Recent developments on the number of ( ≤ k ) - sets , halving lines , and the rectilinear crossing number of Kn . Bernardo

We present the latest developments on the number of (≤ k)-sets and halving lines for (generalized) 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¡k+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 ¡ k+1 2 ¢ + 3 ¡ k−bn/3c+1 2 ¢ + 18 ¡ k−d4n/9e+1 2 ¢ −O (n) . This in turn implies that the pseudolinear (and consequently the rectilinear) crossing number of any perfect generalized configuration on n points is at least 277 729 ¡ n 4 ¢ +O ¡ n ¢ ≥ 0.379972¡n 4 ¢ +O ¡ n ¢ .