On ≤k-Edges, Crossings, and Halving Lines of Geometric Drawings of K _n

Abstract

Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by $\operatorname {cr}(P)$, is the rectilinear crossing number of P. A halving line of P is a line passing through two points of P that divides the rest of the points of P in (almost) half. The number of halving lines of P is denoted by h(P). Similarly, a k-edge, 0≤k≤n/2−1, is a line passing through two points of P and leaving exactly k points of P on one side. The number of ≤k-edges of P is denoted by E _≤k (P). Let $\overline {\mathrm {cr}}(n)$, h(n), and E _≤k (n) denote the minimum of $\operatorname {cr}(P)$, the maximum of h(P), and the minimum of E _≤k (P), respectively, over all sets P of n points in general position in the plane. We show that the previously best known lower bound on E _≤k (n) is tight for k<⌈(4n−2)/9⌉ and improve it for all k≥⌈(4n−2)/9⌉. This in turn improves the lower bound on $\overline {\mathrm {cr}}(n)$ from $0.37968\binom{n}{4}+\varTheta (n^{3})$ to $\frac{277}{729}\binom{n}{4}+\varTheta (n^{3})\geq 0.37997\binom{n}{4}+\varTheta (n^{3})$. We also give the exact values of $\overline {\mathrm {cr}}(n)$ and h(n) for all n≤27. Exact values were known only for n≤18 and odd n≤21 for the crossing number, and for n≤14 and odd n≤21 for halving lines.

1 Introduction

We consider three important well-known problems in Combinatorial Geometry: the rectilinear crossing number, the maximum number of halving lines, and the minimum number of (≤k)-edges of complete geometric graphs on n vertices. All point sets in this paper are in the plane, finite, and in general position.

Let P be a finite set of points in general position in the plane. The rectilinear crossing number of P, denoted by $\operatorname {cr}(P)$, is the number of crossings obtained when all straight line segments joining pairs of points in P are drawn. (A crossing is the intersection of two segments in their interior.) The rectilinear crossing number of n is the minimum number of crossings determined by any set of n points, i.e., $\overline {\mathrm {cr}}(n)=\min\{\operatorname {cr}(P):\vert P\vert=n\}$. The problem of determining $\overline {\mathrm {cr}}(n)$ for each n was posed by Erdős and Guy [18, 20] in the early 1970s. This is equivalent to finding the minimum number of convex quadrilaterals determined by n points, as every pair of crossing segments bijectively corresponds to the diagonals of a convex quadrilateral.

A halving line of P is a line passing through two points of P and dividing the rest in almost half. So when P has n points and n is even, a halving line of P leaves n/2−1 points of P on each side; whereas when n is odd, a halving line leaves (n−3)/2 points on one side and (n−1)/2 on the other. The number of halving lines of P is denoted by h(P). Generalizing a halving line, a k-edge of P, with 0≤k≤n/2−1, is a line through two points of P leaving exactly k points on one side. The number of k-edges of P is denoted by E _k (P). Since a halving line is a (⌊n/2⌋−1)-edge, then E _{⌊n/2⌋−1}(P)=h(P). Similarly, for 0≤k≤n/2−1, E _≤k (P) and E _≥k (P) denote the number of ≤k-edges and ≥k-edges of P, respectively. That is, $E_{\leq k}( P) =\sum_{j=0}^{k}E_{j}( P)$ and $E_{\geq k}( P)=\sum_{j=k}^{\lfloor n/2\rfloor-1}E_{j}( P)=\binom{n}{2}-\sum_{j=0}^{k-1} E_{j}( P)$. Let h(n) and E _≤k (n) be the maximum of h(P) and the minimum of E _≤k (P), respectively, over all sets P of n points. A concept closely related to k-edges is that of k-sets; a k-set of P is a set Q that can be separated from P∖Q with a straight line. Rotating this separating line clockwise until it hits a point on each side yields a (k−1)-edge, and it turns out that this association is bijective. Thus the number of k-sets of P is equal to the number of (k−1)-edges of P. As a consequence, any of the results obtained here for k-edges can be directly translated into equivalent results for (k+1)-sets. Erdős, Lovász, Simmons, and Straus [17, 21] first introduced the concepts of halving lines, k-sets, and k-edges.

Since the introduction of these parameters back in the early 1970s, the determination (or estimation) of $\overline {\mathrm {cr}}(n)$, h(n), and E _≤k (n) have become classical problems in combinatorial geometry. General bounds are known but exact values have only been found for small n. The best known general bounds for the halving lines are $\varOmega (ne^{c\sqrt{\log n}})\leq h(n)\leq O( n^{4/3})$, due to Tóth [26] and Dey [16], respectively. Later on Nivasch [23] found a simpler construction with a smaller constant c. The previously best asymptotic bounds for the crossing number were

$$0.3792\binom{n}{4}+\varTheta \bigl(n^{3}\bigr)\leq \overline {\mathrm {cr}}( n )\leq0.380488\binom{n}{4}+\varTheta \bigl(n^{3}\bigr). $$

(1)

The lower bound is due to Aichholzer et al. [11] and it follows from (2) as we indicate below. The upper bound follows from a recursive construction devised by Ábrego and Fernández-Merchant [6] using a suitable initial construction found by the authors in [2]. The best lower bound for the minimum number of ≤k-edges is

(2)

due to Aichholzer et al. [11]. Further references and related problems can be found in [14].

The last two problems are naturally related, and their connection to the first problem is shown by the following identity, independently proved by Lovász et al. [22] and Ábrego and Fernández-Merchant [7]. For any set P of n points,

(3)

Hence, lower bounds on E _≤k (n) give lower bounds on $\overline {\mathrm {cr}}(n)$.

The majority of our results (all non-constructive parts) are proved in the more general context of generalized configurations of points, where the points in P are joined by pseudosegments rather than straight line segments. Goodman and Pollack [19] established a correspondence between the set of generalized configurations of points and what they called allowable sequences. In Sect. 2, we define allowable sequences, introduce the necessary notation to state the three problems above in the context of allowable sequences, and include a summary of results for these problems in both, the geometric and the allowable sequence context.

The main result in this paper is Theorem 1 in Sect. 3, which bounds E _≥k (P) by a function of E _k−1(P). This result has the following important consequences.

• 1. In Sect. 4, we find exact values of $\overline {\mathrm {cr}}(n)$ and h(n) for n≤27. Exact values were only known for n≤18 and odd n≤21 in the case of $\overline {\mathrm {cr}}(n)$, and for n≤14 and odd n≤21 in the case of h(n). (See Table 1.) We also show that the same values are achieved for the more general case of the pseudolinear crossing number $\widetilde {\mathrm {cr}}(n)$ and the maximum number of halving pseudolines $\widetilde{h}( n)$. (See Sect. 2 for the definitions.)

  Table 1 New exact values. The ^* values were only known in the rectilinear case

• 2. Theorem 2 in Sect. 5 improves the lower bound in (2) for k≥⌈(4n−11)/9⌉. It gives a recursive lower bound whose asymptotic value is given by

  $$E_{\leq k}( n) \geq\binom{n}{2}-\frac{1}{9}\sqrt{1-\frac{2k+2}{n}}\bigl( 5n^{2}+19n-31\bigr),$$

  as shown in Corollary 3.

