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An extended lower bound on the number of ( k)-edges to generalized configurations of points
 

Summary: An extended lower bound on the number of ( k)-edges
to generalized configurations of points
and the pseudolinear crossing number of Kn
B.M. ŽAbrego
J. Balogh
S. FernŽandez-Merchant
J. Lea~nos
G. Salazar
December 13, 2007
Abstract
Recently, Aichholzer, GarcŽia, Orden, and Ramos derived a remarkably improved
lower bound for the number of ( k)-edges in an n-point set, and as an immediate
corollary, an improved lower bound on the rectilinear crossing number of Kn. We
use simple allowable sequences to extend all their results to the more general setting
of simple generalized configurations of points and slightly improve the lower bound
on Sylvester's constant from 0.37963 to 0.379688. In other words, we prove that the
pseudolinear (and consequently the rectilinear) crossing number of Kn is at least
0.379688 n
4 + n3
. We use this to determine the exact pseudolinear crossing

  

Source: Abrego, Bernardo - Department of Mathematics, California State University, Northridge
Fernandez, Silvia - Department of Mathematics, California State University, Northridge

 

Collections: Mathematics