An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.


Introduction
The search for lower bounds for the minimum number of ≤k-edges in sets of n points of the plane for n ≥ 2 k + 2 (e ≤k (n)) is an important task in Combinatorial Geometry, due to its relation with the rectilinear crossing number problem. The most well-known case of the rectilinear crossing number problem aims to find the number cr(P) of crossings in a complete graph with a set of vertices P consisting of n points in the plane (in general position) and edges represented by segments and the minimum number of crossings over P, cr(n) (see the definitions below). The idea of determining cr(n) for each n was firstly considered by Erdös and Guy (see [1,2]). Determining cr(n) is equivalent to finding the minimum number of convex quadrilaterals defined by n points in the plane. These kinds of problems belong to classical combinatorial geometry problems (Erdös-Szekeres problems). The study of cr(n) is also interesting from the point of view of Geometric Probability. It is connected with the Sylvester Four-Point Problem, in which Sylvester studies the probability of four random points in the plane forming a convex quadrilateral.
Nowadays, finding the value of cr(n) continues to be a challenging open problem. The exact value of cr(n) is known for n ≤ 27 and n = 30. The search of lower and upper asymptotic bounds of cr(n) constitutes a relevant task due to its connection with the problem of finding the value of the Sylvester Four-Point Constant q * . In order to define properly q * , it is necessary to consider a convex open set R in the plane with finite area. Let q(R) be the probability that four points chosen randomly from R define a convex quadrilateral. Whence, q * is defined as the infimum of q(R) taken over all open sets R.
In particular, the connection between q * and cr(n) is given by the following expression: For more details, see [3]. The rigorous definitions of the above-presented concepts are the following: Definition 1. Given a finite set of points in general position in the plane P, assume that we join each pair of points of P with a straight line segment. The rectilinear crossing number of P (cr(P) ) is the number of intersections out of the vertices of said segments. The rectilinear crossing number of n ( cr(n)) is the minimum of cr(P) over all the sets P with n points.

Definition 2.
Given a set of points in general position, A = {p 1 , ..., p n } and an integer number k such that 0 ≤ k ≤ n−2 2 , a k-edge of A is a line R that joins two points of A and leaves exactly k points of A in one of the open half-planes (it is named the k-half plane of R).

Definition 3.
Given a set of points in general position, A = {p 1 , ..., p n }, a ≤ k-edge of A is an i-edge of A with i ≤ k.

Notation 1.
We call e k (P) the number of k−edges of the set P and e k (n) the maximum number of e k (P) over all the sets P with n points.
The relation between the number of ≤k-edges of P and cr(P) is given by the expression: where e ≤k (P) is the number of ≤k-edges of the set P with |P| = n (see [4,5]). This implies that This way, improvements of the lower bound of e ≤k (n) for k ≤ n−2 2 − 2 yield an improvement of the lower bound of the rectilinear crossing number of n. The exact value of e ≤k (n) is known for k < 4n−11 9 (see [4,6,7]). For k ≥ 4n−11 9 , the current best lower bound of e ≤k (n) is e ≤k (n) ≥ u k for the sequence u k defined in [6].
Taking into account the asymptotic equivalence of u k , we have For k close to n−2 2 − 2, namely k = n−t 2 for some fixed constant t, the bound (3) gives For these values of k, if we define P as a set for which e ≤k (n) is attained and e s (P) as the number of s-edges of P (see the definitions below), then we have that the identity: (P) together with the current best upper bound of e s (P) (due to Dey, see [8]) yield a lower bound that is asymptotically better than (4). More precisely, in [8] was shown the existence of a constant C ≤ 6.48 such that e s (P) ≤ Cn(s + 1) for s < n−2 2 and e s (P) ≤ Cn n − 1 2 for s = n−2 2 . To do this, Dey in [8] applied the crossing lemma and the following values for E(<= s)(n), the maximum number of (<= s)-edges due to [9] E(<= s)(n) = s(k + 1) for s < (n − 2)/2, E(<= (n − 2)/2)(n) = n(n − 1)/2.
The best values for C are C = 31,827 2 10 1 3 for s < n−2 2 and C = 31,827 2 12 1 3 for s = n−2 2 , for n an even number, if e s (P) ≥ 103n 6 , (see [10,11]). Notice that this condition is satisfied for large n and s close to n 2 due to the best lower bound of e s (n). As an example, for s = n−3 2 we have the upper bound (5) for n ≥ 327 and, for s = n−5 2 , we have the upper bound (5) for n ≥ 329.
This gives: for n an odd number and for n an even number. In this paper we improve in, at most, t 4 the bounds (7) and (8) for k = n−t 2 and some big values of n. In this way, we achieve the best lower bound of e ≤k (n) for these values of k and n. As a consequence, we improve the lower bound of the rectilinear crossing number of K n .
The outline of the rest of the paper is as follows: In Section 2 we give the improvement of the lower bound of e ≤k (n), k = n−t 2 , for the cases t = 7 (n is an odd number) and t = 8 (n is an even number). In Section 3, we generalize the achieved results in Section 2, and in Section 4 we give some concluding remarks.

The Improvement of the Lower Bound
In order to get the improvement of the lower bound of e ≤k (n), we need the following lemma: Lemma 1. Let k and n be positive integers, and let P be a set of n points in general position in the plane. If k < n−2 2 , then e k (n − 1) ≥ n − k − 2 n e k (P) + k + 1 n e k+1 (P).
Proof. Each (k + 1)-edge of P leaves k + 1 points of P in its (k + 1)-half plane, and each k-edge of P leaves n − k − 2 points of P in one of its half-planes. Therefore, the total number of points of P in these planes, allowing repetitions, is (n − k − 2)e k (P) + (k + 1)e k+1 (P), (10) and then there is a point of P, say p n , that belongs to s half-planes with s ≥ n − k − 2 n e k (P) + k + 1 n e k+1 (P).
If we remove p n , then we obtain a set Q = {p 1 , ..., p n−1 } such that the (k + 1)edges of P corresponding to the s half-planes are now k-edges of Q, because they have (k + 1) − 1 = k points of Q in one of the open half-planes.
Moreover, the k-edges of P corresponding to the s half-planes are now k-edges of Q because they still have k points of Q in one of the open half-planes. Therefore, we have that e k (n − 1) ≥ e k (Q) ≥ s ≥ n − k − 2 n e k (P) + k + 1 n e k+1 (P) (12) as desired.
Corollary 1. Let k and n be positive integers, and let P be a set of n points in general position in the plane. If k < n−2 2 , then min{e k (P), e k+1 (P)} ≤ n n − 1 e k (n − 1) .
Proof. Applying Lemma 1, we obtain This implies the desired result. We will apply this improvement to shift the lower bound on the number of ≤k-edges for sets with n points in the cases k = n−7 2 and k = n−8 2 for some values of n. included in Lemma 1 of [6], we obtain that for n ≥ 33,623, the lower bound: is better than the lower bound for e ≤ n−7

2
(n) of [6]. For these values of n, the lower bound (17) sometimes improves (20) by one and is the best current lower bound of e ≤ n−7 2 (n). As an example, we get the improvement for the following odd values of n: