Splitting number and the core model

Abstract

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