On the consistency of local and global versions of chang’s conjecture

We show that for many pairs of infinite cardinals κ > μ⁺ > μ, (κ⁺, κ) ↠ (μ⁺, μ) is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal, that (κ⁺, κ) ↠ (μ⁺, μ) for every successor cardinal κ and every μ < κ, answering a question of Foreman.