Consistency of Chang’s Conjecture for quadruples (ω4, ω3, ω2, ω1) → (ω3, ω2, ω1, ω)

Establish the relative consistency (e.g., from large cardinal hypotheses) of the quadruple instance of Chang’s Conjecture asserting (ω4, ω3, ω2, ω1) → (ω3, ω2, ω1, ω), i.e., determine whether there exists a generic extension in which every structure of size ω4 with two unary predicates of sizes ω3, ω2, ω1 respectively has an elementary substructure of size ω3, ω2, ω1, ω, respectively.

Background

The paper surveys known results on Chang’s Conjecture, highlighting Foreman’s construction (from a 2-huge cardinal) of a model where the triple instance (ω3, ω2, ω1) → (ω2, ω1, ω) holds. Extending such results to quadruples has encountered technical obstacles, notably the problem of “ghost coordinates.”

The authors present a simplified construction for triples using projections associated with term forcing and Easton collapses, but note that extending this approach to the quadruple case presents unresolved difficulties (see the later discussion of ghost coordinates).