  We strengthen the property $\Delta$ of a function $f:[\omega_2]^2\rightarrow
[\omega_2]^{\leq \omega}$ considered by Baumgartner and Shelah. This allows us
to consider new types of amalgamations in the forcing used by Rabus, Juh\'asz
and Soukup to construct thin-very tall compact scattered spaces. We
consistently obtain spaces $K$ as above where $K^n$ is hereditarily separable
for each $n\in\N$. This serves as a counterexample concerning cardinal
functions on compact spaces as well as having some applications in Banach
spaces: the Banach space $C(K)$ is an Asplund space of density $\aleph_2$ which
has no Fr\'echet smooth renorming, nor an uncountable biorthogonal system.
