AD}$, even when $2^ heta > heta$.