A Pseudoline Counterexample to the Strong Dirac Conjecture

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1 A Pseudolne Counterexample to the Strong Drac Conjecture George B. Purdy Ben Lund Dept. of Computer Scence Rutgers Unversty Pscataway, New Jersey, U.S.A. Dept. of Electrcal Engneerng and Computng Systems Unversty of Cncnnat Cncnnat, Oho, U.S.A. Justn W. Smth Dept. of Computer Scence Northern Kentucky Unversty Kentucky, U.S.A. Submtted: Feb 14, 2012; Accepted: Apr 23, 2014; Publshed: May 13, 2014 Mathematcs Subject Classfcaton: 52C10 Abstract We demonstrate an nfnte famly of pseudolne arrangements, n whch an arrangement of n pseudolnes has no member ncdent to more than 4n/9 ponts of ntersecton. Ths shows the Strong Drac conjecture to be false for pseudolnes. We also rase a number of open problems relatng to possble dfferences between the structure of ncdences between ponts and lnes versus the structure of ncdences between ponts and pseudolnes. 1 Introducton A central problem of dscrete geometry s to elucdate the structure of ncdences between ponts and lnes. Untl the recent exploson of applcatons of polynomal methods to problems n ncdence geometry ([7, 16, 19]), the tools most successfully appled to questons about ncdences between ponts and lnes could be mmedately appled to prove equvalent results for ncdences between ponts and pseudolnes. By an arrangement of pseudolnes, we mean a set of smple closed curves n the real projectve plane, any par of whch meet at a sngle crossng pont [10, p. 79]. Gven an arrangement L of lnes n the real projectve plane, P 2, let r(l) be the maxmum number of vertces on any lne of L (a vertex of L s a pont ncdent to at least the electronc journal of combnatorcs 21(2) (2014), #P2.31 1

2 2 lnes of L). In 1951, G. Drac (workng n the dual context of pont sets) conjectured a lower bound on r(l) [6]. Conjecture 1 (Strong Drac). Let L be an arrangement of n lnes n P 2 that do not all pass through a sngle pont. There exsts a constant c such that r(l) n/2 c. In ths paper, we show that the old and wdely beleved Strong Drac conjecture s false when generalzed from arrangements of lnes to arrangements of pseudolnes. We gve an explct example of an nfnte famly of pseudolne arrangements L for whch r(l) 4n/9 for any L L In 1961, Erdős proposed a weaker verson of the Strong Drac conjecture [9]. It was proved ndependently n 1983 both by Beck[2] and by Szemeréd and Trotter[18], and holds for arrangements of pseudolnes. Theorem 2 (Weak Drac). Let L be an arrangement of n pseudolnes n P 2 that do not all pass through a sngle pont. Then r(l) = Ω(n). For straght lnes, the constant n Theorem 2 can be taken to be 1/37, as shown n [17]. The proof n [17] reles on Hrzebruch s nequalty [14], an algebrac result not known to hold for pseudolnes. Tradtonally, the Strong Drac conjecture has been studed from the perspectve of pont sets. In that settng, the conjecture s that any set of n ponts ncludes a pont ncdent to n/2 c lnes spanned by the pont set. However, there are symmetres nherent n known extremal examples that are easer to see n the context of lne arrangements. We brefly revew these examples n Secton 2. In Secton 2.1, we descrbe a technque of vsualzng lne and pseudolne arrangements wth dhedral symmetry by presentng only a sngle wedge, whch can be used to reconstruct the entre arrangement. Ths method was ntroduced by Eppsten on hs blog [8], and further developed by Berman n an nvestgaton of smplcal pseudolne arrangements [3]. In Secton 2.2, we present an nfnte famly of arrangements of pseudolnes, such that an arrangement of n pseudolnes from ths famly has no member ncdent to more than 4n/9 10/9 vertces of the arrangement. The famly of pseudolnes presented here was prevously studed by Berman [3], n the context of smplcal arrangements. Ths s the frst tme an nfnte famly of pseudolnes has been demonstrated to volate the concluson of the Strong Drac conjecture. In Secton 3, we ask a number of questons relatng to the central queston of what dfferences exst between the structure of ncdences between ponts and lnes versus the structure of ncdences between ponts and pseudolnes. the electronc journal of combnatorcs 21(2) (2014), #P2.31 2

