PROGRESS ON DIRAC'S CONJECTURE
PROGRESS ON DIRAC’S CONJECTURE
arXiv:1207.3594v4 [math.CO] 17 Jun 2013
MICHAEL S. PAYNE AND DAVID R. WOOD
Abstract. In 1951, Gabriel Dirac conjectured that every set P of n non-collinear points in the plane contains a point in at least n2 − 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 nc lines determined by P , for some large unspecified constant c. We prove that every set P of n non-collinear points contains a n point in at least 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 ` collinear determines at 1 least 98 n(n − `) lines.
1. Introduction 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 n2 − c1 lines determined by P , for some constant c1 . See reference [3] for examples showing that the n2 bound would be tight. Note that if P is non-collinear and contains at least n2 collinear points, then Dirac’s Conjecture holds. Thus we may assume that P contains at most n2 collinear points, and n > 5. In 1961, Erd˝os [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 cn2 lines determined by P , for some constant c2 . In 1983, the Weak Dirac Conjecture was proved indepedently by Beck [4] and Szemer´edi and Trotter [19], in both cases with c2 unspecified and very large. We prove the Weak Dirac Conjecture with c2 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 determined by P .
n 37
lines
Date: May 2, 2014. Michael Payne is supported by an Australian Postgraduate Award from the Australian Government. Research of David Wood is supported by the Australian Research Council. 1
2
PROGRESS ON DIRAC’S CONJECTURE
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 2 the arrangement of P has at least n37 point-line incidences.
n 37
collinear points,
Proof of Theorem 3 assuming Theorem 4. Let P be a set of n non-collinear points in n collinear points, then every other point is in at the plane. If P contains at least 37 n least 37 lines determined by P (one through each of the collinear points). Otherwise, 2 by Theorem 4, the arrangement of P has at least n37 incidences, and so some point is n lines determined by P . incident with at least 37 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 c3 n(n − `) lines, for some constant c3 . In Section 3 we use the proof of Theorem 4 and some simple lemmas to show that c3 > 1 Similar methods and a bit more effort yield c3 > 93 (see [16] for details).
1 98 .
2. Proof of Theorem 4 The proof of Theorem 4 takes inspiration from the well known proof of Beck’s Theorem [5] as a corollary of the Szemer´edi–Trotter Theorem [19], and also from the simple proof of the Szemer´edi–Trotter Theorem due to Sz´ekely [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 > cr(G) >
103 16 n
edges,
1024 m3 . 31827 n2
In fact, we employ a slight strengthening of the Szemer´edi–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 si be the number of i-lines. Let Gi 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 P contributes i − 1 edges, |E(Gi )| = j>i (j − 1)sj . Part (a) of the following version of the
PROGRESS ON DIRAC’S CONJECTURE
G = G2
G3
3
G4
G5
Figure 1. The graphs G2 , G3 , G4 , G5 in the case of the 5 × 5 grid. Szemer´edi–Trotter Theorem gives a bound on |E(Gi )|, while part (b) is the well known version that bounds the number of j-lines for j > i. Theorem 7 (Szemer´edi–Trotter Theorem). Let α and β be positive constants such that every graph H with n vertices and m > αn edges satisfies m3 . βn2
cr(H) > Let P be a set of n points in the plane. Then (a)
X (j − 1)sj 6 max αn, j>i
and (b)
X
sj 6 max
j>i
Proof. Suppose applied to Gi ,
P
j>i (j
β n2 2(i − 1)2
αn β n2 , i − 1 2(i − 1)3
, .
− 1)sj = |E(Gi )| > αn. Then by the assumed Crossing Lemma
P P ( j>i (j − 1)sj )2 |E(Gi )| (i − 1)2 ( j>i sj )2 |E(Gi )| |E(Gi )|3 cr(Gi ) > = > . βn2 βn2 βn2 On the other hand, since two lines cross at most P j>i sj cr(Gi ) 6 6 2
once, 1 X 2
sj
2
.
j>i
Combining these inequalities yields part (a). Part (b) follows directly from part (a).
