Chang's conjecture may fail at supercompact cardinals (submitted)

Abstract

We prove a revised version of Laver's indestructibility theorem which slightly improves over the classical result. An application yields the consistency of $(κ^+,κ)\notcc(\aleph\_1,\aleph\_0)$ when $\kappa$ is supercompact. The actual proofs show that $\omega \_1$-regressive Kurepa-trees are consistent above a supercompact cardinal even though $\mathrm{MM}$ destroys them on all regular cardinals. This rather paradoxical fact contradicts the common intuition.