Separation with restricted families of sets

نویسندگان

  • Zsolt Lángi
  • Márton Naszódi
  • János Pach
  • Gábor Tardos
  • Géza Tóth
چکیده

Given a finite n-element set X , a family of subsets F ⊂ 2 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| > 2, then one can select ⌈logn⌉ + 1 members of F that separate X . If |F| ≥ α2 for some 0 < α < 1/2, then logn+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 R by convex sets are also considered.

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عنوان ژورنال:
  • J. Comb. Theory, Ser. A

دوره 144  شماره 

صفحات  -

تاریخ انتشار 2016