A Version of $\kappa$-Miller Forcing

Abstract

Let $\kappa$ be an uncountable cardinal such that $2^{<\kappa} = \kappa$ or just ${\rm cf}(\kappa) > \omega$, $2^{2^{<\kappa}}= 2^\kappa$, and $([\kappa]^\kappa, \supseteq)$ collapses $2^\kappa$ to $\omega$. We show under these assumptions the $\kappa$-Miller forcing with club many splitting nodes collapses $2^\kappa$ to $\omega$ and adds a $\kappa$-Cohen real.