Set-theoretic study of measures

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The crucial fact that separates the uncountable from the countable is that the notion of stationariness can be defined for subsets of uncountable sets. Roughly speaking, stationariness is a combinatorial counterpart of having a positive measure.The combinatorics of P_κλ investigates that notion in its greatest generality. Here κ denotes a regular uncountable cardinal and λ is a cardinal at least κ.When κ, is a λ-supercompact cardinal, a normal ultrafilter on P_κλ is an important object to study. It has been an open question whether P_κλ carries a normal ultrafilter with property χ if κ, is λ-supercompact. In [3] we constructed one with a property slightly weaker than this. The proof shows in fact that the Generalized Continuum Hypothesis implies the existence of an ultrafilter with property χ. The latter result should be useful in relative consistency results.In joint work with Shelah [2] we showed that the principle of Stationary Reflection in P_κλ fails if κ is greater than ω1 and λ is greater than κ. Moreover we have made progress in applying the principle of Stationary Reflection. In [4] we gave two extensions of Shelah's result that the Diamond principle holds in P_<ω1>λ In [5] we extended the well-known result of Foreman-Magidor-Shelah that the club filter on P_<ω1>λ can be presaturated.