The proof of Theorem 4 also employs Hirzebruch’s Inequality [10]. Theorem 8 (Hirzebruch’s Inequality). Let P be a set of n points with at most n − 3 collinear. Then X 3 (2i − 9)si . s2 + s3 > n + 4 i>5
Theorem 4 follows from Theorem 6 and the following general result by setting α = 103 16 , 31827 1 β = 1024 , c = 71, and δ = , in which case δ > 36.158 . The value of δ is readily
4
PROGRESS ON DIRAC’S CONJECTURE
calculated numerically since since Xi+1 i>c
i3
=
Xi+1 i>1
i3
−
c−1 X i+1 i=1
i3
= ζ(2) + ζ(3) −
c−1 X i+1 i=1
i3
= 2.847 . . . −
c−1 X i+1 i=1
i3
,
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 m3 cr(H) > . βn2 Fix an integer c > 8 and a real ∈ (0, 12 ). 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 δn2 point-line incidences, where !! β (c − h − 2)(c + 1) X i + 1 1 1 − α − + . δ= h+1 2 c3 i3 i>c
Proof. Let J := {2, 3, . . . , bnc}. Considering the visibility graph G of P and its subgraphs Gi as defined previously, let k be the minimum integer such that |E(Gk )| 6 αn. If there is no such k then let k := bnc + 1. An integer i ∈ J is large if i > k, and is small if i 6 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 PS , PM and PL 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 IS and IM denote the number of small and medium incidences respectively. Let I denote the total number of incidences. Thus, X I= isi . i∈J
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 n2 points are collinear and n > 5, there are no more than n − 3 collinear points. Therefore, P Hirzebruch’s Inequality implies that hs2 + 3h i>5 (2i − 9)si > 0 since h > 0. 4 s3 − hn − h Thus, c X i PS = s2 + 3s3 + 6s4 + si 2 i=5 c c X X 3h i 6 (h + 1)s2 + + 3 s3 + 6s4 + si − hn − h (2i − 9)si 4 2 i=5
i=5
PROGRESS ON DIRAC’S CONJECTURE
5
c X h+1 h+4 3 i−1 9h · 2s2 + · 3s3 + · 4s4 + − 2h + isi − hn . 2 4 2 2 i i=5 h+1 h+4 3 i−1 Setting X := max 2 , 4 , 2 , max56i6c 2 − 2h + 9h implies that i 6
PS 6 XIS − hn .
(1)
9h Considering the second partial derivative with respect to i shows that i−1 2 − 2h + i is maximised for i = 5 or i = c. Some linear optimisation shows that, since c > 8, X is h+1 c−1 9h minimised when h = c(c−2) 5c−18 and X = 2 = 2 − 2h + c .
To bound the number of medium pairs, consider a medium i ∈ J. Since i is not large, P edi–Trotter Theorem, j>i (j − 1)sj > αn. Hence, using parts (a) and (b) of the Szemer´ (2)
X
jsj =
X
j>i
(j − 1)sj +
j>i
X
sj 6
j>i
βn2 βn2 i βn2 + = . 2(i − 1)2 2(i − 1)3 2(i − 1)3
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. ! ! k−1 k−1 X X i PM − XIM = si − X isi 2 i=c+1
=
1 2
k−1 X
i=c+1
(i − 1 − 2X) isi
i=c+1
k−1 1 X = (i − c + Y ) isi 2 i=c+1 k−1 X k−1 X Y 1 jsj + = 2 2 i=c+1 j=i
k−1 X
! isi
.
i=c+1
Applying (2) yields (3)
PM − XIM
β n2 6 4
c+1 Xi+1 Y 3 + c i3
! .
i>c
It remains to bound the number of large pairs: bnc X i n X n α n2 (4) PL = si 6 (i − 1)si = |E(Gk )| 6 . 2 2 2 2 i=k
i>k
Combining (1), (3) and (4), n 1 = (n2 − n) 6 PS + PM + PL 2 2 6 XIS − hn + XIM
β n2 + 4
c+1 Xi+1 Y 3 + c i3 i>c
! +
α n2 . 2
6
PROGRESS ON DIRAC’S CONJECTURE
Thus, I > IS + IM
1 > 2X
β 1 − α − 2
c+1 Xi+1 Y 3 + c i3
!!
i>c
n2 +
2h − 1 n . 2X
The result follows since h > 1.
