Chang’s Conjecture May Fail at Supercompact Cardinals

We prove a revised version of Laver’s indestructibility theorem which slightly improves over the classical result. An application yields the consistency of (κ, κ) ։/ (א1,א0) when κ is supercompact. The actual proofs show that ω1-regressive Kurepatrees are consistent above a supercompact cardinal even though MM destroys them on all regular cardinals. This rather paradoxical fact contradicts the common intuition.