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%I #19 Dec 11 2014 05:57:21
%S 1,3,6,9,13,18,22,27,33,38,44,51,57
%N Maximal number of halving lines for 2n points in plane.
%C 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?
%C 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
%H B. M. Abrego, S. Fernandez-Merchant, J. Leaños and G. Salazar, <a href="http://dx.doi.org/10.1016/j.endm.2008.01.045">The maximum number of halving lines and the rectilinear crossing number of K_n for n <= 27</a>, Electronic Notes in Discrete Mathematics, 30 (2008), 261-266.
%H A. Beygelzimer and S. Radziszowski, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00430-2">On halving line arrangements</a>, Discrete Math., 257 (2002), 267-283.
%H Jeff Erickson, <a href="http://compgeom.cs.uiuc.edu/~jeffe/open/ksets.html">Halving lines and k-sets</a>
%H Tanya Khovanova and Dai Yang, <a href="http://arxiv.org/abs/1210.4959">Halving Lines and Their Underlying Graphs</a>, arXiv:1210.4959 [math.CO], 2012.
%H Tanya Khovanova, Dai Yang, <a href="http://arxiv.org/abs/1304.5658">Connected Components of Underlying Graphs of Halving Lines</a>, arXiv:1304.5658 [math.CO], 2013.
%H Tanya Khovanova, Dai Yang, <a href="http://arxiv.org/abs/1310.3510">Fission of Halving Edges Graphs</a>, arXiv:1310.3510 [math.CO], 2013.
%H Geza Toth, <a href="http://www.cs.bme.hu/~geza/k-set.pdf">Point sets with many k-sets</a>, in Proceedings of the 16th Annual ACM Symposium on Computational Geometry, 2000, pp. 37-42.
%K nonn,more
%O 1,2
%A _N. J. A. Sloane_, Oct 18 2002
%E More terms from Bernardo M Abrego (bernardo.abrego(AT)csun.edu), May 05 2008
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