Progress on Dirac's Conjecture

In 1951, Gabriel Dirac conjectured that every set P of n non-collinear points in the plane contains a point in at least n/2-c lines determined by P, for some constant c. The following weakening was proved by Beck and Szemer\'edi-Trotter: every set P of n non-collinear points contains a point in at least n/c lines determined by P, for some large unspecified constant c. We prove that every set P of n non-collinear points contains a point in at least n/37 lines determined by P. We also give the best known constant for Beck's Theorem, proving that every set of n points with at most k collinear determines at least n(n-k)/98 lines.


I
Let P be a finite set of points in the plane. A line that contains at least two points in P is said to be determined by P . In 1951, Dirac [6] made the following conjecture, which remains unresolved: Conjecture 1 (Dirac's Conjecture). Every set P of n non-collinear points contains a point in at least n 2 − c 1 lines determined by P , for some constant c 1 .
See reference [3] for examples showing that the n 2 bound would be tight. Note that if P is non-collinear and contains at least n 2 collinear points, then Dirac's Conjecture holds. Thus we may assume that P contains at most n 2 collinear points, and n 5. In 1961, Erdős [7] proposed the following weakened conjecture.
Conjecture 2 (Weak Dirac Conjecture). Every set P of n non-collinear points contains a point in at least n c 2 lines determined by P , for some constant c 2 .
In 1983, the Weak Dirac Conjecture was proved indepedently by Beck [4] and Szemerédi and Trotter [19], in both cases with c 2 unspecified and very large. We prove the Weak Dirac Conjecture with c 2 much smaller. (See references [8,9,11,13,17] for more on Dirac's Conjecture.) Theorem 3. Every set P of n non-collinear points contains a point in at least n 37 lines determined by P .
Theorem 3 is a consequence of the following theorem. The points of P together with the lines determined by P are called the arrangement of P .
Theorem 4. For every set P of n points in the plane with at most n 37 collinear points, the arrangement of P has at least n 2 37 point-line incidences.
Proof of Theorem 3 assuming Theorem 4. Let P be a set of n non-collinear points in the plane. If P contains at least n 37 collinear points, then every other point is in at least n 37 lines determined by P (one through each of the collinear points). Otherwise, by Theorem 4, the arrangement of P has at least n 2 37 incidences, and so some point is incident with at least n 37 lines determined by P .
In his work on the Weak Dirac Conjecture, Beck proved the following theorem [4].
Theorem 5 (Beck's Theorem). Every set P of n points with at most collinear determines at least c 3 n(n − ) lines, for some constant c 3 .
In Section 3 we use the proof of Theorem 4 and some simple lemmas to show that c 3 1 98 . Similar methods and a bit more effort yield c 3 1 93 (see [16] for details).

