To prove Theorem 4, we have to get rid of the assumption in Theorem 1 that the curves are pairwise intersecting. We achieve this in two steps. First, in Subection 3.1, we state and prove a separation result between the number of touchings and the number of intersections that does not assume strict pairwise intersection, but still assumes a very dense intersection graph. Then, in Subection 3.2, we apply planar separation arguments to get rid of this milder assumption.
In both of these steps, we lose in the separating function, namely, the exponent of the log log function decreases. We did not attempt to optimize for this exponent, because we believe that even a logarithmic separation should hold, as stated in Conjecture 1.
3.1 Sampling
In this subsection, we prove the following lemma. Note that, like Theorem 4, it is about open Jordan curves, not closed ones.
Lemma 6 Let A be a family of n simple open Jordan curves in general position in the plane.
Let T be the set of touching points between curves of A and let X be the set of intersection points. With h=n2/|T| and f =|X|/|T| we have f72h144 = Ω(log logn).
Proof: As the statement of the lemma is asymptotic, we may assume in the calculations below thatn is sufficiently large.
We select a pair of distinct curves a0, b0 ∈ A. We try to select them so as to satisfy these conditions (see Figure 13):
(a) For m0 =|a0∩b0|we wantm0 ≤120f h.
(b) For m1 =|(a0∪b0)∩X|we want m1 ≤80nf h.
(c) Let m2 be the number of touchings t∈T between two curves in A \ {a0, b0} with both of these curves intersectinga0∪b0 (see Figure 13). We wantm2 ≥ |T|/(20h2).
By selecting the pair (a0, b0) uniformly at random, the expectation of m0 is E[m0] =
|X|/ n2
<3f /h. We have E[m1]≤2|X|/n= 2nf /h. For the expectation of m2 notice that at most |T|/2 touchings are contained in a curve a∈ A with |a∩T| ≤ |T|/(2n). The remaining elements of T (at least |T|/2 of them) are counted in m2 with probability at least |T2|/(5n4) each, yielding E[m2]≥ |T|3/(10n4) =|T|/(10h2).
By Markov’s inequality condition (a) fails with probability less than 1/(40h2) and the same holds for condition (b). Using Markov’s inequality again and the fact that m2 ≤ |T| we have that condition (c) is satisfied with probability at least 1/(20h2). Thus, all three conditions are simultaneously satisfied with some positive probability. We fix such a choice of the curves a0
and b0 and call them the ground curves.
∆ b0
a a0
b
Figure 13: Proof of Lemma 6. We select a pair a0, b0 ∈ Asatisfying three properties including that at least|T|/(20h2) of the touchings involve a pair of curvesa, b∈ A \ {a0, b0}both intersectinga0∪b0. We then choose an open cell ∆⊂R2\(a0∪b0) which contains at least|T|/(2400f h3) of these touchings.
Let the family A′ ⊆ A \ {a0, b0} consist of the curves that intersect at least one of a0 or b0. By property (c), these curves create m2 ≥ |T|/(20h2) touchings.
Let us consider the arrangement of a0 and b0. In case the ground curves are disjoint there is a single cell of this arrangement and its boundary is a0∪b0. We will treat this somewhat peculiar case later. Otherwise, the arrangement has m0 ≤120f h cells, each with a connected boundary. For each open cell ∆ ⊂ R2\(a0 ∪b0), let T∆ ⊂T ∩∆ denote the set of touching points between the curves ofA′ within ∆ (again, see Figure 13). By the pigeon-hole principle, there exists an open cell ∆⊂R2\(a0∪b0) with
|T∆| ≥ |T|/(2400f h3).
We fix such a cell ∆. We consider each connected component of a∩∆ for curves a∈ A′. These are simple Jordan curves in ∆ with at least one end point on the boundary. We make the curves slightly shorter to make sure each has exactly one endpoint on the boundary but they still determine the same set T∆ of touchings. We denote by A′′ the resulting family of m≤n+m1 ≤(80f h+ 1)n curves; see Figure 14 (left).