• 3. Theorem 3 in Sect. 6 improves the lower bound in (1) to

  $$\overline {\mathrm {cr}}(n) \geq\frac{277}{729}\binom{n}{4}+\varTheta \bigl( n^{3}\bigr) \geq0.37997\binom{n}{4}+\varTheta \bigl( n^{3} \bigr).$$

In Sect. 7, and to complement item 2 above, we show that (2) is tight for k<⌈(4n−11)/9⌉. More precisely, we construct sets of points simultaneously achieving equality in (2) for all k<⌈(4n−11)/9⌉.

Several results of this paper appeared (without proofs) in the conference proceedings of LAGOS’07 [3, 4].

2 Allowable Sequences and Generalized Configurations of Points

Any set P of n points in the plane can be encoded by a sequence of permutations of the set [n]={1,2,…,n} as follows. Consider a directed line l. Orthogonally project P onto l and label the points of P from 1 to n according to their order in l. In this order, the identity permutation (1,2,…,n), is the first permutation of our sequence. Note that l can be chosen so that none of the projections overlap. Continuously rotate l counterclockwise. The order of the projections of P onto l changes every time two projections overlap, that is, every time a line through two points of P becomes perpendicular to l. Each time this happens, a new permutation is recorded as part of our sequence. After a 180^∘-rotation of l we obtain a sequence of $\binom{n}{2}+1$ permutations such that the first permutation (1,2,…,n) is the identity, the last permutation (n,n−1,…,2,1) is the reverse of the identity, any two consecutive permutations differ by a transposition of adjacent elements, and any pair of points (labels 1,…,n) transpose exactly once. This sequence is known as a halfperiod of the circular sequence associated to P. The circular sequence of P is then a doubly infinite sequence of permutations obtained by rotating l indefinitely in both directions.

As an abstract generalization of a circular sequence, a simple allowable sequence on [n] is a doubly infinite sequence Π=(…,π ₋₁,π ₀,π ₁,…) of permutations of [n], such that any two consecutive permutations π _i and π _i+1 differ by a transposition τ(π _i ) of neighboring elements, and such that for every j, π _j is the reverse permutation of $\pi_{j+\binom{n}{2}}$. A halfperiod of Π is a sequence of $\binom{n}{2}+1$ consecutive permutations of [n]. As before, any halfperiod of Π uniquely determines Π and all properties for halfperiods mentioned above still hold. Moreover, the halfperiod $\pi=( \pi_{i},\pi_{i+1},\ldots,\pi_{i+\binom{n}{2}})$ is completely determined by the transpositions $\tau(\pi_{i}),\tau(\pi_{i+1}),\ldots,\tau(\pi_{i+\binom{n}{2}-1})$. Note that the sequence (…,τ(π ₋₁),τ(π ₀),τ(π ₁),…) is $\tbinom{n}{2}$-periodic. Thus we indistinctly refer to π as a sequence of permutations or as a sequence of (suitable) transpositions. Allowable sequences that are the circular sequence of a set of points are called stretchable.

A pseudoline is a curve in ℙ², the projective plane, whose removal does not disconnect ℙ². Alternatively, a pseudoline is a simple curve in the plane that extends infinitely in both directions. A simple generalized configuration of points consists of a set of $\binom{n}{2}$ pseudolines and n points in the plane such that each pseudoline passes through exactly two points, and any two pseudolines intersect exactly once.

Circular and allowable sequences were first introduced by Goodman and Pollack [19]. They proved that not every allowable sequence is stretchable and established a correspondence between allowable sequences and generalized configurations of points.

The three problems at hand can be extended to generalized configurations of points, or equivalently, to simple allowable sequences. In this new setting, a transposition of two points in positions k and k+1, or n−k and n−k+1 in a simple allowable sequence Π corresponds to a (k−1)-edge. We say that such transposition is a k-transposition, or respectively, an (n−k)-transposition, and if 1≤k≤n/2 all these transpositions are called k-critical. Therefore E _k (Π), E _≤k (Π), and E _≥k (Π) correspond to the number of (k+1)-critical, (≤k+1)-critical, and (≥k+1)-critical transpositions in any halfperiod of Π. A halving line of Π is a ⌊n/2⌋-transposition, and thus h(Π)=E _{⌊n/2⌋−1}(Π). Identity (3), which relates the number of k-edges to the crossing number, was originally proved for allowable sequences. In this setting, a pseudosegment is the segment of a pseudoline joining two points in a generalized configuration of points, and $\operatorname {cr}( \varPi )$ is the number of pseudosegment-crossings in the generalized configuration of points that corresponds to the allowable sequence Π. All these definitions and functions coincide with their original counterparts for P when Π is the circular sequence of P. However, when $\overline {\mathrm {cr}}(n)$, h(n), and E _≤k (n) are minimized or maximized over all allowable sequences on [n] rather than over all sets of n points, the corresponding quantities may change and therefore we use the notation $\widetilde {\mathrm {cr}}(n),\widetilde{h}(n)$, and $\widetilde{E}_{\leq k}(n)$. Because n-point sets correspond to the stretchable simple allowable sequences on [n], it follows that $\widetilde {\mathrm {cr}}(n)\leq \overline {\mathrm {cr}}(n)$, $\widetilde{h}(n)\geq h(n)$, and $\widetilde{E}_{\leq k}( n) \leq E_{\leq k}( n)$. Tamaki and Tokuyama [25] extended Dey’s upper bound for allowable sequences to $\widetilde{h}(n)=O(n^{4/3})$. Ábrego et al. [1] proved that the lower bound for E _≤k (n) in (2) is also a lower bound on $\widetilde{E}_{\leq k}( n)$. They used this bound to extend (and even slightly improve) the corresponding lower bound on $\overline {\mathrm {cr}}(n)$ to $\widetilde {\mathrm {cr}}(n)$.

Our main result, Theorem 1 in Sect. 3, concentrates on the central behavior of allowable sequences. We bound E _≥k (Π) by a function of E _k−1(Π). As a consequence, we improve (or match) the upper bounds on $\widetilde{h}(n)$ for n≤27, and thus the lower bounds on $\widetilde {\mathrm {cr}}(n)$ in the same range. This is sufficient to match the corresponding best known geometric constructions [9] for h(n) and $\overline {\mathrm {cr}}(n)$. This shows that for all n≤27, $\widetilde{h}(n)=h( n)$ and $\widetilde {\mathrm {cr}}(n)=\overline {\mathrm {cr}}(n)$ whose exact values are summarized in Table 1.

3 The Central Theorem

In this section, we present our main theorem. Given a halfperiod $\pi=( \pi_{0},\pi_{1},\pi_{2},\ldots, \pi_{\binom{n}{2}})$ of an allowable sequence and an integer 1≤k<n/2, the k-center of the permutation π _j , denoted by C(k,π _j ), is the set of elements in the middle n−2k positions of π _j . Let L ₀, C ₀=C(k,π ₀), and R ₀ be the set of elements in the first k, middle n−2k, and last k positions, respectively, of the permutation π ₀. Define

$$s ( k,\pi ) =\min \biggl\{ \bigl \vert C_{0}\cap C ( k,\pi_{i} ) \bigr \vert :0\leq i\leq\binom{n}{2} \biggr\}.$$

Note that s(k,π)≤n−2k−1 because at least one of the n−2k elements of C ₀ must leave the k-center.

