9 papers
A Topological Rainbow Ramsey Theorem
Hannes Jakob, Jing Zhang
We show that it is consistent relative to the existence of suitable large cardinals that for any countable-to-one coloring , there exists a closed subset $A\subs…
On Shelah's Approachability Ideal
Hannes Jakob, Alejandro Poveda
We solve a long-standing open problem of Shelah regarding the \emph{Approachability Ideal} . Given a singular cardinal , a regular cardinal $μ\in (\mathrm{cf}(γ),…
Total Failure of Approachability at Successors of Singulars of Countable Cofinality
Hannes Jakob
Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal of countable cofinality and every regular uncountable …
Failure of Approachability at the Successor of the first Singular for any Cofinality
Hannes Jakob, Maxwell Levine
We solve two long-standing open problems regarding the combinatorics of . We answer a question of Shelah by showing that it is consistent for any that $\mat…
On Friedman's Property
Hannes Jakob
We define forcing orders which add witnesses to the failure of various forms of Friedman's Property. These posets behave similarly to the forcing order adding a nonreflecting stati…
Distinguishing Internally Club and Approachable on an Infinite Interval
Hannes Jakob, Maxwell Levine
Krueger showed that PFA implies that for all regular , there are stationarily many that are internally club but not internally approachable. From…
Cascading Variants of Internal Approachability
Hannes Jakob
We construct models in which there are stationarily many structures that exhibit different variants of internal approachability at different levels. This answers a question of Fore…
Slender Trees and the Approximation Property
Hannes Jakob
We obtain a relatively simple criterion for when a forcing has the -approximation property, generalizing a result of Unger. Afterwards we apply this criterion to construct…
Disjoint Stationary Sequences on an Interval of Cardinals
Hannes Jakob
We answer a question of Krueger by obtaining -- from countably many Mahlo cardinals -- a model where there is a disjoint stationary sequence on for every . In…