We can slightly inflate the boundary of ∆ to a simple closed Jordan curve c ⊂ ∆ with c intersecting each curvea∈ A′′ exactly once, close to the end point of a on the boundary of ∆ and with all the touching pointsT∆ on the side ∆′ of c contained in ∆. Let us enumerate the curves inA′′ asA′′={a1, a2, . . . , am} such that the intersection pointspi =ai∩c appear on c in this cyclic order.
b0
∆
′c pi ai
a0
pj aj p1 p2
pm
pℓ+1
pj pℓ
∆
′ai
pm p1 pi
bi
bj
aj
Figure 14: Proof of Lemma 6 – constructing the familiesF and G. Left: We obtain a new family A′′
by trimming each curve ofAto ∆. We then slightly inflate the boundary of ∆ to a simple closed Jordan curvec which encloses a region ∆′⊂∆ and meets eachai∈ A′′ at a single pointpi. Right: We choose a random index 1≤ℓ≤m and augment eachai∈ A′′ with a curvebi outside ∆′, with the property that bi and bj intersect at a single point if and only if 1 ≤i, j ≤ℓ or ℓ < i, j ≤m, and otherwise they are disjoint.
We pick a parameter 1 ≤ ℓ ≤ m = |A′′| and form the family F by slightly modifying the curvesaifor 1≤i≤ℓand formGby slightly modifyingaiforℓ < i≤m. The slight modification consists of keeping ai∩∆′ and attaching a curve bi to it that starts atpi and is disjoint from
∆′; see Figure 14 (right). We choose these additional curves such that (i) the curvesbi and bi′ are disjoint if i≤ ℓ < i′, (ii) the distinct curves bi and bi′ intersect exactly once if i, i′ ≤ℓ or i, i′ > ℓ and (iii) the curves bi are in general position. Clearly, such curves bi exist.
The family F consist of at mostm pairwise intersecting Jordan curves and the same is true forG. With adding dummy curves we can actually assume that both families consist of m+n curves.
Let us choose ℓ uniformly at random. The touching between the curves ai and ai′ with i < i′ will remain a touching between the corresponding modified curves (one in F, the other one in G) if we have i≤ℓ < i′. This happens with probability (i′ −i)/m. Among the |T∆| ≥
|T|/(2400f h3) such touchings at most half can be between curvesai andai′ withi < i′ < i+x forx=|T|/(4800f h3m). Each touching point of the other half ofT∆remains a touching points between a curve inF and a curve inGwith probability at leastx/m. Thus, the expected number of touchings between a curve inF and curve in Gis at least (|T∆|/2)·(x/m). We choose ℓsuch that the actual number of these touchings is at least this expectation.
Some of the intersection points between curves inF ∪G come fromX. We have one remaining intersection point between any two curves inF and also between any two curves inG for a total of O(m2+n2) additional intersection points.
We are almost ready to apply Theorem 2 to finish the proof. The only hurdle yet to clear is to pass from open Jordan curves to closed ones. This can simply be done by slightly inflating each curve. The process can be done in such a way that (i) touching curves remain touching, (ii) intersecting curves remain intersecting, (iii) the general position property is preserved and (iv) the number of intersection points is multiplied by at most 4. With this we obtain families F′ and G′, each consisting of m+n pairwise intersecting simple closed Jordan curves with the total number of intersections between curves inF ∪ GbeingO(|X|+m2+n2) =O(f2h2n2) and with the number of tangencies between a curve inF and a curve inG at least (|T∆|/2)·(x/m) = Ω(f−4h−10n2). Applying Theorem 2 to these families F′ and G′ yields the statement of the lemma.
Finally, we have to consider the special case when the ground curves a0 and b0 chosen in the first step of our proof are disjoint. In this case the arrangement of the ground curves has a single cell ∆ =R2\(a0∪b0). We defineA′′ exactly as in the general case, so A′′ consists of curves contained in ∆ with one end point on one of the ground curves. We distinguish “type-a”
or “type-b” curves in A′′ according to whether it has an end point on a0 or on b0.
If at least one third of the touchings between two curves ofA′′are between two type-a curves, then we simply ignoreb0and the type-b curves and consider the cell ∆∗ =R2\a0 and the type-a curves. We can finish the proof as in the general case as ∆∗ has a connected boundary.
If at least one third of the touchings between two curves ofA′′are between two type-b curves, then we proceed analogously.
If none of the above two cases hold, then we concentrate on the touchings between a type-a and a type-b curve: at least one third of all touchings between curves of A′′ must be like this.
The situation is even simpler in this case with no need for any random choice. We obtainF by modifying slightly the type-a curves and we obtain G by modifying slightly the type-b curves.
For this we have to separately inflate the two ground curves. We finish the proof as in the general case. ♠
3.2 Separation
The main ingredient we need for the proof of Theorem 4 is the following separator theorem of Fox and Pach [FP08] for intersection graphsof families of Jordan arcs.
Theorem 5 For any collection Aof n Jordan arcs in the plane in general position with a total of x intersection points, there is a subset B ⊆ A of cardinality O(√
x) so that A \ B can be divided into disjoint subsets A1 and A2 of cardinality at most 2n/3 each with the property that no arc ofA1 meets an arc of A2.
