On the Splitting Number at Regular Cardinals

Abstract

Let $\kappa$,$\lambda$ be regular uncountable cardinals such that $\lambda >{\kappa }^{+}$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(\kappa )=\lambda$ starting from a ground model in which $o(\kappa )=\lambda$ and prove that assuming $\neg 0^{\P }$, $s(\kappa )=\lambda$ implies that $o(\kappa )\ge \lambda$ in the core model.