3. A constant for Beck’s Theorem 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 `) lines.
1 98 n(n−
The following lemma, due to Kelly and Moser [11], follows directly from Melchior’s InP equality [14], which states that s2 > 3 + i>4 (i − 3)si . 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 . Lemma 11 (Kelly–Moser). If P is not collinear, then 3L > 3 + I, and since I = E + L, also 2L > 3 + E. 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 − `). n Proof of Theorem 10. Assume ` is the size of the largest collinear subset of P . If ` > 49 1 1 then E > 49 n(n − `) by Lemma 12 and thus L > 98 n(n − `) by Lemma 11. On the n δ 1 other hand, suppose ` 6 49 . Setting 2 = 3 and c = 67 in Theorem 9 gives 6 49 and 1 1 1 1 2 δ > 32.57 . So I > 32.57 n > 32.57 n(n − `) and thus L > 98 n(n − `) by Lemma 11.
A more direct approach similar to the methods used in the proof of Theorem 9 can be 1 shown to improve Theorem 10 slightly to yield 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.
PROGRESS ON DIRAC’S CONJECTURE
7
Observation 13 (Beck). If P is not collinear, then at least half the lines determined by P contain 3 points or less. Proof. By Lemma 11, 3s2 + 3s3 + 3
X i>4
si >
X
isi > 2s2 + 2s3 + 4
i>2
X
si .
i>4
Thus 2(s2 + s3 ) >
X
si ,
i>2
as desired.
Corollary 14. Every set P of n points with at most ` collinear determines at least 1 196 n(n − `) lines each with at most 3 points. References [1] Martin Aigner and G¨ unter M. Ziegler. Proofs from The Book. Springer, 3rd edn., 2004. MR: 2014872. [2] Mikl´ os Ajtai, Vaˇsek Chv´atal, Monroe M. Newborn, and Endre Szemer´edi. Crossing-free subgraphs. In Theory and practice of combinatorics, vol. 60 of NorthHolland Math. Stud., pp. 9–12. North-Holland, 1982. MR: 806962. [3] Jin Akiyama, Hiro Ito, Midori Kobayashi, and Gisaku Nakamura. Arrangements of n points whose incident-line-numbers are at most n/2. Graphs Combin., 27(3):321– 326, 2011. doi: 10.1007/s00373-011-1023-4. [4] J´ ozsef Beck. On the lattice property of the plane and some problems of Dirac, Motzkin and Erd˝os in combinatorial geometry. Combinatorica, 3(3-4):281–297, 1983. doi: 10.1007/BF02579184. MR: 0729781. [5] Wikipedia contributors. Beck’s theorem (geometry). Wikipedia, the free encyclopedia, 2010. http://en.wikipedia.org/wiki/Beck’s_theorem_(geometry). [6] Gabriel A. Dirac. Collinearity properties of sets of points. Quart. J. Math., Oxford Ser. (2), 2:221–227, 1951. doi: 10.1093/qmath/2.1.221. MR: 0043485. [7] Paul Erd˝ os. Some unsolved problems. Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl., 6:221–254, 1961. http://www.renyi.hu/~p_erdos/1961-22.pdf. MR: 0177846. [8] Paul Erd˝ os and George Purdy. Some combinatorial problems in the plane. J. Combin. Theory Ser. A, 25(2):205–210, 1978. doi: 10.1016/0097-3165(78)90085-7. MR: 0505545. [9] Paul Erd˝ os and George Purdy. Two combinatorial problems in the plane. Discrete Comput. Geom., 13(3-4):441–443, 1995. doi: 10.1007/BF02574054. MR: 1318787. [10] Friedrich Hirzebruch. Singularities of algebraic surfaces and characteristic numbers. In The Lefschetz Centennial Conference, Part I, vol. 58 of Contemp. Math., pp. 141–155. Amer. Math. Soc., 1986. doi: 10.1090/conm/058.1/860410. MR: 860410. [11] Leroy M. Kelly and William O. J. Moser. On the number of ordinary lines determined by n points. Canad. J. Math., 10:210–219, 1958. doi: 10.4153/CJM-1958-024-6. MR: 0097014.
