Given a finite n-element set X, a family of subsets F⊂2X is said to separate X if any two elements of X are separated by at least one member of F. It is shown that if |F|>2n−1, then one can select ⌈logn⌉+1 members of F that separate X. If |F|≥α2n for some 0<α<1/2, then logn+O(log1αloglog1α) members of F are always sufficient to separate all pairs of elements of X that are separated by some member of 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 Rd by convex sets are also considered.
- Erdős–Szekeres theorem
- Search theory
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics