A Pseudoline Counterexample to the Strong Dirac Conjecture

We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of n pseudolines has no member incident to more than 4n/9 points of intersection. This shows the " Strong Dirac " conjecture to be false for pseudolines. We also raise a number of open problems relating to possible differences between the structure of incidences between points and lines versus the structure of incidences between points and pseudolines.


Introduction
Given an arrangement L of lines in the real projective plane, P 2 , let r(L) be the maximum number of vertices of L on any line of L. In 1951, G. Dirac (working in the dual context of point sets) conjectured a lower bound on r(L).
Conjecture 1 (Strong Dirac). Let L be an arrangement of n lines in P 2 that do not all pass through a single point. There exists a constant c such that r(L) ≥ n/2 − c.
In 1961, Erdős proposed a weaker version of the Strong Dirac conjecture [9]. It was proved independently in 1983 both by Beck [3] and by Szemerédi and Trotter [14], and holds for arrangements of pseudolines. An arrangement of pseudolines in the real projective plane is an arrangement of simple closed curves, any pair of which meet at a single crossing point.
Theorem 2 (Weak Dirac). Let L be an arrangement of n pseudolines in P 2 that do not all pass through a single point. Then r(L) = Ω(n).
In this paper, we show that Conjecture 1 does not hold for arrangements of pseudolines, and that analogs of Theorem 2 hold for various types of arrangements.
There are relatively few combinatorial properties that are known to hold for line arrangements and not for pseudoline arrangements. The results in this paper show that the techniques used to prove the Weak Dirac theorem may be applied more generally than they have been in the past, but probably can't be used to prove the Strong Dirac conjecture.
The paper is organized as follows. Section 2 focuses on the Strong Dirac conjecture. Traditionally, the Strong Dirac conjecture has been studied from the perspective of point sets. In this setting, the conjecture is that any set of n points includes a point incident to n/2 − c lines spanned by the point set. There are symmetries inherent in known extremal examples, that are easier to see in the dual setting. We initially review previous results from the dual perspective used in this paper.
In Section 2.1, we describe a technique of visualizing line and pseudoline arrangements with dihedral symmetry by presenting only a single wedge, which can be used to reconstruct the entire arrangement. This method was introduced by Eppstein, on his blog [8], and further developed by Berman in an investigation of simplicial pseudoline arrangements [4].
In Section 2.2, we present an infinite family of arrangements of pseudolines, such that an arrangement of n pseudolines from this family has no member incident to more than 4n/9 − 10/9 vertices of the arrangement. The family of pseudolines presented was previously studied by Berman [4], in the context of simplicial arrangements. This is the first time an infinite family of pseudolines has been demonstrated to violate the conclusion of the Strong Dirac conjecture.
Section 3 focuses on showing that the relationship between incidence bounds and lower bounds on the maximum number of intersection points on any line carries over to other types of arrangements. This relationship was used by both Beck [3] and by Szmerédi and Trotter [14] to prove the Weak Dirac. Szemerédi and Trotter proved an asymptotically tight lower bound on the number of incidences between a set of points and lines in the plane.
Theorem 3 (Szmerédi, Trotter [14]). Given a set of n lines in P 2 , no more than O(n 2 /k 3 + n/k) points are incident to k or more of the lines.
Beck proved a similar, weaker, bound. Interestingly, even though his incidence bound was not tight, Beck was able to use it to prove an asymptotically tight Weak Dirac theorem.
In Section 3.1, we formally define the class of arrangements we will consider. Informally, we consider arrangements of curves in a plane (not necessarily over an ordered field), each pair of which intersects some fixed number of times. We also prove incidence and Dirac-type bounds that follow directly from the specified combinatorics; these are analogous to results proved for line arrangements prior to 1983.
The Szemerédi-Trotter theorem is now one of many upper bounds on the number of incidences between finite sets P, L of geometric objects. In Section 3.2, we show that, for arrangements that have the combinatorial properties specified in Section 3.1, these bounds may be used to derive results analogous to the Weak Dirac theorem. In Section 3.3, we use the result of Section 3.2 to show that • maximally intersecting families of n "well-behaved" curves, as defined by Pach and Sharir [12], include a member incident to Ω(n) points of intersection; • families of n (d/2)-flats in P d (real projective d-space), with each pair of flats intersecting at a point, include a member incident to Ω(n 1− ) points of intersection, for any constant > 0; and • there exists a constant δ > 0 such that families of n lines in F p P 2 (finite projective plane), with n < p, include a member to Ω(n 1/2+δ ) points of intersection.

