
COMMENTS

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 equalsized 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 pseudohalving 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


REFERENCES

B. M. Abrego, S. FernandezMerchant, J. LeaĆ±os 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), 261266.
A. Beygelzimer and S. Radziszowski, On halving line arrangements, Discrete Math., 257 (2002), 267283.
T. Khovanova and D. Yang, Halving Lines and Their Underlying Graphs, arXiv preprint arXiv:1210.4959, 2012.
Tanya Khovanova, Dai Yang, Connected Components of Underlying Graphs of Halving Lines, arXiv:1304.5658 [math.CO]
T. Khovanova, D. Yang, Fission of Halving Edges Graphs, arXiv preprint arXiv:1310.3510, 2013
Geza Toth, "Point sets with many ksets", in Proceedings of the 16th Annual ACM Symposium on Computational Geometry, 2000, pp. 3742.
