A note on hyperseparating set systems

We say that a set system $\mathcal{F}$ is $k$-completely hyperseparating if for any vertex $v$, there are at most $k$ sets in $\mathcal{F}$ with intersection $\{v\}$. We determine the minimum size of such set systems on an $n$-element underlying set, generalizing a very recent result for $k=2$ by Batíková, Kepka, and Nemĕc. We say that $\mathcal{F}$ is $k$-hyperseparating if for any vertex $v$, there are at most $k$ sets in $\mathcal{F}$ such that no other vertex is contained by exactly the same sets out of these $k$ sets. We determine the minimum size of $2$-hyperseparating set systems on an $n$-element underlying set.