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 μ is a strong limit singular cardinal, ${\displaystyle \theta ={\left(2^{cf(\mu )}\right)}^{+}}$ and ${\displaystyle 2^{\mu }={\mu }^{+}}$ then there are Boolean algebras ${\displaystyle B_1,B_2}$ such that ${\displaystyle c\left(B_1\right)=\mu ,c\left(B_2\right)<\theta butc(B_1\cdot B_2)={\mu }^{+}}$. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if ${\displaystyle B}$ is a ccc Boolean algebra and ${\displaystyle {\mu }^{{\beth }_{\omega }}\le \lambda =cf(\lambda )\le 2^{\mu }}$ then ${\displaystyle B}$ satisfies the λ-Knaster condition (using the "revised GCH theorem").