Repeated application of this result yields the following.
Lemma 7 Let Abe a collection ofn simple Jordan curves in general position in the plane. Let d be maximum number of intersection points in a curve of A and let T be an arbitrary subset of the intersection points with |T|2≥nd3. There exist a subset A0 ⊆ A of Θ(n2d3/|T|2) curves such that Ω(nd3/|T|) of the points in T are intersection points of curves of A0.
Proof: Let us set a threshold parameter 1≤M ≤n. We split theAby finding a small separator subsetB ⊂ Aand partitioningA \B intoA1 andA2, as described in Theorem 5. We recursively apply this procedure to the families A1 and A2, stopping only when we obtain subsets of size less than M. Let B′ denote the set of all separator arcs that are removed at any invocation of the recursion and let A′i for i∈ I stand for the final partition of A \ B′. Note that, by the properties of the separation, the curves inA′i do not intersect curves from other partsA′j,j6=i.
When we split a set A′ ⊆ A of size |A′| = m ≥ M, we find a subset B′ ⊆ A′ with size
|B′|= O(√
dm), as the curves in A′ determine at most dm intersection points. The resulting parts ofA′\ B′ have size at most 2m/3.
The final parts produced by the above partitioning satisfy |A′i| < M, since this was our halting condition. Taking into account that these parts were obtained by splitting a subset of size at leastM, they also satisfy|A′i| ≥M/3−O(√
M d).
Consider all the different parts A′ obtained in intermediate steps of the above partitioning process. Those sets of size |A′| =m in an interval (3/2)iM ≤m < (3/2)i+1M for some fixed integer i are clearly pairwise disjoint, so their number is at most n/((3/2)iM). We find a separator of sizeO(p
(3/2)iM d) for each one of them. The total contribution of this interval to the size ofB′ isO((3/2)−i/2M−1/2nd1/2). We sum these contributions for the integers 0≤i≤ log3/2n, and obtain |B′|=O(np
d/M).
We set M = Cn2d3/|T|2 with a large constant C. We may assume this yields M ≤ n, for otherwise the choice A0 = A satisfies the requirements of the lemma. It is clear from our bound on |B′| that if C is large enough, we have |B′| ≤ |T|/(2d). Analogously, from
|A′i| ≥M/3−O(√
M d) that holds for all i∈I, we get|A′i| ≥M/4 if C is large enough.
During the partition, we lose at most |B′|d≤ |T|/2 intersections between the curves of A, that is, for at least half of the points t ∈ T, neither of the two curves of A through t are in B′. Both of these curves are therefore in the same part A′i as curves from distinct parts are disjoint. By the pigeonhole principle, there is a partA′i0 with at least|T|/(2|I|) of the points of T, showing up as intersection points between curves of A′i0. The choice A0 =A′i0 satisfies the requirements of the lemma since |A′i0| = Θ(M) = Θ(n2d3/|T|2) and |T|/(2|I|) = Ω(nd3/|T|), where we use that|A′i| ≥M/4 and therefore|I| ≤n/(M/4) =O(|T|2/(nd3)). ♠
Proof of Theorem 4: As the statement of the theorem is asymptotic in nature we may assume that |T|/n is sufficiently large. Note that |T|/n < n, so this also means that n is sufficiently large. We introduce the notation f =|X|/|T|.
We first reduce the maximum number of intersection points on a curve inAto d=⌊|X|/n⌋. To this end, we break each arc a ∈ A into sub-arcs a1, . . . , ah all of which, with the possible exception of the last one, contain exactly d intersection points. This splitting yields a family A′ ofn′ = Θ(n) Jordan arcs without modifying the setX of intersection points or the setT of touching points.
If |T|2 < n′d3 we apply Lemma 6 to A′. The lemma claims that f72h144 = Ω(log logn′) for h = n′2/|T|. We further have |T|2 < n′d3 = O(|X|3/n2) yielding h < O(f3) and thus f72h144 =O(f504). The statement of the theorem follows.
