On the strengths and weaknesses of weak squares

• Let T be a countable theory with a distinguished predicate R. A model of T is said to be of type (λ,μ) if the cardinality of the model is λ and the cardinality of the model’s interpretation of R is μ. For cardinals α,β, γ, and δ, (α,β) → (γ, δ) is the assertion that for every countable theory T , if T has a model of type (α,β), then it has a model of type (γ, δ). Chang showed that under GCH, (א1,א0) → (κ+, κ) holds for every regular cardinal κ. Jensen later showed that under GCH+square, (א1,א0) → (κ+, κ) holds for every singular cardinal κ as well.