Introduction

The term “square” refers not just to one but to an entire family of combinatorial principles. The strongest is denoted by “◻” or by “Global ◻,” and there are many interesting weakenings of this notion. Before introducing any particular square principle, we provide some motivating applications. In this section, the term “square” will serve as a generic term for “some particular square principle.”

• Jensen introduced square principles based on work regarding the fine structure of L. In his first application, he showed that, in L, there exist κ-Suslin trees for every uncountable cardinal κ that is not weakly compact.

• 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. […]