 
Summary: Recent developments on the number of ( k)sets, halving
lines, and the rectilinear crossing number of Kn.
Bernardo M. ´Abrego
Silvia Fern´andezMerchant *
Jes´us Lea~nos
Gelasio Salazar
Resumen
We present the latest developments on the number of ( k)sets and halving lines for (gen
eralized) configurations of points; as well as the rectilinear and pseudolinear crossing numbers
of Kn. In particular, we define perfect generalized configurations on n points as those whose
number of ( k)sets is exactly 3
ˇk+1
2
˘
for all k n/3. We conjecture that for each n there
is a perfect configuration attaining the maximum number of ( k)sets and the pseudolinear
crossing number of Kn. We prove that for any k n/2 the number of ( k)sets is at least
3
ˇk+1
2
