A Version of $κ$-Miller Forcing

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