Combinatorial separation axioms and cardinal invariants

Forcing Axioms and Cardinal Arithmetic

We survey some recent results on the impact of strong forcing axioms such as the Proper Forcing Axiom PFA and Martin’s Maximum MM on cardinal arithmetic. We concentrate on three combinatorial principles which follow from strong forcing axioms: stationary set reflection, Moore’s Mapping Reflection Principle MRP and the P-ideal dichotomy introduced by Abraham and Todorčević which play the key rol...

Cardinal invariants and compactifications

We prove that every compact space X is a Čech-Stone compactification of a normal subspace of cardinality at most d(X)t(X), and some facts about cardinal invariants of compact spaces.

Tameness from Large Cardinal Axioms

We show that Shelah’s Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply t...

On bD-Sets and Associated Separation Axioms

Isolating Cardinal Invariants

There is an optimal way of increasing certain cardinal invariants of the