Strong Dirac
In 1951, Dirac conjectured that among any set of n non-collinear points, P , there must exist a point incident to at least n 2 lines spanned by P [7]. This bound can be attained for odd n when the points lie on two intersecting lines. Typically, Dirac's original conjecture is stated in a slightly weaker form (i.e., the "Strong Dirac"). 1 In [1], Akiyama et al. show that the n 2 bound (i.e., the Strong Dirac conjecture with c = 0) can be attained for all sufficiently large n except those of the form 12k + 11 (which they left as an open problem). However, there exists a family of configurations, with an arbitrarily large number of points, for which the conjecture is false for c = 0. This infinite family of counterexamples is due to Felsner and contains 6k + 7 points with none incident to more than 3k + 2 spanned lines when k is even, and 3k + 3 when k is odd. [6, p. 313] The dual form for this family is demonstrated in Figure 1. Figure 1: The dual of Felsner's arrangement with 6k + 7 = 31 lines (including the line at infinity) and no line incident to more than 3k + 2 = 14 points of intersection.

∞
No infinite family of arrangements of n such that each member has fewer than n/2 − 3/2 intersection points, but Grünbaum found several small arrangements with that property [10,11]. The line arrangement A [25,5] in [11] is the smallest member of the infinite family of pseudoline arrangements presented below.

Wedge presentation of symmetric pseudoline arrangements
A beautiful feature of Figure 1 is its symmetry. This drawing has the symmetry of a regular hexagon (i.e., the dihedral group D 6 ). While studying simplicial pseudoline arrangements (ones in which each planar face has three sides), Eppstein observed that arrangements with dihedral symmetry can be generated, similar to a kaleidoscope, from the contents of a single "wedge" [8]. Figure 2 shows a single wedge from Felsner's arrangement.
He noted that the entire path of a line through an arrangement can be traced by considering that line to be "bouncing", like a laser beam bouncing off mirrors, from one side of the wedge to the other. (Notice that in Figure 2 the beams must "retrace" their path after the third bounce.) In fact for straight-line arrangements, this bouncing must follow the law of reflection: the angle of incidence equals the angle of reflection. By applying basic trigonometry, one may deduce for straight-line arrangements the number and locations of the bounces as a function of the wedge angle and the beam's initial angle of incidence.
To generate an arrangement from a wedge, the wedge must have an angle of π/k for some positive integer k ≥ 2. The arrangement is produced by alternately rotating and duplicating the wedge or its mirror image, k times each, so that they fill the plane.
For pseudoline arrangements, the "bouncing" beams need not obey the law of reflection. As with Felsner's arrangement a beam might retrace its path after the k 2 th bounce. Berman, in [4], further develops Eppstein's "kaleidoscope" method to construct and classify many types of symmetric simplicial pseudoline arrangements (including the one presented in Section 2.2).

Pseudoline counterexample to Strong Dirac
Theorem 4. For any j ∈ N + , there exists an arrangement of n = 18j +7 pseudolines such that no pseudoline is incident to more than 8j + 2 vertices.
Proof. We will describe the construction of a wedge for a pseudoline arrangement for arbitrary j, and show that it has the claimed number of pseudolines and intersection property. We refer to Berman [4, Fig.11] for a proof that the described wedge actually represents a pseudoline arrangement.
For an arbitrary j, the wedge angle will be π/(6j + 2). There are four distinct symmetry classes of pseudolines, plus the line at infinity. Two of these will be represented by the sides of the wedge; we will call these the top and bottom edges. Two will be represented by beams; we will call these the red and blue beams.
Let r i be the point at which the i th bounce of the red beam occurs, counting from infinity. Likewise, let b i be the point at which the i th bounce of the blue beam occurs. After the beams reach the points r 3j+1 or b 3j+1 , respectively, the beams "retrace" their paths. More specifically, for any j, r k = r 6j+2−k and b k = b 6j+2−k .
We call r 3j+1 and b 3j+1 the "terminating points" for their respective beams. Prior to reaching its terminating point, every third bounce of the blue beam coincides with a bounce of the red beam (i.e., r i = b 3i for i ≤ j). The two beams are parallel to the bottom edge before the first bounce, and both b 1 and r 1 are on the top edge.
We will proceed by induction. For j = 1, the theorem holds; the arrangement generated from this wedge contains 3(6j + 2) + 1 = 25 pseudolines, each of which incident to at most 8j + 2 = 10 vertices. See Figure  3 for the wedge, and Figure 4 for the associated arrangement.
Assume that the theorem holds for j − 1. While maintaining for the points each existing bounce their distances to the corner of the wedge from the previous case, we reduce the wedge angle to π/(6j +2). In order to produce from this new wedge a valid arrangement, we must specify how to construct {r 3j−1 , r 3j , r 3j+1 } and {b 3j−1 , b 3j , b 3j+1 } for their respective beams.
We begin by extending the red by placing r 3j−1 , r 3j , and r 3j+1 on alternating sides of the wedge, each slightly closer to the corner of the wedge than the previous. This extension adds only 6 vertices to its associated red lines.
To extend the blue beam, we must cross the red beam placing b 3j−1 on the opposite side of the wedge. The subsequent point, b 3j , coincides with r j . Finally, place b 3j+1 at an appropriate location on the opposite  side of the edge, slightly farther from the corner than r j+1 . This extension adds a total of eight vertices for the associated blue lines, and two more for the red lines.
We must now consider the additional vertices formed on the sides of the wedge (which correspond to the axes of symmetry). To one set of axes, we added eight vertices each; to the other, we added only six each. See Figure 5 for the j = 2 case, i.e., the first complete "extension".

