Improved Bounds for the Number of (≤ k)-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn

  • József Balogh
  • Gelasio Salazar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)


We use circular sequences to give an improved lower bound on the minimum number of (≤ k)-sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number □(S) of convex quadrilaterals determined by the points in S is at least \(0.37553\binom{n}{4} + O(n^3)\). This in turn implies that the rectilinear crossing number \(\overline{\hbox{\rm cr}}(K_n)\) of the complete graph K n is at least \(0.37553\binom{n}{4} + O(n^3)\). These improved bounds refine results recently obtained by Ábrego and Fernández-Merchant, and by Lovász, Vesztergombi, Wagner and Welzl.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • József Balogh
    • 1
  • Gelasio Salazar
    • 2
  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.IICO-UASLPSan Luis PotosiMexico

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