Theorem 1

Let Π be an allowable sequence on [n] and π any halfperiod of Π. If s=s(k,π), then

$$E_{\geq k} ( \varPi ) \leq ( n-2k-1 ) E_{k-1} ( \varPi ) -\frac{s}{2} \bigl( E_{k-1} ( \varPi )-n+1 \bigr).$$

Proof

For presentation purposes, we divide this proof into subsections.

Let Π be an allowable sequence on [n] and $\pi=(\pi_{0},\pi_{1},\pi_{2},\ldots,\pi_{\binom{n}{2}})$ any halfperiod of Π, s=s(k,π), and K=E _k−1(π).

Suppose that $\pi_{i_{1}},\pi_{i_{2}},\ldots,\pi_{i_{K}}$ is the subsequence of permutations in π obtained when the k-critical transpositions $\tau(\pi_{i_{1}}),\tau(\pi_{i_{2}}),\ldots,\tau(\pi_{i_{K}})$ of π occur (in this order). For simplicity we write τ _j instead of $\tau(\pi_{i_{j}})$. These permutations partition π into K+1 parts B ₀(π),B ₁(π),B ₂(π),…,B _K (π) called blocks, where B _j (π)={π _l :i _j ≤l<i _j+1} for 1≤j≤K−1, B ₀(π)={π _l :0≤l<i ₁}, and $B_{K}( \pi) =\{ \pi_{l}:i_{K}\leq l\leq\binom{n}{2}\}$. Denote by p _j the point that enters the k-center of $\pi_{i_{j}}$ with τ _j . We say that a (≥k+1)-critical transposition in B _j (π), 1≤j≤K, is an essential transposition if it involves p _j or if it occurs before τ ₁, and a nonessential transposition otherwise.

Rearrangement of π

We claim that, to bound E _≥k (Π), we can assume that all (≥k+1)-critical transpositions of π are essential transpositions. To show this, in case π has nonessential transpositions, we modify π so that the obtained halfperiod λ satisfies E _j (π)=E _j (λ) for all j<k, and thus E _≥k (π)=E _≥k (λ); and either λ has only essential transpositions or the last nonessential transposition of λ occurs in an earlier permutation than the last nonessential transposition of π. Applying this procedure sufficiently many times, we end with a halfperiod λ all of whose (≥k+1)-critical transpositions are essential and such that E _j (π)=E _j (λ) for all j<k, and thus E _≥k (π)=E _≥k (λ).

This is how λ is constructed. Suppose B _j (π) is the last block of π that contains nonessential transpositions. Define λ as the halfperiod that coincides with π everywhere except for the (≥k+1)-transpositions in B _j (π). All nonessential transpositions in B _j (π) take place right before τ _j in λ, and right after τ _j occurs, all essential transpositions in B _j (π) occur consecutively in B _j (λ) but probably in an order different from B _j (π), so that the final position of p _j is the same in B _j (π) and B _j (λ). Note that in fact the last permutations of the blocks B _j (π) and B _j (λ) are equal.

Classification of k-Critical Transpositions

From now on, we assume that π only has essential transpositions. We classify the k-critical transpositions as follows (see Fig. 1): τ _j is an arriving transposition if p _j ∈C ₀. An arriving transposition is a-augmenting if it increments the number of elements in C ₀ in the k-center from a−1 to a, and it is neutral otherwise. We say that τ _j is a returning transposition if it is a k-transposition and p _j ∈R ₀, or if it is an (n−k)-transposition and p _j ∈L ₀. That is, p _i is “getting back” to its starting region. Similarly, τ _j is a departing transposition if it is a k-transposition and p _j ∈L ₀, or if it is an (n−k)-transposition and p _j ∈R ₀. That is, p _j is “getting away” from its original region. We say that a departing transposition τ _j is a cutting transposition, if τ _j is a k-transposition and the next k-critical transposition that involves p _j is an (n−k)-transposition; or if τ _i is an (n−k)-transposition and the next k-critical transposition that involves p _j is a k-transposition. All other departing transpositions are called stalling.

Fig. 1

Classification of essential k-critical transpositions

Finally, we define the weight of a k-critical transposition τ _j , denoted by w(τ _j ), as the number of (≥k+1)-critical transpositions in B _j (π) that are not between two elements of C ₀. Transpositions with weight at most n−2k−1−s are called light. All other transpositions are heavy.

Let $A,N,R,C,S_{\text{light}}$, and $S_{\text{heavy}}$ be the number of augmenting, neutral, returning, cutting, light stalling, and heavy stalling transpositions, respectively. Then $K=A+N+R+C+S_{\text{light}}+S_{\text {heavy}}$.

Bounding E _≥k (Π)

Observe that the k-center of all permutations in B ₀(π) remains unchanged. It follows that all (≥k+1)-critical transpositions of B ₀(π) are between elements of C ₀. Thus $\sum_{j=1}^{K}w( \tau_{j})$ counts all (≥k+1)-critical transpositions except those between two elements of C ₀. There are $\binom{n-2k}{2}$ transpositions between elements of C ₀, but each neutral transposition corresponds to a k-critical (not (≥k+1)-critical) transposition between two elements of C ₀. Thus

$$E_{\geq k}( \varPi ) \leq\binom{n-2k}{2}-N+\sum_{j=1}^{K}w( \tau_{j}). $$

(4)

Bounds for the Weight of a k-Critical Transposition

We bound the weight of a transposition depending on its class (departing, returning, etc.), as well as the number of transpositions within a class, if necessary. For j≥1 all (≥k+1)-critical transpositions in B _j (π) involve p _j and thus w(τ _j )≤n−2k−1. However, since the weight of τ _j does not count transpositions between two elements of C ₀, and there are always at least s elements of C ₀ in the k-center, then w(τ _j )≤n−2k−s whenever τ _j is arriving (because p _j ∈C ₀). Moreover, if τ _j is a-augmenting, then w(τ _j )≤n−2k−a. If τ _j is a returning transposition, then p _j has already been transposed with all the elements of C ₀ that are in the k-center of $\pi_{i_{j}}$. Since there are at least s such elements, then w(τ _j )≤n−2k−1−s. Summarizing,

$$w ( \tau_{j} ) \leq \begin{cases}n-2k-1&\text{for all }\tau_{j},\\n-2k-s,&\text{if }\tau_{j}\text{ is neutral,}\\n-2k-a,&\text{if }\tau_{j}\text{ is }a\text{-augmenting,}\\n-2k-1-s,&\text{if }\tau_{j}\text{ is light stalling or returning.}\end{cases} $$

(5)

Bounding C

We bound the number of cutting transpositions. Since the first (last) k elements of π ₀ are the last (first) elements of $\pi_{\binom{n}{2}}$, then the 2k elements not in C ₀ must participate in at least one cutting transposition. That is, C≥2k. Note that, if p∉C ₀ participates in c≥2 cutting transpositions, then there must be at least c−1 returning transpositions of p. In other words, there must be at least C−2k≥0 returning transpositions. There are C cutting transpositions and at least n−2k−s arriving transpositions (at least one a-augmenting arriving transposition for each s+1≤a≤n−2k). Then K−C−(n−2k−s) counts all other k-critical transpositions, including in particular all returning transpositions. Thus K−C−(n−2k−s)≥C−2k, that is,

$$2C \leq4k+K-n+s. $$

(6)

Augmenting and Heavy Stalling Transpositions

We keep track of the augmenting and heavy stalling transpositions together. To do this, we consider the bipartite graph G whose vertices are the augmenting and the heavy stalling transpositions. The augmenting transposition τ _l is adjacent in G to the heavy stalling transposition τ _j if j<l, p _j is in the k-center of all permutations in blocks B _j to B _l , one transposition from τ _j and τ _l is a k-transposition and the other is an (n−k)-transposition, and p _l does not swap with p _j in B _l (π). We bound the degree of a vertex in G.