Weak Dirac
In this section, we show that incidence bounds imply analogs to the Weak Dirac theorem for a variety of different types of arrangements.

α-Curve combinatorics
First, we will define exactly the class of arrangements we will consider. Definition 5. Let G = (L, P, I) be a connected, simple, bipartite graph on vertex sets L and P and edge set I. If each pair of elements in L is connected to exactly α elements of P , then G is an α-curve combinatorics. This is a generalization of a line combinatorics, as defined by Bartolo et. al. [2]. It can be thought of as the unordered combinatorics of a set of curves in the plane, with each pair of curves intersecting α times. Definition 6. For any curve combinatorics G = (L, P, I), let t k (G) be the number of elements of P with degree k and, let r(G) be the maximum degree of any element in L.
Some bounds on t k and r follow from the definition of α-curve combinatorics. The following upper bounds on t k are a generalization of Lemma 2.1 in [3].
Proof. 1. We will count triplets (l, l , p), with l, l ∈ L, p ∈ P , and (l, p), (l , p) ∈ I, in two different ways. There are n 2 pairs of elements in L; each of these pairs is adjacent to α elements in P . Thus, there are α n 2 such triplets. There are at least t k (G) elements of P having degree k or more; each of these is adjacent to k 2 or more pairs of lines. Thus, the number of such triplets is at least t k (G) k 2 . Solving the inequality yields the result.

2.
Let P = {p 1 , p 2 , ..., p t }, and let L i ⊆ L be the elements of L that are adjacent to p i .
Dirac's original proof of the Ω(n 1/2 ) bound on r relied only on combinatorial arguments [7]. The following is a generalization of his argument.
Theorem 8. Let G = (L, P, I) be a α-curve combinatorics with |L| = n such that no set of α elements of P is adjacent to every member of L. Then r(G) = Ω(n 1/(α+1) ).
Proof. Let g be the maximum degree of any element of L. Let h be the maximum number of elements of L adjacent to any subset P α ⊂ P of size α.
We will show that g ≥ h. Let P ⊂ P be a subset of size α with each member adjacent to the same set L ⊂ L of size h. By our hypothesis, there exists a l ∈ L not adjacent to every element of P . By the definition of an α-curve combinatorics, there must be α paths of length 2 between l and each element of L . However, no two elements of L can both be adjacent to the same element p ∈ (P \ P ), otherwise there will be a K 2,α+1 . Thus, l must be adjacent to at least h elements of P , and thus g ≥ h.
Assume h = n x , for some value x. Let l ∈ L have degree g. Since l has α paths of length 2 to each l ∈ L \ l, g α h ≥ n − 1, which implies g ≥ n (1−x)/α . We've demonstrated that g ≥ max(n x , n (1−x)/α ) which has its minimum value when x = 1/(α + 1). This completes the proof.
Proof. For any pair {l , l } ∈ L × L, l = l , let P {l ,l } ⊂ P be the elements to which both l and l are adjacent. Note that |P {l ,l } | = α for all pairs {l , l }. Let l d be the number of pairs {l , l } such that min(deg(p) : p ∈ P {l ,l } ) = d. Counting pairs of elements in L two ways, (1) Proposition 10. For any c < 1, there exists a v such that Proof. Let 2 j = Ω(n ζ ). The total number of points with degree 2 j (i.e., t 2 j ) is bounded above by C(n 2+δ /2 j(2+ ) + n/2 j ) for some constant C. In addition, an element of P with degree 2 j+1 is adjacent to at most 2 j+1 2 pairs of elements in L. Thus, .
Since δ/ ≤ γ, Choosing a sufficiently large v ensures that the sum is less than c n 2 , completing the proof of Proposition 10.
Proof. Proposition 10 together with equation 1 immediately implies that either 1.
Select a point p 1 ∈ P with deg(p 1 ) ≥ n/2 v . Let L 1 ⊆ L be the set of nodes adjacent to p 1 . Let P 1 be the set of elements in P \ {p 1 } that are adjacent to some element in L 1 . Construct G 1 = (L 1 , P 1 , I 1 ) as the induced subgraph formed by L 1 and P 1 . Clearly, G 1 is a (α − 1)-curve combinatorics.
This process may be repeated until either: • we find a subset P j ⊆ P , j ≤ α, with |P j | = Ω((n/2 jv ) 2−γ ) = Ω(n 2−γ ), or • we can construct a complete bipartite subgraph of G containing the nodes This completes the proof of Proposition 11.
By Proposition 11, there are two cases to consider. First, assume |P | = Ω(n 2−γ ). Since every point of P is adjacent to at least one line of L, the pigeonhole principle implies that there must be a line adjacent to Ω(n 1−γ ) points of P , proving the theorem. Now, assume there exists a complete bipartite subgraph G = (L , P , I ), G ⊆ G with |L | = Ω(n) and |P | = α. If |L | = n, then G is a K n,α , proving the theorem. Otherwise, there exists an element l ∈ (L \ L ) not adjacent to all α elements of P . By the definition of a curve combinatorics, l must have α paths of length 2 to every element of L . However, since no two elements of L can both be incident to the same point in P \ P (otherwise a forbidden subgraph would exist), l must be adjacent to |L | = Ω(n) elements of P \ P .
Let C be a set of simple curves in the plane. The set C is said to have β degrees of freedom and multiplicity-type α, if • for β points there are at most α curves passing through all of them, and • any pair of curves intersect in at most α points.
These two conditions correspond, respectively, to a forbidden K β+1,α and a forbidden K 2,α+1 in the bipartite graph encoding the incidences between such a set of curves and a set of points.
Pach and Sharir proved the following upper bound on the number of incidences between "well-behaved" curves and points [12].
Theorem 12. Let P be a set of m points and let C be a set of n simple curves all lying in the plane. If C has β degrees of freedom and multiplicity-type α, then the number of incidences between P and C is O α,β m β/(2β−1) n (2β−2)/(2β−1) + m + n .
We can use this with Lemma 9 to prove the following corollary.
Corollary 13. Let C be an arrangement of n simple curves all lying in the plane. If C has β degrees of freedom, multiplicity-type α, and every pair of curves intersects exactly α times, then either all of the curves intersect in a single set of α points, or r(C) = Ω α,β (n).
Proof. Let t k = t k (C). The corollary will follow immediately from Lemma 9 and the following upper bound on the number of curves incident to k or more points: Since there are at most α n 2 points, this inequality clearly holds for small k. Otherwise, by Theorem 12 we have Thus, either t k = O(n/k) or t k = O(n 2 /k 2+ ), where = 1/(α − 1).

