Given two Banach spaces $Z$ and $X$, can we find a Banach space $Y$ that contains $X$ as an uncomplemented subspace and $Y/X = Z$? We will mention two instances of this problem connected to set theoretic questions. When $X = c_0$ and $Z=C(K)$ is a space of continuous functions on a nonmetric compactum, the answer may be negative under $MA_{\omega_1}$ but it is always positive under CH (joint work with W. Marciszewski and G. Plebanek). When $X = \ell_\infty/c_0$ and $Z=c_0(\mathfrak{c})$, the answer is positive provided splitting chains exist in $\mathcal{P}(\omega)/fin$ (joint work with P. Borodulin-Nadzieja, F. Cabello, D. Chodounsk\'{y} and O. Guzm\'{a}n)