Note as to size-minimal hypercompletly separating systems

TL;DR

This paper characterizes the minimal size of systems of subsets that can uniquely separate elements of a finite set, providing an exact formula for their minimal cardinality.

Contribution

It establishes a precise formula for the size of minimal hypercompletely separable systems for finite sets.

Findings

The minimal size of such systems is given by the ceiling of (1 + sqrt(8s+1))/2.

The formula applies to all finite sets with size s.

The result characterizes the structure of size-minimal separating systems.

Abstract

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