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PROGRESS ON DIRAC’S CONJECTURE
[12] F. Thomson Leighton. Complexity Issues in VLSI. MIT Press, 1983. [13] Ben D. Lund, George B. Purdy, and Justin W. Smith. Some results related to a conjecture of Dirac’s. 2012. arXiv: 1202.3110. ¨ [14] E. Melchior. Uber Vielseite der projektiven Ebene. Deutsche Math., 5:461–475, 1941. MR: 0004476. [15] J´anos Pach, Radoˇs Radoiˇci´c, G´abor Tardos, and G´eza T´ oth. Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom., 36(4):527–552, 2006. doi: 10.1007/s00454-006-1264-9. MR: 2267545. [16] Michael S. Payne. Combinatorial geometry of point sets with collinearities. PhD dissertation, University of Melbourne, Department of Mathematics and Statistics, in preparation. [17] George Purdy. A proof of a consequence of Dirac’s conjecture. Geom. Dedicata, 10(1-4):317–321, 1981. doi: 10.1007/BF01447430. MR: 608148. [18] L´aszl´ o A. Sz´ekely. Crossing numbers and hard Erd˝os problems in discrete geometry. Combin. Probab. Comput., 6(3):353–358, 1997. doi: 10.1017/S0963548397002976. MR: 1464571. [19] Endre Szemer´edi and William T. Trotter, Jr. Extremal problems in discrete geometry. Combinatorica, 3(3-4):381–392, 1983. doi: 10.1007/BF02579194. MR: 729791.
Department of Mathematics and Statistics The University of Melbourne Melbourne, Australia E-mail address: [email protected]
School of Mathematical Sciences Monash University Melbourne, Australia E-mail address: [email protected]
arXiv:1207.3594v4 [math.CO] 17 Jun 2013
MICHAEL S. PAYNE AND DAVID R. WOOD
Abstract. In 1951, Gabriel Dirac conjectured that every set P of n non-collinear points in the plane contains a point in at least n2 − 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 nc lines determined by P , for some large unspecified constant c. We prove that every set P of n non-collinear points contains a n point in at least 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 ` collinear determines at 1 least 98 n(n − `) lines.
1. Introduction 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 n2 − c1 lines determined by P , for some constant c1 . See reference [3] for examples showing that the n2 bound would be tight. Note that if P is non-collinear and contains at least n2 collinear points, then Dirac’s Conjecture holds. Thus we may assume that P contains at most n2 collinear points, and n > 5. In 1961, Erd˝os [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 cn2 lines determined by P , for some constant c2 . In 1983, the Weak Dirac Conjecture was proved indepedently by Beck [4] and Szemer´edi and Trotter [19], in both cases with c2 unspecified and very large. We prove the Weak Dirac Conjecture with c2 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 determined by P .
n 37
lines
Date: May 2, 2014. Michael Payne is supported by an Australian Postgraduate Award from the Australian Government. Research of David Wood is supported by the Australian Research Council. 1
2
PROGRESS ON DIRAC’S CONJECTURE
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 2 the arrangement of P has at least n37 point-line incidences.
n 37
collinear points,
Proof of Theorem 3 assuming Theorem 4. Let P be a set of n non-collinear points in n collinear points, then every other point is in at the plane. If P contains at least 37 n least 37 lines determined by P (one through each of the collinear points). Otherwise, 2 by Theorem 4, the arrangement of P has at least n37 incidences, and so some point is n lines determined by P . incident with at least 37 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 c3 n(n − `) lines, for some constant c3 . In Section 3 we use the proof of Theorem 4 and some simple lemmas to show that c3 > 1 Similar methods and a bit more effort yield c3 > 93 (see [16] for details).