If |T|2 ≥n′d3, then we apply Lemma 7 to the collection A′ and the set of touching points T. Let A′0 ⊆ A′ be the collection whose existence is claimed by this lemma. The size |A′0| of this collection is n′0 = Θ(n′2d3/|T|2) = Θ(f2|X|/n). The family determines x′0 ≤ n′0d = O(f2|X|2/n2) intersections among which t′0 = Ω(n′d3/|T|) = Ω(f|X|2/n2|) are touchings. We apply Lemma 6 to the familyA′0. Withf0′ =x′0/t′0=O(f) andh′0 =n′02/t′0 =O(f3), the lemma states f0′72h′1440 = Ω(log logn′0). Here f0′72h′1440 = O(f504) and n′0 =θ(f2|X|/n) = Ω(|X|/n) = Ω(|T|/n). The statement of the theorem follows again. ♠
References
[ANPPSS04] P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir, and S. Smorodinsky, Lenses in arrangements of pseudocircles and their applications, J. ACM51(2004), 139–186.
[AgS05] P. K. Agarwal and M. Sharir, Pseudo-line arrangements: duality, algorithms, and ap-plications,SIAM J. Comput. 34 (2005), no. 3, 526–552.
[ACNS82] M. Ajtai, V. Chv´atal, M. Newborn, E. Szemer´edi, Crossing-free subgraphs. In: The-ory and Practice of Combinatorics, North-Holland Mathematics Studies 60, North-Holland, Amsterdam, 1982, pp. 9-12.
[An70] E. M. Andreev, Convex polyhedra of finite volume in Lobaˇcevski˘i space,Mat. Sb. (N.S.), 83(1970), no. 125, 256 - 260.
[ArS02] B. Aronov and M. Sharir, Cutting circles into pseudo-segments and improved bounds for incidences, Discrete Comput. Geom. 28(2002), no. 4, 475–490.
[Ch1] T. M. Chan, On levels in arrangements of curves, Discrete Comput. Geom. 29 (2003), 375–393. (Also inProc. 41th IEEE Sympos. Found. Comput. Sci. (FOCS), 2000, pp. 219–227.) [Ch2] T. M. Chan, On levels in arrangements of curves, II: a simple inequality and its conse-quence,Discrete Comput. Geom.34(2005), 11–24. (Also inProc. 44th IEEE Sympos. Found.
Comput. Sci. (FOCS), 2003, pp. 544–550.)
[Ch3] T. M. Chan, On levels in arrangements of curves, III: further improvements, Proc. 24th ACM Symposium on Computational Geometry (SoCG), 2008, pp. 85–93.
[Cha00] B. Chazelle, The Discrepancy Method. Randomness and Complexity, Cambridge Uni-versity Press, Cambridge, 2000.
[CEGSW90] K. L. Clarkson, H. Edelsbrunner, L. J. Guibas, M. Sharir, and E. Welzl, Combi-natorial complexity bounds for arrangements of curves and spheres,Discrete Comput. Geom.
5 (1990), no. 2, 99–160.
[CS89] K. L. Clarkson and P. W. Shor, Applications of random sampling in computational geometry. II, Discrete Comput. Geom.4 (1989), no. 5, 387–421.
[De98] T. K. Dey, Improved bounds for planar k-sets and related problems, Discrete Comput.
Geom. 19(1998), no. 3, 373–382.
[Dv10] Z. Dvir, Incidence theorems and their applications, Found. Trends Theor. Comput. Sci.
6 (2010), no. 4, 257–393 (2012).
[Ed87] H. Edelsbrunner,Algorithms in Combinatorial Geometry. EATCS Monographs on The-oretical Computer Science, 10, Springer-Verlag, Berlin, 1987.
[ESZ16] J. S. Ellenberg, J. Solymosi, and J. Zahl, New bounds on curve tangencies and orthog-onalities, Disc. Anal. 22(2016), 1 – 22.
[EH89] P. Erd˝os and A. Hajnal, Ramsey-type theorems, Discrete Appl. Math. 25 (1989), no.
1-2, 37–52.
[Er46] P. Erd˝os, On sets of distances of n points,Amer. Math. Monthly 53(1946), 248–250.
[FFPP10] J. Fox, F. Frati, J. Pach, and R. Pinchasi: Crossings between curves with many tangencies, in: WALCOM: Algorithms and Computation, Lecture Notes in Comput. Sci.5942, Springer-Verlag, Berlin, 2010, 1–8. Also in: An Irregular Mind, Bolyai Soc. Math. Stud. 21, J´anos Bolyai Math. Soc., Budapest, 2010, 251–260.
[FP08] J. Fox and J. Pach, Separator theorems and Tur´an-type results for planar intersection graphs, Adv. in Math.219 (2008), 1070–1080.
[FP10] J. Fox and J. Pach, A separator theorem for string graphs and its applications,Combin.
Probab. Comput. 19 (2010), 371–390.
[FP12] J. Fox and J. Pach, String graphs and incomparability graphs,Adv. in Math.230 (2012), 1381–1401.