Let τ _j be a heavy stalling transposition. If p _j ∈L ₀ (the case p _j ∈R ₀ is equivalent), then τ _j is a k-transposition. Because p _j moves to the right exactly w(τ _j )>n−2k−1−s positions within B _j (π), it follows that the k-center right before τ _j+1 occurs (i.e., the k-center of $\pi_{i_{j+1}-1}$) has at most n−2k−1−w(τ _j )<s points of C ₀ to the right of p _j . Also, since τ _j is stalling, the next time that p _j leaves the k-center is by a k-transposition τ _j+b . This means that the k-center right before τ _j+b occurs (i.e., the k-center of $\pi_{i_{j+b}-1}$) has at least s points of C ₀ to the right of p _j . Thus, between τ _j and τ _j+b there must be at least s−(n−2k−1−w(τ _j )) arriving (n−k)-transpositions τ _l such that p _l remains to the right of p _j in B _l (π), i.e., p _l does not swap with p _j in B _l (π). These transpositions are adjacent to τ _j and thus the degree of τ _j in G is at least w(τ _j )−(n−2k−1−s). Hence,

$$\bigl \vert E(G) \bigr \vert \geq\sum_{\tau_{j}\text{ heavy stalling}} \bigl( w ( \tau_{j} ) - ( n-2k-1-s ) \bigr),$$

where E(G) is the set of edges of G.

Let τ _l be an a-augmenting transposition. Since p _l ∈C ₀, and weights do not count transpositions between two elements of C ₀, then at most n−2k−a−w(τ _l ) points in L ₀∪R ₀ do not swap with p _l in B _l (π). Only these points are possible p _j s such that τ _j is adjacent to τ _l . Thus the degree of τ _l in G is at most n−2k−a−w(τ _l )≤n−2k−1−s−w(τ _l ).

Note that there is at least one a-augmenting transposition for each s+1≤a≤n−2k. This is because the k-center of at least one permutation of π contains exactly s elements of C ₀ (by definition of s), and the k-center of $\pi_{\binom{n}{2}}$ contains exactly n−2k elements of C ₀ (since it coincides with C ₀). Then the number of elements in the k-center must be eventually incremented from s to n−2k. For each s+1≤a≤n−2k, we use n−2k−a−w(τ _l ) to bound the degree of one a-augmenting transposition. For all other augmenting transpositions we use the bound n−2k−1−s−w(τ _l ). Hence

The previous two inequalities imply that

(7)

Final Calculations

We use (5) and (7) to bound $\sum_{i=1}^{K}w( \tau_{i}) -N$.

By (4),

Finally, by (6),

$$E_{\geq k} ( \varPi ) \leq ( n-2k-1 ) K-\frac{s}{2} ( K-n+1 ).\\[-18pt]$$

 □

4 New Exact Values of h(n), $\widetilde {h}(n)$, $\overline {\mathrm {cr}}(n)$, and $\widetilde {\mathrm {cr}}(n)$ for n≤27

We start by stating a relaxed version of Theorem 1, which we use in the special case when k=⌊n/2⌋−1.

Corollary 1

Let Π be a simple allowable sequence on [n] and π any halfperiod of Π. If s=s(k,π), then

Proof

There are at least n−2k−s elements of C ₀ that leave the k-center, so there are at least n−2k−s arriving transpositions. In addition, there are at least 2k departing transpositions, one per element not in C ₀. It follows that E _k−1(Π)≥2k+(n−2k−s)=n−s. The first inequality now follows directly from Theorem 1. Finally, s≤n−2k−1 for all halfperiods of Π which yields the second inequality. Another consequence is that E _k−1(Π)≥n−s≥2k+1, which is in fact the minimum possible value of E _k−1 (cf. [22]). □

The previous corollary implies the following result for halving lines.

Corollary 2

If Π is a simple allowable sequence on [n] and n≥6, then

$$h ( \varPi ) \leq \begin{cases}\bigl\lfloor\frac{1}{2}\binom{n}{2}-\frac{1}{2}E_{\leq n/2-3} (\varPi ) \bigr\rfloor&\textit{if }n\text{ \textit{is even},} \\\bigl\lfloor\frac{2}{3}\binom{n}{2}-\frac{2}{3}E_{\leq (n-1 ) /2-3} ( \varPi ) +\frac{1}{3} \bigr\rfloor&\text{\textit{if} }n\text{ \textit{is odd}.}\end{cases} $$

Proof

The inequality follows from Corollary 1 using k=⌊n/2⌋−1, and the identities E _{≥⌊n/2⌋−1}(Π)=h(Π) and $E_{\leq\lfloor n/2\rfloor-3}(\varPi )+E_{\lfloor n/2\rfloor -2}(\varPi )+h(\varPi )=\binom{n}{2}$. □

The exact values of h(n) were previously known only for even n≤14 or odd n≤21 [8, 13]. The exact values of $\overline {\mathrm {cr}}(n)$ were previously known only for even n≤18 or odd n≤21 [11]. Inequality (2) is also valid in the context of allowable sequences [1]. This bound for k=⌊n/2⌋−3 and Corollary 2 give the new upper bounds for $\widetilde{h}(n)$ in Table 1. We also obtained the new lower bounds for $\widetilde{\mathrm{cr}}(n)$ reported in Table 1. To do this we used identity (3) together with the new values of $\widetilde{h}(n)$, the identity $E_{\leq\lfloor n/2\rfloor-2}( \varPi ) =\binom{n}{2} -h(\varPi )$, and inequality (2) for k≤⌊n/2⌋−3. For example, if n=24 then $E_{\leq10}(\varPi )=\binom{24}{2}-h(\varPi )\geq 276-\widetilde{h}(24)=225$ and by (2), the vector (E _≤0(Π),E _≤1(Π),…,E _≤9(Π)) is bounded below entry-wise by (3,9,18,30,45,63,84,108,138,174), so (3) implies that $\widetilde{\mathrm{cr}}(24) =\sum_{k=0}^{10}(21-2k)E_{\leq k}(\varPi )-\frac{3}{4}\binom{24}{3}\geq3699$.

All the bounds shown in Table 1 are attained by Aichholzer’s et al. constructions [9], and thus Table 1 actually shows the exact values of $\widetilde{h}( n)$, h(n), $\widetilde{\mathrm{cr}}( n)$, and $\overline{\mathrm{cr}}( n)$ for n in the specified range.

The previous argument to obtain new upper bounds for h(n) does not improve the general bound h(n)≤cn ^4/3. However, even with the current best constant c=(31287/8192)^1/3<1.5721 [8, 24], our bound is better when n is even in the range 8≤n≤184.

5 New Lower Bound for the Number of ≤k-Edges

In this section, we obtain a new lower bound for the number of ≤k-edges. Our emphasis is on finding the best possible asymptotic result as well as the best bounds that apply to the small values of n for which the exact value is unknown. Theorem 2 provides the exact result that can be applied to small values of n, whereas Corollary 3 gives the best known asymptotic behavior.

Let m=⌈(4n−11)/9⌉. For each n, define the following recursive sequence:

The following is the new lower bound on E _≤k (n). It follows from Theorem 1.

