Weak Square Sequences and Special Aronszajn Trees

A classical theorem of set theory is the equivalence of the weak square principle μ 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. Recall the weak square principle μ for an infinite cardinal μ, which asserts the existence of a sequence 〈Cα : α ∈ μ ∩ Lim〉 satisfying: (1) for all c ∈ Cα, c is a club subset of α with order type at most μ; (2) |Cα| ≤ μ; (3) for all c ∈ Cα, if β ∈ lim(c) then c ∩ β ∈ Cβ . For a regular uncountable cardinal κ, a tree (T,ω) denote the class of limit ordinals of uncountable cofinality. For a set of ordinals a, ot(a) is the order type of a, and lim(a) is the set of ordinals β such that sup(a ∩ β) = β. 2010 Mathematical Subject Classification: 03E05.