Jonsson and Magidor filters
Omer Ben-Neria, Shimon Garti
TL;DR
The paper investigates filter-enhanced square-bracket partition relations focusing on Jónssonity and Magidority in both AC and AD settings. It develops lifting lemmas for Jónsson filters at singular limits via Rowbottom-type arguments and defines a Jónsson-degree $\alpha_J(\mathscr{F})$ to compare filter strength. In the Magidor realm, it analyzes Kleinberg sequences under AD, showing non-Magidority at finite levels but Magidority at the limit when the strong partition cardinal is large enough, and establishes a corresponding $\omega$-Magidority for $\aleph_1$-based cases. Finally, it demonstrates that Magidor filters can be forced over singular cardinals in AD models, first by locating Magidor ultrafilters below $\Theta$ and then by Prikry-type forcing to produce a singular cardinal carrying a Magidor filter in the extension, highlighting preservation issues for AD. Collectively, the results connect determinacy-based partition properties with forcing techniques to realize Magidority and Jónssonity in new contexts, and raise open questions about strong Magidor filters and the limits of forcing in AD environments.
Abstract
We study the filter versions of square bracket partition relations, focusing on Jonssonicity and Magidority. We show that the singular cardinals in a Kleinberg sequence above some strong partition cardinal are not Magidor, but the limit of the sequence is Magidor. This is done under AD. We also force over a model of AD to obtain a singular cardinal carrying a Magidor filter.