[FPT11] J. Fox, J. Pach, and C. D. T´oth, Intersection patterns of curves,J. London Math. Soc.
83 (2011), 389-406.
[G] L. Guth, Polynomial partitioning for a set of varieties, manuscript.
[GK10] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem, Adv. Math.225 (2010), no. 5, 2828–2839.
[G13] L. Guth, Unexpected applications of polynomials in combinatorics, in: The Mathematics of Paul Erd˝os. I, Springer, New York, 2013, 493–522.
[G15] L. Guth, Distinct distance estimates and low degree polynomial partitioning, Discrete Comput. Geom. 53(2015), no. 2, 428–444.
[GK15] L. Guth and N. H. Katz, On the Erd˝os distinct distances problem in the plane,Ann. of Math. (2) 181 (2015), no. 1, 155–190.
[HW87] D. Haussler and E. Welzl, ε-nets and simplex range queries, Discrete Comput. Geom.
2 (1987), no. 2, 127–151.
[KLPS86] K. Kedem, R. Livn´e, J. Pach, and M. Sharir, On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles, Discrete Comput. Geom.1 (1) (1986), 59 – 71.
[Koe36] P. Koebe, Kontaktprobleme der Konformen Abbildung, Ber. S¨achs. Akad. Wiss.
Leipzig, Math.-Phys. Kl.88 (1936), 141 - 164.
[KST54] T. K˝ov´ari, V. S´os, and P. Tur´an, On a problem of K. Zarankiewicz, Colloquium Math.
3 (1954), 50 - 57.
[Le83] T. Leighton, Complexity Issues in VLSI, Foundations of Computing Series, MIT Press, Cambridge, 1983.
[MaT06] A. Marcus and G. Tardos, Intersection reverse sequences and geometric applications, J. Combin. Theory Ser. A 113(2006), no. 4, 675–691.
[Ma14] J. Matouˇsek, Near-optimal separators in string graphs, Combin. Probab. Comput. 23 (2014), no. 1, 135–139.
[Mu02] D. Mubayi: Intersecting curves in the plane,Graphs Combin.18(2002), no. 3, 583–589.
[PRT15] J. Pach, N. Rubin, and G. Tardos, On the Richter-Thomassen conjecture about pair-wise intersecting closed curves, in: Proc. 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015, San Diego), SIAM, 2015, 1506–1515. Also Combin. Prob. Comput.
25(2016), no. 6, 941 – 958.
[PRT16] J. Pach, N. Rubin, and G. Tardos, Beyond the Richter-Thomassen Conjecture, in:
Proc. 27th Annual ACM-SIAM Symposium on Discrete Algorithms(SODA 2016, Arlington), SIAM, 2016, 957–968.
[PaS98] J. Pach and M. Sharir, On the number of incidences between points and curves,Combin.
Probab. Comput. 7(1998), no. 1, 121–127.
[PaS09] J. Pach and M. Sharir,Combinatorial Geometry and its Algorithmic Applications. The Alcal´a lectures. Mathematical Surveys and Monographs, 152, American Mathematical Society, Providence, RI, 2009.
[PST12] J. Pach, A. Suk, and M. Treml, Tangencies between families of disjoint regions in the plane,Comput. Geom. 45(2012), no. 3, 131–138.
[RiT95] R. B. Richter and C. Thomassen, Intersections of curve systems and the crossing number of C5×C5,Discrete Comput. Geom. 13(1995), no. 2, 149–159.
[Sa99] G. Salazar: On the intersections of systems of curves, J. Combin. Theory Ser. B 75 (1999), no. 1, 56–60.
[ShA95] M. Sharir and P. K. Agarwal: Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, Cambridge, 1995.
[SSZ15] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between points and circles, Combin. Probab. Comput.24 (2015), no. 3, 490–520.
[SzT83a] E. Szemer´edi and W. T. Trotter, Jr., Extremal problems in discrete geometry, Com-binatorica3 (1983), no. 3-4, 381–392.
[SzT83b] E. Szemer´edi and W. T. Trotter, Jr., A combinatorial distinction between the Eu-clidean and projective planes, European J. Combin.4 (1983), no. 4, 385–394.
[TT98] H. Tamaki and T. Tokuyama, How to cut pseudoparabolas into segments, Discrete Comput. Geom. 19(1998), no. 2, 265–290.
[Thu97] W. Thurston, Three-dimensional Geometry and Topology, Vol. 1, Silvio Levy Ed., Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997.
[Tut70] W. T. Tutte, Toward a theory of crossing numbers, J. Combinatorial Theory 8 (1970), 45 - 53.