Note as to size-minimal hypercompletly separating systems

If $S$ is a non-empty finite set, $\mathrm{\mid }S\mathrm{\mid }=s$, then a system $\mathcal{A}$ of subsets of $S$ is a size-minimal hypercompletely separable system (i.e., for every $a\in S$ there are $A,B\in \mathcal{A}$ such that $A\cap B=\{a\}$) if and only if $\mathrm{\mid }\mathcal{A}\mathrm{\mid }=\lceil \frac{1+\sqrt{8s+1}}{2}\rceil$.