B}})$, while $(\omega, (2^\theta)^+)\notin{{\rm Depth}}_{\rm Sr}({{\mathbb B}})$ (see Monk [@M2 Problem 14]; we refer the reader to Chapter 4 of Monk’s book [@M2] for the motivation and background of this problem). Here we will partially answer this question, showing that it is consistent that there is such ${{\mathbb B}}$ and $\theta$. The question if that can be done in ZFC remains open.

Our consistency result is obtained by forcing, and the construction of the required forcing notion is interesting [*per se*]{}. We use the method of [*historic forcing*]{} which was first applied in Shelah and Stanley [@ShSt:258]. The reader familiar with [@ShSt:258] will notice several correspondences between the construction here and the method used there. However, we do not relay on that paper and our presentation here is self-contained.

Let us describe how our historic forcing notion is built. So, we fix two (regular) cardinals $\theta,\lambda$ and our aim is to force a Boolean algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$ such that $|{\dot{{{\mathbb B}}}^\theta_\lambda}|=\lambda^+$ and for every subalgebra ${{\mathbb B}}\subseteq{\dot{{{\mathbb B}}}^\theta_\lambda}$ of size $\lambda^+$ we have ${{\rm Depth}}({{\mathbb B}})=\theta$. The algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$ will be generated by $\langle x_i:i\in\dot{U}\rangle$ for some set $\dot{U}\subseteq\lambda^+$. A condition $p$ will be an approximation to the algebra 