Theorem 2

For any n and k such that m−1≤k≤(n−3)/2,

$$E_{\leq k}(n)\geq u_{k}.$$

Proof

We need the following two lemmas to estimate the growth of the sequence u _k with respect to n and k. For presentation purposes, we defer their proofs to the end of the section.

Lemma 1

For any k such that m−1≤k≤(n−5)/2,

$$3\sqrt{1-\frac{2k+9/2}{n}}<\frac{\binom{n}{2}-u_{k}}{\binom{n}{2}-u_{m-1}} \leq3\sqrt{1-\frac{2k+2}{n}}. $$

(8)

Lemma 2

For any k such that m≤k≤(n−5)/2,

$$3\sqrt{1-\frac{2k+9/2}{n}} \biggl( \binom{n}{2}-u_{m-1} \biggr)\geq ( n-1 ) ( n-2k-3 ).$$

We prove the stronger statement $\widetilde{E}_{\leq k}( n) \geq u_{k}$. Let Π be an allowable sequence on [n] and π any of its halfperiods. We proceed by induction on k. If k=m−1 the result holds by Inequality (2), proved in the more general context of allowable sequences [1]. Assume that k≥m and E _≤k−1(Π)≥u _k−1. Let s=s(k+1,π); by Theorem 1,

$$E_{\geq k+1} ( \varPi ) \leq ( n-2k-3 ) E_{k} ( \varPi ) -\frac{s}{2} \bigl( E_{k} ( \varPi ) - ( n-1 ) \bigr).$$

If s=0 or E _k (Π)≥n−1, then E _≥k+1(Π)≤(n−2k−3)E _k (Π). Thus

$$\binom{n}{2}-E_{\leq k}(\varPi )\leq ( n-2k-3 ) \bigl( E_{\leq k} (\varPi )-E_{\leq k-1}(\varPi ) \bigr),$$

and by induction

which implies that E _≤k (Π)≥u _k by definition of u _k . Now assume s>0 and E _k (Π)<n−1. Because E _k (Π)≥2k+3 (see the proof of Corollary 1), it follows that k≤(n−5)/2. By Theorem 1,

Recall that s=s(k+1,π)≤n−2k−3. Because E _k (Π)<n−1, it follows that

Therefore

$$E_{\leq k}(\varPi )=\binom{n}{2}-E_{\geq k+1}(\varPi )\geq \binom {n}{2}-(n-1) ( n-2k-3 ).$$

By Lemma 2,

$$E_{\leq k}(\varPi )\geq\binom{n}{2}-3\sqrt{1-\frac{2k+9/2}{n}} \biggl(\binom {n}{2}-u_{m-1} \biggr),$$

and by Lemma 1, E _≤k (Π)≥u _k for all allowable sequences Π on [n]. Therefore $E_{\leq k}(n)\geq \widetilde{E}_{\leq k}(n)\geq u_{k}$.  □

Corollary 3

For any n and k such that m−1≤k≤(n−2)/2,

$$E_{\leq k}(n)\geq\binom{n}{2}-\frac{1}{9}\sqrt{1-\frac{2k+2}{n}} \bigl( 5n^{2}+19n-31 \bigr).$$

Proof

Let Π be an allowable sequence on [n]. If k=⌊n/2⌋−1, then $E_{\leq\lfloor n/2\rfloor-1}( \varPi )=\binom{n}{2}$. For k<⌊n/2⌋−1, it follows that n≥3 and from Theorem 2 and Lemma 1,

$$E_{\leq k}(\varPi )\geq u_{k}\geq\binom{n}{2}-3\sqrt{1-\frac{2k+2}{n}} \biggl( \binom{n}{2}-u_{m-1} \biggr).$$

Considering the possible residues of n modulo 9, it can be verified that for n≥3,

$$u_{m-1} \geq\frac{17}{54} n^2-\frac{65}{54} n+\frac{31}{27}\quad \bigl(\text{equality if $n\equiv3$ (mod 9)}\bigr).$$

Therefore $E_{\leq k}(n)\geq\widetilde{E}_{\leq k}(n)\geq\binom{n}{2}-\frac{1}{9}\sqrt{1-\frac{2k+2}{n}}(5n^{2}+19n-31)$. □

5.1 Proofs of Lemmas 1 and 2

Proof of Lemma 1

The integer range [m−1,(n−5)/2] is empty for n≤5. Assume n≥6 and proceed by induction on k. If k=m−1, then $3\sqrt{1-(2m+5/2)/n}\leq1\leq3\sqrt{1-2m/n}$ is equivalent to ⌈(4n−11)/9⌉≤4n/9≤⌈(4n−11)/9⌉+5/4 which holds in general. Assume that k≥m and that (8) holds for k−1. From the definition of u _k and the induction hypothesis,

and $(n-2k-3)\sqrt{1-2k/n}\;/ (n-2k-2)\leq\sqrt{1-(2k+2)/n}$ because k≤(n−5)/2, which proves the second inequality in (8). Similarly, from the definition of u _k and the induction hypothesis,

Hence, to prove the second inequality in (8), it is enough to show that $3( \binom{n}{2}-u_{m-1}) d>1$, where

$$d=\frac{n-2k-3}{n-2k-2}\sqrt{1-\frac{2k+5/2}{n}}-\sqrt{1-\frac{2k+9/2}{n}} $$

(9)

is always positive because k≤(n−5)/2. First note that

which implies that

$$3 \biggl( \binom{n}{2}-u_{m-1} \biggr) \geq\frac{1}{9}\bigl( 5n^{2}-25n+4 \bigr). $$

(10)

Multiplying the easily verified inequality

$$1>\frac{ ( n-2k-3 ) \sqrt{n-2k-5/2}+ ( n-2k-2 )\sqrt{n-2k-9/2}}{ ( 2n-4k-5 ) \sqrt{n-2k-5/2}}$$

by (9), yields

Since (4n−11)/9≤k≤(n−5)/2, then 3≤n−2k−2≤(n+4)/9. Thus

$$d> \biggl( 1-\frac{1}{12} \biggr) \frac{27}{ ( n+4 )\sqrt{n ( n-1/2 ) }}=\frac{99}{4 ( n+4 )\sqrt{n ( n-1/2 ). }}$$

This inequality, together with (10), imply that for all n≥6,

$$3 \biggl( \binom{n}{2}-u_{m-1} \biggr) d>\frac{11}{4}\biggl( \frac {5n^{2}-25n+4}{ ( n+4 ) \sqrt{n ( n-1/2 )}} \biggr) >1.$$

 □

Proof of Lemma 2

For each n≤40 the integer range [m,(n−5)/2] is either empty or contains only k=⌊(n−5)/2⌋. For these cases, the inequality can easily be verified. Assume n≥41, it follows from (10) that

$$9 \biggl( 1-\frac{2k+9/2}{n} \biggr) \biggl( \binom{n}{2}-u_{m-1}\biggr)^{2}\geq\frac{(n-2k-9/2) (5n^{2}-25n+4 )^{2}}{81n}.$$

Since k≤(n−5)/2, then

$$n-2k-9/2\geq\frac{(n-2k-3)^{2}}{n-2k+3}.$$

Also k≥m≥(4n−11)/9 implies n−2k+3≤(n+49)/9 and thus

$$\frac{(n-2k-9/2) ( 5n^{2}-25n+4 )^{2}}{81n}\geq\frac {(n-2k-3)^{2} ( 5n^{2}-25n+4 )^{2}}{9n (n+49 )}.$$

Finally, for n≥41,

$$\frac{ ( 5n^{2}-25n+4 )^{2}}{9n ( n+49 )}\geq(n-1)^{2},$$

and consequently

$$9 \biggl( 1-\frac{2k+9/2}{n} \biggr) \biggl( \binom{n}{2}-u_{m-1}\biggr)^{2}\geq(n-1)^{2}(n-2k-3)^{2}.$$

 □

6 New Lower Bound on $\overline {\mathrm {cr}}(n)$

In this section, we use Corollary 3 to get the following new lower bound on $\overline {\mathrm {cr}}(n)$.