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The proof of Theorem 4 takes inspiration from the well known proof of Beck's Theorem [5] as a corollary of the Szemerédi-Trotter Theorem [19], and also from the simple proof of the Szemerédi-Trotter Theorem due to Székely [18], which in turn is based on the Crossing Lemma.
The crossing number of a graph G, denoted by cr(G), is the minimum number of crossings in a drawing of G. The following lower bound on cr(G) was first proved by Ajtai et al.
[2] and Leighton [12] (with worse constants). A simple proof with better constants can be found in [1]. The following version is due to Pach et al. [15].
Theorem 6 (Crossing Lemma). For every graph G with n vertices and m 103 16 n edges, In fact, we employ a slight strengthening of the Szemerédi-Trotter Theorem formulated in terms of visibility graphs. The visibility graph G of a point set P has vertex set P , where vw ∈ E(G) whenever the line segment vw contains no other point in P (that is, v and w are consecutive on a line determined by P ).
For i 2, an i-line is a line containing exactly i points in P . Let s i be the number of i-lines. Let G i be the spanning subgraph of the visibility graph of P consisting of all edges in j-lines where j i; see Figure 1 for an example. Note that since each i-line contributes i − 1 edges, |E(G i )| = j i (j − 1)s j . Part (a) of the following version of the Szemerédi-Trotter Theorem gives a bound on |E(G i )|, while part (b) is the well known version that bounds the number of j-lines for j i.
Theorem 7 (Szemerédi-Trotter Theorem). Let α and β be positive constants such that every graph H with n vertices and m αn edges satisfies Let P be a set of n points in the plane. Then Then by the assumed Crossing Lemma On the other hand, since two lines cross at most once, Combining these inequalities yields part (a). Part (b) follows directly from part (a).
Theorem 8 (Hirzebruch's Inequality). Let P be a set of n points with at most n − 3 collinear. Then Theorem 4 follows from Theorem 6 and the following general result by setting α = 103 16 , β = 31827 1024 , c = 71, and δ = , in which case δ 1 36.158 . The value of δ is readily calculated numerically since since where ζ is the Riemann zeta function.
Theorem 9. Let α and β be positive constants such that every graph H with n vertices and m αn edges satisfies Fix an integer c 8 and a real ∈ (0, 1 2 ). Let h := c(c−2) 5c−18 . Then for every set P of n points in the plane with at most n collinear points, the arrangement of P has at least δn 2 point-line incidences, where Proof. Let J := {2, 3, . . . , n }. Considering the visibility graph G of P and its subgraphs G i as defined previously, let k be the minimum integer such that |E(G k )| αn.
If there is no such k then let k := n + 1. An integer i ∈ J is large if i k, and is small if i c. An integer in J that is neither large nor small is medium.
An i-pair is a pair of points in an i-line. A small pair is an i-pair for some small i.
Define medium pairs and large pairs analogously, and let P S , P M and P L denote the number of small, medium and large pairs respectively. An i-incidence is an incidence between a point of P and an i-line. A small incidence is an i-incidence for some small i. Define medium incidences analogously, and let I S and I M denote the number of small and medium incidences respectively. Let I denote the total number of incidences. Thus, The proof procedes by establishing an upper bound on the number of small pairs in terms of the number of small incidences. Analogous bounds are proved for the number of medium pairs, and the number of large pairs. Combining these results gives the desired lower bound on the total number of incidences.
For the bound on small pairs, Hirzebruch's Inequality is useful. Since at most n 2 points are collinear and n 5, there are no more than n − 3 collinear points. Therefore, Hirzebruch's Inequality implies that hs 2 + 3h 4 s 3 − hn − h i 5 (2i − 9)s i 0 since h > 0. Thus, Considering the second partial derivative with respect to i shows that i−1 2 − 2h + 9h i is maximised for i = 5 or i = c. Some linear optimisation shows that, since c 8, X is minimised when h = c(c−2) 5c−18 and X = h+1 To bound the number of medium pairs, consider a medium i ∈ J. Since i is not large, j i (j − 1)s j > αn. Hence, using parts (a) and (b) of the Szemerédi-Trotter Theorem, Given the factor X in the bound on the number of small pairs in (1), it helps to introduce the same factor in the bound on the number of medium pairs. It will be convenient to define Y := c − 1 − 2X.

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Beck proved Theorem 5 as part of his work on Dirac's Conjecture [4]. Theorem 9 from the previous section and Lemmas 11 and 12 below can be used to give the best known constant in Beck's Theorem.
Theorem 10. Every set P of n points with at most collinear determines at least 1 98 n(n− ) lines.
The following lemma, due to Kelly and Moser [11], follows directly from Melchior's Inequality [14], which states that s 2 3 + i 4 (i − 3)s i . As before, I is the total number of incidences in the arrangement of P . Let E be the total number of edges in the visibility graph of P , and let L be the total number of lines in the arrangement of P .
When there is a large number of collinear points, the following lemma becomes stronger than Theorem 9.
Lemma 12. Let P be a set of n points in the plane such that some line contains exactly points in P . Then the visibility graph of P contains at least (n − ) edges.
Proof. Let S be the set of collinear points in P . For each point v ∈ S and for each point w ∈ P \ S, count the edge incident to w in the direction of v. Since S is collinear and w is not in S, no edge is counted twice. Thus E |S| · |P \ S| = (n − ).
Proof of Theorem 10. Assume is the size of the largest collinear subset of P . If A more direct approach similar to the methods used in the proof of Theorem 9 can be shown to improve Theorem 10 slightly to yield 1 93 n(n − ) lines. The details are omitted, but can be found in [16].
Beck's Theorem is often stated as a bound on the number of lines with few points. In his original paper Beck [4] mentioned briefly in a footnote that Lemma 11 implies the following.
Observation 13 (Beck). If P is not collinear, then at least half the lines determined by P contain 3 points or less.