Cellularity of free products of Boolean algebras (or topologies)

The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if mu is a strong limit singular cardinal, theta = (2(cf(mu)))(+) and 2(mu) = mu (+) then there are Boolean algebras B(1),B(2) such that c(B(1)) = mu, c(B(2)) < <theta> but c(B(1)*B(2)) = mu (+). Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if B is a ccc Boolean algebra and mu (beth omega) less than or equal to lambda = cf(lambda) less than or equal to 2(mu) then B satisfies the lambda -Knaster condition (using the "revised GCH theorem").