The number of small semispaces of a finite set of points in the plane
Journal of Combinatorial Theory Series A
More on k-sets of finite sets in the plane
Discrete & Computational Geometry
Results on k-sets and j-facets via continuous motion
Proceedings of the fourteenth annual symposium on Computational geometry
On the crossing number of complete graphs
Proceedings of the eighteenth annual symposium on Computational geometry
On the rectilinear crossing number of complete graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Crossing numbers of random graphs
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
Abstract order type extension and new results on the rectilinear crossing number
Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
New results on lower bounds for the number of (≤k)-facets
European Journal of Combinatorics
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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 Kn 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.