Sufficiency of three-stage Easton collapses and existence of a projection for forcing the triple Chang property

Determine whether the three-stage Easton collapse forcing E(µ, κ) ∗ E(κ, j(κ)) ∗ E(j(κ), j2(κ)) alone forces the triple instance of Chang’s Conjecture (µ+3, µ+2, µ+) → (µ+2, µ+, µ); and ascertain whether there exists a projection from E(µ, j(κ)) ∗ E(j(κ), j2(κ)) to E(µ, κ) ∗ E(κ, j(κ)) ∗ E(j(κ), j2(κ)).

Background

The main theorem achieves the triple instance of Chang’s Conjecture by employing a more elaborate forcing construction that interleaves iteration and product via the ⋆ operation, together with Easton collapses. This refinement is introduced because it yields the projections and master conditions necessary for the argument.

The authors explicitly state that it is unknown whether the simpler composition of three Easton collapses suffices to force the desired triple instance or whether there is a direct projection from E(µ, j(κ)) ∗ E(j(κ), j2(κ)) to the three-step product E(µ, κ) ∗ E(κ, j(κ)) ∗ E(j(κ), j2(κ)). This uncertainty motivates their introduction of the ⋆ operation and the accompanying projection.