Separation with restricted families of sets

TL;DR

This paper investigates the minimum size of set families needed to separate all pairs of elements in a finite set, providing bounds based on the family's size and extending results to multiple sets, VC-dimension, and convex sets.

Contribution

It establishes new bounds on the number of family members required for separation, generalizes to multiple sets, and explores related geometric and VC-dimension scenarios.

Findings

Families larger than 2^{n-1} require only log n + 1 members for separation.

For families of size at least α 2^n, about log n + O(log(1/α) log log(1/α)) members suffice.

Results extend to simultaneous separation, VC-dimension bounded families, and convex set separations.

Abstract

Given a finite $n$-element set $X$, a family of subsets ${\mathcal F}\subset 2^X$ is said to separate $X$ if any two elements of $X$ are separated by at least one member of $\mathcal F$. It is shown that if $|\mathcal F|>2^{n-1}$, then one can select $\lceil\log n\rceil+1$ members of $\mathcal F$ that separate $X$. If $|\mathcal F|\ge \alpha 2^n$ for some $0<\alpha<1/2$, then $\log n+O(\log\frac1{\alpha}\log\log\frac1{\alpha})$ members of $\mathcal F$ are always sufficient to separate all pairs of elements of $X$ that are separated by some member of $\mathcal F$. This result is generalized to simultaneous separation in several sets. Analogous questions on separation by families of bounded Vapnik-Chervonenkis dimension and separation of point sets in ${\mathbb{R}}^d$ by convex sets are also considered.