Abstract
It is known that if $\kappa $ is inaccessible in $V$, and $W$ is an outer model of $V$ such that $(\kappa ^+)^V = (\kappa ^+)^W$, and $\mathop{\rm cf} ^W \!(\kappa ) = \omega $, then $\square _{\kappa ,\omega }$ holds in $W$. Many strengthenings
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