, Volume 7, Issue 1, pp 109-123

An upper bound on the number of planarK-sets

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Abstract

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.

The research of J. Pach was supported in part by NSF Grant CCR-8901484 and by Grant OTKA-1418 from the Hungarian Foundation for Scientific Research. The research of W. Steiger and E. Szemerédi was supported in part by NSF Grant CCR-8902522. All authors express gratitude to the NSF DIMACS Center at Rutgers.