(d/2)-Flats in R d
Solymosi and Tao [13] proved an incidence bound for d/2 dimensional algebraic varieties in R d under some pseudoline-like conditions. Their theorem can be used to prove a corresponding bound on r for such varieties.
Rather than stating their result in its full generality, we'll consider only the special case of p-flats in R d , with d ≥ 2p.
Theorem 14. Let > 0, p ≥ 1, and d ≥ 2p. Let P be a set of m points, and let L be a set of n p-flats in R d such that any two distinct flats in L intersect in at most one point. Then, the number of incidences between L and P is O p, m 2/3+ n 2/3 + m + n . Proof. Let t k = t k (L). The corollary will follow immediately from Lemma 9 and the following upper bound on the number of p-flats incident to k or more points: t k = O p, (n 2+ /k 3+ + n/k).
Since there are at most n 2 points, this inequality holds for small k. From Theorem 14, kt k = O p, (t 2/3+ k n 2/3 + n + t k ).

Finite planes without too many lines
We can also use the following theorem of Bourgain, Katz, and Tao [5]. Let F p P 2 be the projective finite plane over the field with p elements, for p prime.
Theorem 16 (Bourgain, Katz, Tao). Let P and L be points and lines in F p P 2 with cardinality |P |, |L| ≤ N ≤ p. Then the number of incidences between L and P is for some universal constant > 0.
A significant question is whether (and by how much) the bound on r(L) for line arrangements differs from that for pseudoline arrangements. Problem 2. Is it possible to prove a lower bound on r(L) that holds for line arrangements and not for pseudoline arrangements?
One feature of the family of pseudoline arrangements presented in Section 2.2 is that (n − 1)/3 lines are all incident to a single vertex. A natural question is whether this is an essential feature of any pseudoline counterexample to the Strong Dirac 2 . Problem 3. Is there an infinite family of arrangements of n pseudolines, such that • no vertex of any arrangement in the family is incident to Ω(n) pseudolines, and • no member of any arrangement is incident to more than n/(2 + ) pseudolines for some > 0? Both Felsner's example, and the example presented in Section 2.2 have a high degree of symmetry. Assuming that the Strong Dirac holds for line arrangements, it may be easier to prove for the special case of symmetric line arrangements.