1 98 .
2. Proof of Theorem 4 The proof of Theorem 4 takes inspiration from the well known proof of Beck’s Theorem [5] as a corollary of the Szemer´edi–Trotter Theorem [19], and also from the simple proof of the Szemer´edi–Trotter Theorem due to Sz´ekely [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 > cr(G) >
103 16 n
edges,
1024 m3 . 31827 n2
In fact, we employ a slight strengthening of the Szemer´edi–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 si be the number of i-lines. Let Gi 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 P contributes i − 1 edges, |E(Gi )| = j>i (j − 1)sj . Part (a) of the following version of the
PROGRESS ON DIRAC’S CONJECTURE
G = G2
G3
3
G4
G5
Figure 1. The graphs G2 , G3 , G4 , G5 in the case of the 5 × 5 grid. Szemer´edi–Trotter Theorem gives a bound on |E(Gi )|, while part (b) is the well known version that bounds the number of j-lines for j > i. Theorem 7 (Szemer´edi–Trotter Theorem). Let α and β be positive constants such that every graph H with n vertices and m > αn edges satisfies m3 . βn2
cr(H) > Let P be a set of n points in the plane. Then (a)
X (j − 1)sj 6 max αn, j>i
and (b)
X
sj 6 max
j>i
Proof. Suppose applied to Gi ,
P
j>i (j
β n2 2(i − 1)2
αn β n2 , i − 1 2(i − 1)3
, .
− 1)sj = |E(Gi )| > αn. Then by the assumed Crossing Lemma
P P ( j>i (j − 1)sj )2 |E(Gi )| (i − 1)2 ( j>i sj )2 |E(Gi )| |E(Gi )|3 cr(Gi ) > = > . βn2 βn2 βn2 On the other hand, since two lines cross at most P j>i sj cr(Gi ) 6 6 2
once, 1 X 2
sj
2
.
j>i
Combining these inequalities yields part (a). Part (b) follows directly from part (a).
The proof of Theorem 4 also employs Hirzebruch’s Inequality [10]. Theorem 8 (Hirzebruch’s Inequality). Let P be a set of n points with at most n − 3 collinear. Then X 3 (2i − 9)si . s2 + s3 > n + 4 i>5
Theorem 4 follows from Theorem 6 and the following general result by setting α = 103 16 , 31827 1 β = 1024 , c = 71, and δ = , in which case δ > 36.158 . The value of δ is readily
4
PROGRESS ON DIRAC’S CONJECTURE
calculated numerically since since Xi+1 i>c
i3
=
Xi+1 i>1
i3
−
c−1 X i+1 i=1
i3
= ζ(2) + ζ(3) −
c−1 X i+1 i=1
i3
= 2.847 . . . −
c−1 X i+1 i=1
i3
,
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 m3 cr(H) > . βn2 Fix an integer c > 8 and a real ∈ (0, 12 ). 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 δn2 point-line incidences, where !! β (c − h − 2)(c + 1) X i + 1 1 1 − α − + . δ= h+1 2 c3 i3 i>c
Proof. Let J := {2, 3, . . . , bnc}. Considering the visibility graph G of P and its subgraphs Gi as defined previously, let k be the minimum integer such that |E(Gk )| 6 αn. If there is no such k then let k := bnc + 1. An integer i ∈ J is large if i > k, and is small if i 6 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 PS , PM and PL 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 IS and IM denote the number of small and medium incidences respectively. Let I denote the total number of incidences. Thus, X I= isi . i∈J
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 n2 points are collinear and n > 5, there are no more than n − 3 collinear points. Therefore, P Hirzebruch’s Inequality implies that hs2 + 3h i>5 (2i − 9)si > 0 since h > 0. 4 s3 − hn − h Thus, c X i PS = s2 + 3s3 + 6s4 + si 2 i=5 c c X X 3h i 6 (h + 1)s2 + + 3 s3 + 6s4 + si − hn − h (2i − 9)si 4 2 i=5
i=5
PROGRESS ON DIRAC’S CONJECTURE
5
c X h+1 h+4 3 i−1 9h · 2s2 + · 3s3 + · 4s4 + − 2h + isi − hn . 2 4 2 2 i i=5 h+1 h+4 3 i−1 Setting X := max 2 , 4 , 2 , max56i6c 2 − 2h + 9h implies that i 6
PS 6 XIS − hn .
