On independent subsets of Boolean algebras (Q1312170)

The authors give a partial solution to a problem of the reviewer: Let \(\kappa\) be a regular cardinal and let \(\{L_ i: i<\kappa\}\) be a family of linear orderings with first element such that no \(L_ i\) contains a strictly decreasing sequence of length \(\kappa^ +\). Then the independence of the direct product \(\prod_{i<\kappa}\text{Intalg}(L_ i)\) is at most \(2^ \kappa\). More recently (January 1993), S. Shelah solved the problem completely by deriving the same conclusion with no special assumption on the linear orders.