1 research outputs found
New lower bounds for the number of -edges and the rectilinear crossing number of
We provide a new lower bound on the number of -edges of a set of
points in the plane in general position. We show that for the number of -edges is at least
which, for , improves the previous best lower
bound in [J. Balogh, G. Salazar, Improved bounds for the number of ()-sets, convex quadrilaterals, and the rectilinear crossing number of ].
As a main consequence, we obtain a new lower bound on the rectilinear crossing
number of the complete graph or, in other words, on the minimum number of
convex quadrilaterals determined by points in the plane in general
position. We show that the crossing number is at least which improves the previous bound of in [J. Balogh, G. Salazar, Improved bounds for the number of ()-sets, convex quadrilaterals, and the rectilinear crossing number of ]
and approaches the best known upper bound in [O.
Aichholzer, H. Krasser, Abstract order type extension and new results on the
rectilinear crossing number].Comment: 14 pages, 4 figures, submitted to Discrete and Computational
Geometry, fixed a gap in the proof of Theorem 1