Discrete & Computational Geometry

, Volume 7, Issue 2, pp 109-123

First online:

An upper bound on the number of planarK-sets

  • János PachAffiliated withMathematical Institute, Hungarian Academy of ScienceCourant Institute, New York University
  • , William SteigerAffiliated withComputer Science Department, Rutgers University
  • , Endre SzemerédiAffiliated withMathematical Institute, Hungarian Academy of ScienceComputer Science Department, Rutgers University

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Given a setS ofn points, a subsetX of sizek is called ak-set if there is a hyperplane Π that separatesX fromS−X. We prove thatO(nk/log*k) is an upper bound for the number ofk-sets in the plane, thus improving the previous bound of Erdös, Lovász, Simmons, and Strauss by a factor of log*k.