Separating weak partial square principles

We introduce the weak partial square principles pλ,κ and p λ,<κ, which combine the ideas of a weak square sequence and a partial square sequence. We construct models in which weak partial square principles fail. The main result of the paper is that λ,κ does not imply λ,<κ. Let λ be a regular uncountable cardinal and let κ ≤ λ be a nonzero cardinal. Recall the square principle λ,κ (respectively λ,<κ), which asserts the existence of a sequence 〈Cα : α ≤ λ, α limit〉 satisfying that every Cα is a nonempty family of no more than κ many (respectively less than κ many) club subsets of α, each with order type at most λ, such that for all c ∈ Cα and γ ∈ lim(c), c ∩ γ ∈ Cγ . The principle λ,1 is the same as the square principle λ, and λ,λ is the same as the weak square principle λ; both of these principles are due to Jensen [3]. The other principles were introduced by Schimmerling [7]. Jensen [2] proved that for nonzero cardinals μ < κ ≤ λ, λ,κ does not imply λ,μ. Now consider an uncountable cardinal λ and a regular cardinal μ ≤ λ. A set S ⊆ λ∩cof(μ) is said to carry a partial square if there exists a sequence 〈cα : α ∈ S〉 such that each cα is a club subset of α with order type μ, and if γ is a common limit point of cα and cβ , then cα ∩γ = cβ ∩γ. Shelah [8] proved that if λ is regular, then there are densely many stationary subsets of λ ∩ cof(<λ) which carry a partial square. As a consequence, consistency results on partial squares usually focus on stationary subsets of λ ∩ cof(λ). In this paper we introduce principles which combine the weak square principles of Jensen and Schimmerling with the idea of a partial square sequence. Let λ be an uncountable cardinal, let μ ≤ λ be regular, and let κ ≤ λ be a nonzero cardinal. A set S ⊆ λ ∩ cof(μ) is said to carry a κ-weak partial square (respectively <κ-weak partial square) if there exists a sequence 〈cα : α ∈ S〉 satisfying: (1) for all α ∈ S, cα is a club subset of α with order type μ; (2) for all β < λ, the set {cα ∩ β : α ∈ S, β ∈ lim(cα)} has size less than or equal to κ (respectively less than κ). For a regular uncountable cardinal λ, let pλ,κ (respectively p λ,<κ) be the statement that there exists a stationary subset of λ ∩ cof(λ) which carries a κ-weak partial square (respectively <κ-weak partial square). The goal of the paper is to construct models in which weak partial square principles fail, and to separate the different weak partial square principles. For this purpose we prove in Sections 1 and 3 two propositions which show that certain kinds of forcing posets cannot add threads to sequences of families of clubs. In