In this series of two talks, we will discuss the
(non)existence of certain types of gaps in the Boolean algebra $P(\omega)/fin$.
Specifically, we will demonstrate that there are no $(\kappa,\lambda)$-gaps for $\kappa$,
$\lambda\le\omega$, and then construct a Hausdorff gap, i.e., an $(\omega_1, \omega_1)$-gap.
Additionally, we will present a characterisation of the bounding number
in terms of a Rothberger gap. Finally as an application, we will use the
Hausdorff gap to demonstrate the existence of a universal measure zero
set. All results mentioned above will be proven within ZFC. In the end,
we will also briefly discuss the notion of indestructible gaps and
describe the situation under PFA.