Theorem 3

$\overline {\mathrm {cr}}(n)\geq\frac{277}{729}\binom{n}{4}+\varTheta (n^{3})>0.379972\binom {n}{4}+\varTheta (n^{3})$.

Proof

We actually prove that the right hand side is a lower bound on $\widetilde {\mathrm {cr}}(n)$. According to (3), if Π is an allowable sequence on [n], then

$$\operatorname {cr}( \varPi ) =\binom{n}{4} \Biggl( 24 {\sum _{k=0}^{ \lfloor n/2 \rfloor-1}} \frac{1}{n} \biggl( 1-\frac{2k}{n} \biggr) \frac{E_{\leq k} (\varPi ) }{n^{2}} \Biggr) +\varTheta \bigl(n^{3} \bigr). $$

(11)

Using inequality (2) for 0≤k≤m−1 gives

$$\frac{E_{\leq k} ( \varPi ) }{n^{2}}\geq\frac{3}{2} \biggl( \frac{k}{n} \biggr)^{2}+\frac{3}{2}\max \biggl( 0,\frac{k}{n}-\frac{1}{3} \biggr)^{2}-\varTheta \biggl( \frac{1}{n}\biggr).$$

Similarly, if m≤k≤⌊n/2⌋−1, then by Corollary 3,

$$\frac{E_{\leq k} ( \varPi ) }{n^{2}}\geq\frac{1}{2}-\frac{5}{9}\sqrt{1-\frac{2k}{n}}+\varTheta \biggl( \frac{1}{n} \biggr).$$

These two inequalities in their corresponding ranges applied to (11) give a Riemann sum (on the variable x=k/n) that can be estimated using the corresponding integral. Note that the error terms are uniformly bounded by Θ(n ³). Therefore,

 □

Table 2 gives the best lower bounds for $\widetilde {\mathrm {cr}}( n)$ in the range 28≤n≤99 that follow from using (3) with the bound in either (2) or the new bound from Theorem 2, with the notable exception of n=30 for which $\widetilde {\mathrm {cr}}(30)=9726$ was recently determined by Cetina et al. [15].

Table 2 New lower bounds for $\widetilde {\mathrm {cr}}(n)$

7 A Point Set with Few ≤k-Edges for Every k≤4n/9−1

Combining (2) and Theorem 2, we obtain the best known lower bound for E _≤k (n). If n is a multiple of 9 and k≤(4n/9)−1, then this bound reads

(12)

Our aim in this section is to show that this bound is tight for n≥27. This improves on the construction in [10], where tightness for (12) is proved for k≤(5n/12).

We recursively construct, for each integer r≥3, a 9r-point set S _r such that for every k≤(4n/9)−1, E _≤k (S _r ) equals the right-hand side of (12).

7.1 Constructing the Sets S _r

If a and b are distinct points, then ℓ(ab) denotes the line spanned by a and b, and $\overline{ab}$ denotes the closed line segment with endpoints a and b, directed from a toward b. Let θ denote the clockwise rotation by an angle of 2π/3 around the origin. At this point the reader may want to take a sneak preview at Fig. 2, where S ₃ is sketched.

Fig. 2

The 27-point set S ₃. The points $a_{\infty},a'_{\infty}, b_{\infty}, b'_{\infty}, c_{\infty}$, and $c'_{\infty}$ do not belong to S ₃

For each r≥3 the set S _r is naturally partitioned into nine sets of size r: A _r ={a ₁,…,a _r }, $A_{r}'=\{a_{1}',\ldots,a_{r}'\}$, $A_{r}''$, and their respective 2π/3 and 4π/3 rotations around the origin. The elements of $A_{r}''$ are not labeled because they change in each iteration. For i=1,…,r, we let $b_{i}=\theta(a_{i}), b_{i}'=\theta(a_{i}'),c_{i}=\theta^{2}(a_{i})$, and $c_{i}'=\theta^{2}(a_{i})$. Thus if we let B _r ={b ₁,…,b _r }, $B_{r}'=\{b_{1}',\ldots,b_{r}'\}$, $B_{r}''=\theta(A_{r}'')$, C _r ={c ₁,…,c _r }, $C_{r}'=\{c_{1}',\ldots,c_{r}'\}$, and $C_{r}''=\theta^{2}(A''_{r})$, then we obtain $B_{r}\cup B_{r}'\cup B_{r}''$ (respectively, $C_{r}\cup C_{r}'\cup C_{r}''$) by applying θ (respectively, θ ²) to $A_{r}\cup A_{r}'\cup A_{r}''$. We refer to this property as the 3-symmetry of S _r .

As we mentioned before, the construction of the sets S _r is recursive. For r≥3, we obtain A _r+1 and $A_{r+1}'$ by adding suitable points a _r+1 to A _r and $a'_{r+1}$ to $A_{r}'$. Keeping 3-symmetry, this determines B _r+1, $B'_{r+1}$, C _r+1, and $C'_{r+1}$. However, the set $A''_{r+1}$ is not obtained by adding a point to $A''_{r}$, but instead is defined in terms of $B_{r+1},B'_{r+1}, C_{r+1}$, and $C'_{r+1}$; this explains why we have not listed the elements in $A''_{r},B''_{r}$, and $C''_{r}$.

Before moving on with the construction, we remark that the sets S _r contain subsets of more than two collinear points. As will become clear from the construction, the points can be slightly perturbed to general position, so that the number of ≤k-edges remains unchanged for every k≤4n/9−1.

We start by describing S ₃, see Fig. 2. First we explicitly fix A ₃ and $A_{3}'$: a ₁=(−700,−50), a ₂=(−410,150), a ₃=(−436,144), $a'_{1}=( -1300, 20 )$, $a'_{2}=(-1200,-10 )$, and $a'_{3}=(-1170,-14 )$. Thus B ₃, $B_{3}'$, C ₃, and $C_{3}'$ also get determined. For the points in $A_{3}''$ we do not give their exact coordinates, instead we simply ask that they satisfy the following: all the points in $A_{3}''$ lie on the x-axis, and are sufficiently far to the left of $A_{3} \cup A_{3}'$ so that if a line ℓ ₁ passes through a point in $A_{3}''$ and a point in $S_{3}\setminus{(B_{3}''\cup C_{3}'')}$, and a line ℓ ₂ passes through two points in $S_{3}\setminus{A_{3}''}$, then the slope of ℓ ₁ is smaller in absolute value than the slope of ℓ ₂, i.e., ℓ ₁ is closer (in slope) to a horizontal line, than ℓ ₂.

