Fat subsets of P kappa (lambda)

Abstract

For a subset of a cardinal greater than ω1, fatness is strictly stronger than stationarity and strictly weaker than being closed unbounded. For many regular cardinals, being fat is a sufficient condition for having a closed unbounded subset in some generic extension. In this work we characterize fatness for subsets of Pκ(λ). We prove that for many regular cardinals κ and λ, a fat subset of Pκ(λ) obtains a closed unbounded subset in a cardinal-preserving generic extension. Additionally, we work out the conflict produced by two different definitions of fat subset of a cardinal, and introduce a novel (not model-theoretic) proof technique for adding a closed unbounded subset to a fat subset of a cardinal.