Separation with restricted families of sets

Lángi, Zsolt and Naszódi, Márton and Pach, J. and Tardos, G. and Tóth, G. (2016) Separation with restricted families of sets. JOURNAL OF COMBINATORIAL THEORY SERIES A, 144. pp. 292-305. ISSN 0097-3165

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Abstract

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 [log n] + 1 members of F that separate X. If |F| ≥ α2n for some 0 < α < 1/2, then log n + O(log 1 α log log 1 α ) 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.

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