We need to define six auxiliary points not in S _r : a _∞=ℓ(a ₂ a ₃)∩ℓ(c ₂ c ₃) and $a'_{\infty}= \ell(a'_{2} a'_{3})\cap\ell(a_{2} a_{3})$. As expected, let b _∞=θ(a _∞), c _∞=θ ²(a _∞), $b'_{\infty}=\theta(a'_{\infty})$, and $c'_{\infty}=\theta^{2}(a'_{\infty})$.

We now describe how to get S _r+1 from S _r . The crucial step is to define the points b _r+1 and $a'_{r+1}$ to be added to B _r and $A'_{r}$ to obtain B _r+1 and $A'_{r+1}$, respectively. Then we construct $A''_{r+1}$ and applying θ and θ ² to B _r+1, $A'_{r+1}$, and $A''_{r+1}$, we obtain the rest of S _r+1.

Suppose that for some r≥3, the set S _r has been constructed so that the following properties hold for t=r (this is clearly true for the base case r=3):

• (I) The points a ₂,…,a _t appear in this order along $\overline{a_{2} a_{\infty}}$.

• (II) The points $a_{2}',\ldots,a_{t}'$ appear in this order along $\overline{a'_{2}a'_{\infty}}$.

• (III) For all i=2,…,t−1 and j=2,…,t, $\ell(a'_{i} a_{j})$ intersects the interior of $\overline{b_{i} b_{i+1}}$.

• (IV) For all j=2,…,t, $\ell(a'_{t} a_{j})$ intersects the interior of $\overline{b_{t} b_{\infty}}$.

Now we add b _r+1 and $a'_{r+1}$. Place b _r+1 anywhere on the open line segment determined by b _∞ and the intersection point of $\ell(a_{r}'a_{2})$ with $\overline{b_{r}b_{\infty}}$. (The existence of this intersection point is guaranteed by (IV), see Fig. 3.) Place $a'_{r+1}$ anywhere on the open line segment determined by $a'_{\infty}$ and the intersection point of ℓ(b _r+1 a _∞) with $\overline{a'_{r}a'_{\infty}}$. (This intersection exists because $a_{\infty}', a_{\infty},a_{2}$, and b _∞ are collinear and appear in this order along $\ell(a_{\infty}' b_{\infty})$, the line $\ell(a_{\infty}' b_{\infty})$ separates b _r+1 from $a_{r}'$, and the line $\ell(a_{r}'a_{2})$ separates b _r+1 from a _∞, see Fig. 4.) Thus B _r+1 and $A'_{r+1}$ and consequently $A_{r+1}, C_{r+1}, B'_{r+1}$, and $C'_{r+1}$, are defined. It is straightforward to check that (I)–(IV) hold for t=r+1.

Fig. 3

b _r+1 is placed in between b _r and b _∞, above the line $\ell(a'_{r}a_{2})$

Fig. 4

$a'_{r+1}$ is placed in between $a'_{r}$ and $a'_{\infty}$, below the line ℓ(a _∞ b _r+1)

It only remains to describe how to construct $A''_{r+1}$. As we mentioned above, this set is not a superset of $A''_{r}$, instead it gets defined analogously to $A''_{3}$: we let the points in $A_{r+1}''$ lie on the x-axis, and sufficiently far to the left of $A_{r+1} \cup A_{r+1}'$, so that if ℓ ₁ passes through a point in $A_{r+1}''$ and through a point in $S_{r+1}\setminus{(B_{r+1}''\cup C_{r+1}'')}$, and ℓ ₂ spans two points in $S_{r+1}\setminus{A_{r+1}''}$, then the slope of ℓ ₁ is smaller in absolute value than the slope of ℓ ₂.

7.2 Calculating E _≤k (S _r )

We fix r≥3, and proceed to determine E _≤k (S _r ) for each k, 0≤k≤4r−1. It is now convenient to label the elements of $A''_{r}, B''_{r}$, and $C''_{r}$. Let $a''_{1},a''_{2},\ldots,a''_{r}$ be the elements of $A''_{r}$, ordered as they appear from left to right along the negative x-axis. As expected, let $b''_{i}=\theta(a''_{i})$ and $c''_{i}=\theta^{2}(a''_{i})$, for i=1,…,r.

We call a k-edge bichromatic if it joins two points with different label letters (i.e., if it is of the form ab,bc, or ac); otherwise, a k-edge is monochromatic. A monochromatic edge is of type aa if it is of the form ℓ(a _i a _j ) for some integers i,j; edges of types aa′,aa″,a′a′,a′a″,a″a″ (and their counterparts for b and c) are similarly defined. Finally, we say that an edge of any of the types aa,aa′,aa″,a′a′,a′a″, or a″a″ is of type A; edges of types B and C are similarly defined. We let $E^{\mathrm{bic}}_{\le k}$ (respectively, $E^{\mathrm{mono}}_{\le k}$) stand for the number of bichromatic (respectively, monochromatic) ≤k-edges, so that $E_{\le k}(S_{r}) =E^{\mathrm{bic}}_{\le k}(S_{r}) +E^{\mathrm{mono}}_{\le k}(S_{r})$.

We say that a finite point set P is 3-decomposable if it can be partitioned into three equal-size sets $\overline{A}$, $\overline{B}$, and $\overline{C}$ satisfying the following: there is a triangle T enclosing P such that the orthogonal projections of P onto the three sides of T show $\overline{A}$ between $\overline{B}$ and $\overline{C}$ on one side, $\overline{B}$ between $\overline{A}$ and $\overline{C}$ on another side, and $\overline{C}$ between $\overline{A}$ and $\overline{B}$ on the third side (see [2]). We say that $\{\overline{A},\overline{B},\overline{C}\}$ is a 3-decomposition of P. It is easy to see that if we let $\overline{A}:=A_{r}\cup A_{r}'\cup A_{r}''$, $\overline{B}:=B_{r}\cup B_{r}'\cup B_{r}''$, and $\overline{C}:=C_{r}\cup C_{r}'\cup C_{r}''$, then $\{\overline{A}, \overline{B}, \overline{C}\}$ is a 3-decomposition of S _r : indeed, it suffices to take an enclosing triangle of S _r with one side orthogonal to the line spanned by the points in A″, one side orthogonal to the line spanned by the points in B″, and one side orthogonal to the line spanned by the points in C″. Thus, it follows from Claim 1 in [2] (where it is proved in the more general setting of allowable sequences) that

(13)

We now count the monochromatic ≤k-edges. By 3-symmetry, it suffices to focus on those of type A.

It is readily checked that for all i and j distinct integers, ℓ(a _i a _j ), $\ell(a'_{i} a'_{j})$, and $\ell(a''_{i} a''_{j})$ are k-edges for some k>4r−1. The same is true for $\ell(a_{i} a'_{j})$ whenever i and j are not both equal to 1 (when i≠1 and j≠1 this follows from (III) and (IV)), while $\ell(a_{1}a'_{1})$ is a (4r−1)-edge. Now, for 1≤i≤r and 2≤j≤r, the line $\ell(a_{i}'' a'_{j})$ separates 4r+i−j points of S _r from the rest. That is, $\ell(a_{i}''a'_{j})$ is a (4r+i−j)-edge if i−j≤r/2−1 or a (5r−2−i+j)-edge if i−j>r/2−1. Similarly, $\ell(a_{i}''a'_{1})$, $\ell(a_{i}'' a_{j})$, and $\ell(a_{i}'' a_{1})$ separate 4r+i−2, 3r+i+j−3, and 3r+i−1 points of S _r from the rest, respectively. In conclusion,

• (i) for 1≤s≤r, the number of (3r−1+s)-edges of types a′a″ or aa″ is 2s;

• (ii) there is exactly one (4r−1)-edge of type aa′; and

• (iii) all other edges of type A are k-edges for some k>4r−1.

