Splitting number and the core model

Abstract

We can generalize the definition of {\it splitting number } $s(\kappa )$ for $\kappa$ uncountable regular: $s(\kappa )=min\{ |\Cal S|:\Cal S\subset \Cal P(\kappa ) \forall a\in \kappa ^\kappa \exists b\in \Cal S |a\cap b|=|a\setminus b|=\kappa\}$ However,$\exists \kappa>\aleph_0$ $s(\kappa )>\kappa ^+$ becomes a considerable hypothesis,shown consistent from a supercompact.We show that it implies inner models of $\exists \alpha :o(\alpha )=\alpha ^{++}$