(1)
9h Considering the second partial derivative with respect to i shows that i−1 2 − 2h + i is maximised for i = 5 or i = c. Some linear optimisation shows that, since c > 8, X is h+1 c−1 9h minimised when h = c(c−2) 5c−18 and X = 2 = 2 − 2h + c .
To bound the number of medium pairs, consider a medium i ∈ J. Since i is not large, P edi–Trotter Theorem, j>i (j − 1)sj > αn. Hence, using parts (a) and (b) of the Szemer´ (2)
X
jsj =
X
j>i
(j − 1)sj +
j>i
X
sj 6
j>i
βn2 βn2 i βn2 + = . 2(i − 1)2 2(i − 1)3 2(i − 1)3
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. ! ! k−1 k−1 X X i PM − XIM = si − X isi 2 i=c+1
=
1 2
k−1 X
i=c+1
(i − 1 − 2X) isi
i=c+1
k−1 1 X = (i − c + Y ) isi 2 i=c+1 k−1 X k−1 X Y 1 jsj + = 2 2 i=c+1 j=i
k−1 X
! isi
.
i=c+1
Applying (2) yields (3)
PM − XIM
β n2 6 4
c+1 Xi+1 Y 3 + c i3
! .
i>c
It remains to bound the number of large pairs: bnc X i n X n α n2 (4) PL = si 6 (i − 1)si = |E(Gk )| 6 . 2 2 2 2 i=k
i>k
Combining (1), (3) and (4), n 1 = (n2 − n) 6 PS + PM + PL 2 2 6 XIS − hn + XIM
β n2 + 4
c+1 Xi+1 Y 3 + c i3 i>c
! +
α n2 . 2
6
PROGRESS ON DIRAC’S CONJECTURE
Thus, I > IS + IM
1 > 2X
β 1 − α − 2
c+1 Xi+1 Y 3 + c i3
!!
i>c
n2 +
2h − 1 n . 2X
The result follows since h > 1.
3. A constant for Beck’s Theorem 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 `) lines.
1 98 n(n−
The following lemma, due to Kelly and Moser [11], follows directly from Melchior’s InP equality [14], which states that s2 > 3 + i>4 (i − 3)si . 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 . Lemma 11 (Kelly–Moser). If P is not collinear, then 3L > 3 + I, and since I = E + L, also 2L > 3 + E. 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 − `). n Proof of Theorem 10. Assume ` is the size of the largest collinear subset of P . If ` > 49 1 1 then E > 49 n(n − `) by Lemma 12 and thus L > 98 n(n − `) by Lemma 11. On the n δ 1 other hand, suppose ` 6 49 . Setting 2 = 3 and c = 67 in Theorem 9 gives 6 49 and 1 1 1 1 2 δ > 32.57 . So I > 32.57 n > 32.57 n(n − `) and thus L > 98 n(n − `) by Lemma 11.
A more direct approach similar to the methods used in the proof of Theorem 9 can be 1 shown to improve Theorem 10 slightly to yield 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.
PROGRESS ON DIRAC’S CONJECTURE
7
Observation 13 (Beck). If P is not collinear, then at least half the lines determined by P contain 3 points or less. Proof. By Lemma 11, 3s2 + 3s3 + 3
X i>4
si >
X
isi > 2s2 + 2s3 + 4
i>2
X
si .
i>4
Thus 2(s2 + s3 ) >
X
si ,
i>2
as desired.
