Special subsets of^cf(μ) μ Boolean algebras and Maharam measure algebras

The original theme of the paper is the existence proof of "there is η̄ = 〈η_α: α < λ〉 which is a (λ, J)-sequence for Ī = 〈I_i: i < δ〉, a sequence of ideals". This can be thought of as a generalization to Luzin sets and Sierpinski sets, but for the product Π_i<δ dom(I_i), the existence proofs are related to pcf. The second theme is when does a Boolean algebra B have a free caliber λ(i.e., if X ⊆ B and |X| = λ, then for some Y ⊆ X with |Y| = λ and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product of free Boolean algebras, and κ-cc Boolean algebras. A central case is λ = (_ω)⁺, or more generally, λ = μ⁺ for μ strong limit singular of "small" cofinality. A second one is μ = μ^<κ <λ <2μ; the main case is λ regular but we also have things to say on the singular case. Lastly, we deal with ultraproducts of Boolean algebras in relation to irr(-) and s(-) etc.