Quadruples and cubes

Source abstract

We prove, in $\mathsf{ZFC}$, that the $λ$-terraced cube relation fails whenever $λ$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $λ$. If $λ$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.