Corollary 14. Every set P of n points with at most ` collinear determines at least 1 196 n(n − `) lines each with at most 3 points. References [1] Martin Aigner and G¨ unter M. Ziegler. Proofs from The Book. Springer, 3rd edn., 2004. MR: 2014872. [2] Mikl´ os Ajtai, Vaˇsek Chv´atal, Monroe M. Newborn, and Endre Szemer´edi. Crossing-free subgraphs. In Theory and practice of combinatorics, vol. 60 of NorthHolland Math. Stud., pp. 9–12. North-Holland, 1982. MR: 806962. [3] Jin Akiyama, Hiro Ito, Midori Kobayashi, and Gisaku Nakamura. Arrangements of n points whose incident-line-numbers are at most n/2. Graphs Combin., 27(3):321– 326, 2011. doi: 10.1007/s00373-011-1023-4. [4] J´ ozsef Beck. On the lattice property of the plane and some problems of Dirac, Motzkin and Erd˝os in combinatorial geometry. Combinatorica, 3(3-4):281–297, 1983. doi: 10.1007/BF02579184. MR: 0729781. [5] Wikipedia contributors. Beck’s theorem (geometry). Wikipedia, the free encyclopedia, 2010. http://en.wikipedia.org/wiki/Beck’s_theorem_(geometry). [6] Gabriel A. Dirac. Collinearity properties of sets of points. Quart. J. Math., Oxford Ser. (2), 2:221–227, 1951. doi: 10.1093/qmath/2.1.221. MR: 0043485. [7] Paul Erd˝ os. Some unsolved problems. Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl., 6:221–254, 1961. http://www.renyi.hu/~p_erdos/1961-22.pdf. MR: 0177846. [8] Paul Erd˝ os and George Purdy. Some combinatorial problems in the plane. J. Combin. Theory Ser. A, 25(2):205–210, 1978. doi: 10.1016/0097-3165(78)90085-7. MR: 0505545. [9] Paul Erd˝ os and George Purdy. Two combinatorial problems in the plane. Discrete Comput. Geom., 13(3-4):441–443, 1995. doi: 10.1007/BF02574054. MR: 1318787. [10] Friedrich Hirzebruch. Singularities of algebraic surfaces and characteristic numbers. In The Lefschetz Centennial Conference, Part I, vol. 58 of Contemp. Math., pp. 141–155. Amer. Math. Soc., 1986. doi: 10.1090/conm/058.1/860410. MR: 860410. [11] Leroy M. Kelly and William O. J. Moser. On the number of ordinary lines determined by n points. Canad. J. Math., 10:210–219, 1958. doi: 10.4153/CJM-1958-024-6. MR: 0097014.
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PROGRESS ON DIRAC’S CONJECTURE
[12] F. Thomson Leighton. Complexity Issues in VLSI. MIT Press, 1983. [13] Ben D. Lund, George B. Purdy, and Justin W. Smith. Some results related to a conjecture of Dirac’s. 2012. arXiv: 1202.3110. ¨ [14] E. Melchior. Uber Vielseite der projektiven Ebene. Deutsche Math., 5:461–475, 1941. MR: 0004476. [15] J´anos Pach, Radoˇs Radoiˇci´c, G´abor Tardos, and G´eza T´ oth. Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom., 36(4):527–552, 2006. doi: 10.1007/s00454-006-1264-9. MR: 2267545. [16] Michael S. Payne. Combinatorial geometry of point sets with collinearities. PhD dissertation, University of Melbourne, Department of Mathematics and Statistics, in preparation. [17] George Purdy. A proof of a consequence of Dirac’s conjecture. Geom. Dedicata, 10(1-4):317–321, 1981. doi: 10.1007/BF01447430. MR: 608148. [18] L´aszl´ o A. Sz´ekely. Crossing numbers and hard Erd˝os problems in discrete geometry. Combin. Probab. Comput., 6(3):353–358, 1997. doi: 10.1017/S0963548397002976. MR: 1464571. [19] Endre Szemer´edi and William T. Trotter, Jr. Extremal problems in discrete geometry. Combinatorica, 3(3-4):381–392, 1983. doi: 10.1007/BF02579194. MR: 729791.
Department of Mathematics and Statistics The University of Melbourne Melbourne, Australia E-mail address: [email protected]
School of Mathematical Sciences Monash University Melbourne, Australia E-mail address: [email protected]