It follows that the number of ≤k-edges of type A is

• (a) 0, for k≤3r−1;

• (b) $2\sum_{s=1}^{k-(3r-1)} s=2 {\tbinom{k-3r+2}{2}}$, for 3r≤k≤4r−2;

• (c) $1 + 2\sum_{s=1}^{(4r-1)-(3r-1)} s = 2 {\tbinom{r+1}{2}} +1$, for k=4r−1.

By 3-symmetry, for each integer k there are exactly as many (≤k)-edges of type A as there are of type B, and of type C. Therefore

(14)

Because $E_{\le k}(S_{r}) =E^{\mathrm{bic}}_{\le k}(S_{r}) +E^{\mathrm{mono}}_{\le k}(S_{r})$, it follows by (13) and (14) that E _≤k (S _r ) equals the right-hand side of (12).

8 Concluding Remarks

The inequality in Theorem 1 is best possible. That is, there are n-point sets P whose simple allowable sequence Π gives equality in the inequality of Corollary 1:

$$E_{\geq k}( \varPi ) = ( n-2k-1) E_{k-1}( \varPi ) + \binom{s}{2}.$$

We present two constructions. The first has s=n−2k−1 and consists of 2k+1 points which are the vertices of a regular polygon and n−2k−1 central points very close to the center of the polygon. This construction was given in [22] to show that E _k−1≥2k+1 is best possible. Indeed, note that the (k−1)-edges of P correspond to the larger diagonals of the polygon, and so E _k−1(Π)=2k+1; moreover, any edge formed by two points in the central part or one point in the central part and a vertex of the polygon determine a ≥k-edge. Thus $E_{\geq k}( \varPi )=\tbinom{n-2k-1}{2}+(2k+1)(n-2k-1)$, which achieves the desired equality.

The second construction has s=0 and thus it can only be achieved when k≥n/3. Consider a (2t+1)-regular polygon where each vertex is replaced by a set of m points on a small segment pointing in the direction of the center of the polygon. Let Π be the allowable sequence corresponding to this point set, n=(2t+1)m, and k=tm. It is straightforward to verify that E _k−1(Π)=(2t+1)m and $E_{\geq k} (\varPi )=2(2t+1)\tbinom{m}{2}$. Thus E _≥k (Π)=(m−1)E _k−1(Π)=(n−2k−1)E _k−1(Π).

Prior to this work, there were two results that provided a lower bound for E _≤k (P) based on the behavior of values of k close to n/2. First, Welzl [27] as a particular case of a more general result proved that E _≤k (P)≥F ₁(k,n), where

$$F_1(k,n)=\binom{n}{2}-2n \Biggl( \sum _{j=k+1}^{n/2}k \Biggr)^{1/2}<\binom{n}{2}-\frac{\sqrt{2}}{2}n^{3/2}\sqrt{n-2k}.$$

Second, Balogh and Salazar [12] proved that E _≤k (P)≥F ₂(k,n), where F ₂(k,n) is a function that, for n/3≤k≤n/2, satisfies

$$F_2(k,n) < \binom{n}{2}- \frac{13\sqrt{3}}{36}n^{3/2}\sqrt{n-2k}+o\bigl(n^{2}\bigr).$$

By direct comparison, it follows that both F ₁(k,n) and F ₂(k,n) are smaller than the bound in Corollary 3. Thus our bound is better than these two previous bounds.

A nice feature of Theorem 1 is that it can give better bounds for E _≤k (n) and k large enough, and for $\overline {\mathrm {cr}}(n)$, provided someone finds a better bound than inequality (2) for E _≤k (n) when 4n/9<k<n/2. For example, Ábrego et al. [5] considered 3-regular point sets P. These are point sets with the property that for 1≤j≤n/3, the jth depth layer of P has exactly 3 points of P. A point p∈P is in the jth depth layer if p belongs to a (j−1)-edge but not to a (≤j−2)-edge of P. If n is a multiple of 18, they proved the following lower bound:

$$E_{\leq k}(P) \geq3\binom{k+2}{2}+3\binom{k+2- n/3}{2}+18\binom{k+2- 4n/9}{2}. $$

(15)

This is better than the bound in Theorem 2 for k>4n/9, however using Theorem 1 it is possible to find an even better lower bound when k≥17n/36. We construct a new recursive sequence u′ starting at m=17n/36 given by

The value of m=17n/36 is the smallest possible for which $u^{\prime}_{m}$ is greater than the right-hand side of (15). Following the proof of Theorem 2 it is possible to show that $E_{\leq k}(P) \geq u^{\prime}_{k}$ for 17n/36≤k<n/2. Thus, if we could show that (15) holds for arbitrary point sets P, then we know that bound will no longer be tight for k≥17n/36. From equivalent statements to Lemmas 1 and 2, it follows that $u^{\prime}_{k} \sim \tbinom{n}{2}-(7\sqrt{2} n^{2}/18)\sqrt{1-2k/n}$. This in turn improves the crossing number of 3-regular point sets P to $\overline {\mathrm {cr}}(P)\geq0.380024\tbinom{n}{4}+\varTheta (n^{3})$.

In [2] we considered other class of point sets called 3-decomposable. These are point sets P for which there is a triangle T enclosing P and a balanced partition A, B, and C of P, such that the orthogonal projections of P onto the sides of T show A between B and C on one side, B between A and C on another side, and C between A and B on the third side. For 3-decomposable sets P we were able to prove a lower bound consisting of an infinite series of binomial coefficients:

$$E_{\leq k}(P) \geq3\binom{k+2}{2}+3\binom{k+2-n/3}{2} +3\sum _{j=2}^{\infty}j(j+1)\binom{k+2-c_{j}n}{2}, $$

(16)

where c _j =1/2−1/(3j(j+1)).

Our main result does not improve this lower bound, however it gives an interesting heuristic that provides some evidence about the potential truth of this inequality for unrestricted point sets P. If we assume that the sum of the first t+1 terms in the right-hand side of (16) is a lower bound for E _≤k (P), then, just as we outlined in the previous paragraph for t=2, Theorem 1 gives a better bound when k is big enough. This happens to be precisely when k≥c _t+1 n, which is also the value of k for which the next term in the sum of (16) gives a nonzero contribution.

It was also shown in [2] that (16) implies the following bound for 3-decomposable sets P:

$$\overline {\mathrm {cr}}(P) \geq \frac{2}{27}\bigl(15-\pi^2\bigr)\binom{n}{4}+\varTheta \bigl(n^3\bigr)>0.380029\binom {n}{4}+\varTheta \bigl(n^3\bigr). $$

(17)

Theorem 1 does not improve the $\tbinom{n}{4}$ coefficient, but it improves the speed of convergence. For instance, using Theorem 1 together with the first 30 terms of (16) gives a better bound than the one obtained solely from the first 101 terms of (16).

Finally, we reiterate our conjectures from [2] that inequalities (16) and (17) are true for unrestricted point sets P. We in fact conjecture that for every k and n, the class of 3-decomposable sets contains optimal sets for both E _≤k (n) and $\overline {\mathrm {cr}}(n)$.