Separation with restricted families of sets

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 $\log n+O(\log \frac{1}{\alpha }\log \log \frac{1}{\alpha })$ 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.

13 pages