A note on hyperseparating set systems

TL;DR

This paper investigates minimal sizes of specialized set systems called hyperseparating and hyperseparating systems, extending recent results and providing new bounds for these combinatorial structures.

Contribution

It generalizes recent results for 2-completely hyperseparating systems and determines minimal sizes for 2-hyperseparating systems, advancing understanding of these combinatorial set systems.

Findings

Determined the minimum size of $k$-completely hyperseparating set systems for general $k$.

Established the minimum size of 2-hyperseparating set systems on $n$-element sets.

Extended recent results to broader classes of hyperseparating set systems.

Abstract

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 Bat\'ikov\'a, Kepka, and Nem\u{e}c. 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.