Special subsets of {}^{cf(mu)} mu, Boolean algebras and Maharam measure algebras

The original theme of the paper is the existence proof of ``there is < eta_alpha : alpha < lambda > which is a (lambda,J)-sequence for < I_i:i, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and Sierpinski sets, but for the product prod_{i< delta} Dom(I_i), the existence proofs are related to pcf. The second theme is when does a Boolean algebra B has free caliber lambda (i.e. if X subseteq B and |X|= lambda, then for some Y subseteq 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 lambda = (beth_omega)^+ or more generally, lambda = mu^+ for mu strong limit singular of ``small'' cofinality. A second one is mu = mu^{< kappa}< lambda < 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.