1,2

Let S be a set of n points in the plane. A halving line is a line through two points in S that splits the remaining points into two equal-sized subsets. How many halving lines can S have?

The values n = 8, 9, 10, 11, 12 and 13 were obtained by Abrego et al. The same values hold also for the maximum number of pseudo-halving lines in a generalized configuration of 2n points. The next unknown value, n = 14 (i.e. the maximum number of halving lines among 28 points), is either 63 or 64. - Bernardo M Abrego (bernardo.abrego(AT)csun.edu), May 05 2008

A. Beygelzimer and S. Radziszowski, On halving line arrangements, Discrete Math., 257 (2002), 267-283.

Geza Toth, "Point sets with many k-sets", in Proceedings of the 16th Annual ACM Symposium on Computational Geometry, 2000, pp. 37-42.

B. M. Abrego, S. Fernandez-Merchant, J. Lea[nonascii characters here] and G. Salazar, The maximum number of halving lines and the rectilinear crossing number of K_n for n <= 27, Electronic Notes in Discrete Mathematics, 30 (2008), 261-266.

Jeff Erickson, Halving lines and k-sets

Sequence in context: A080060 A004131 A032782 this_sequence A129403 A154287 A092847

Adjacent sequences: A076520 A076521 A076522 this_sequence A076524 A076525 A076526

nonn

N. J. A. Sloane (njas(AT)research.att.com), Oct 18 2002

More terms from Bernardo M Abrego (bernardo.abrego(AT)csun.edu), May 05 2008