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)