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We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of $P_{\kappa }\left(X\right)$, where κ ⊆ X. Suppose μ < κ, ${\mu }^{<\mu }=\mu$, and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in ${\left[H\left(\lambda \right)\right]}^{\mu ⁺}$ which are internally club but not internally approachable.

A classical theorem of set theory is the equivalence of the weak square principle ${\square }_{\mu }*$ with the existence of a special Aronszajn tree on μ⁺. We introduce the notion of a weak square sequence on any regular uncountable cardinal, and prove that the equivalence between weak square sequences and special Aronszajn trees holds in general.

We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on ω₂ using finite conditions.