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 <(eta)over bar> = (eta(alpha): alpha < lambda) which is a (lambda, J)-sequence for (I) over bar = (I(i) : i < delta), a sequence of ideals". This can be thought of as a generalization to Luzin sets and Sierpinski sets, but for the product Pi(i<delta) dom(I(i)), the existence proofs are related to pcf. The second theme is when does a Boolean algebra B have a free caliber h (i.e., if X subset of or equal to B and /X/ = lambda, then for some Y subset of or equal to X with /Y/ = lambda and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product of free Boolean algebras, and kappa-cc Boolean algebras. A central case is lambda = (beth(omega))(+), or more generally, lambda = mu(+) for mu strong limit singular of "small" cofinality. A second one is mu = mu(<K) < 2(mu); the main case is lambda 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. (C) 1999 Published by Elsevier Science B.V. All rights reserved.