3 2 Strong Drac conjecture In 1951, Drac conjectured that among any set of n non-collnear ponts, P, there must exst a pont ncdent to at least n lnes spanned by P [6]. Ths bound can be attaned for 2 odd n when the ponts le on two ntersectng lnes. Typcally, Drac s orgnal conjecture s stated n a slghtly weaker form (.e., the Strong Drac ). In [1], Akyama et al. show that the n bound (.e., the Strong Drac conjecture 2 wth c = 0) can be attaned for all suffcently large n except those of the form 12k + 11 (whch they left as an open problem). However, there exsts a famly of pont sets, wth an arbtrarly large number of ponts, for whch the conjecture s false for c = 0. Ths nfnte famly of counterexamples s due to Felsner and contans 6k + 7 ponts wth none ncdent to more than 3k + 2 spanned lnes when k s even, and 3k + 3 when k s odd. [4, p. 313] The dual form for ths famly s demonstrated n Fgure 1. Fgure 1: The dual of Felsner s arrangement wth 6k + 7 = 31 lnes (ncludng the lne at nfnty) and no lne ncdent to more than 3k + 2 = 14 ponts of ntersecton. No nfnte famly of arrangements of n lnes s known such that each member has fewer than n/2 3/2 ntersecton ponts, but Grünbaum found several small arrangements wth that property [12, 13]. The lne arrangement A[25,5] n [13] s the smallest member of the nfnte famly of pseudolne arrangements presented below. 2.1 Wedge presentaton of symmetrc pseudolne arrangements A beautful feature of Fgure 1 s ts symmetry. Ths drawng has the symmetry of a regular hexagon (.e., the dhedral group D 6 ). Whle studyng smplcal pseudolne arrangements (ones n whch each planar face has three sdes), Eppsten observed that the electronc journal of combnatorcs 21(2) (2014), #P2.31 3

4 arrangements wth dhedral symmetry can be generated, smlar to a kaledoscope, from the contents of a sngle wedge [8]. Fgure 2 shows a sngle wedge from Felsner s arrangement. Fgure 2: A sngle wedge from Felsner s arrangement. He noted that the entre path of a lne through an arrangement can be traced by consderng that lne to be bouncng, lke a laser beam bouncng off mrrors, from one sde of the wedge to the other. (Notce that n Fgure 2 the beams must retrace ther path after the thrd bounce.) In fact for straght-lne arrangements, ths bouncng must follow the law of reflecton: the angle of ncdence equals the angle of reflecton. By applyng basc trgonometry, one may deduce for straght-lne arrangements the number and locatons of the bounces as a functon of the wedge angle and the beam s ntal angle of ncdence. To generate an arrangement from a wedge, the wedge must have an angle of π/k for some postve nteger k 2. The arrangement s produced by alternately rotatng and duplcatng the wedge or ts mrror mage, k tmes each, so that they fll the plane. For pseudolne arrangements, the bouncng beams need not obey the law of reflecton. As wth Felsner s arrangement a beam mght retrace ts path after the k 2 th bounce. Berman, n [3], further develops Eppsten s kaledoscope method to construct and classfy many types of symmetrc smplcal pseudolne arrangements (ncludng the one presented n Secton 2.2). 2.2 Pseudolne counterexample to Strong Drac conjecture Theorem 3. For any j N +, there exsts an arrangement of n = 18j + 7 pseudolnes such that no pseudolne s ncdent to more than 8j + 2 vertces. We wll descrbe the constructon of a wedge for a pseudolne arrangement for arbtrary j, and show that t has the clamed number of pseudolnes and ntersecton property. We refer to Berman [3, Fg.11] for a proof that the descrbed wedge actually represents a pseudolne arrangement. the electronc journal of combnatorcs 21(2) (2014), #P2.31 4

5 For an arbtrary j, the wedge angle wll be π/(6j+2). There are four dstnct symmetry classes of pseudolnes, plus the lne at nfnty. Two of these wll be represented by the sdes of the wedge; we wll call these the top and bottom edges. Two wll be represented by beams; we wll call these the red and blue beams. See Fgure 3. Let r (j) be the pont at whch the th bounce of the red beam occurs along one of the two edges, countng from nfnty. Lkewse, let b (j) be the pont at whch the th bounce of the blue beam occurs. When the mpled value of j s obvous, we smply refer to these ponts as r and b. After the beams reach the ponts r 3j+1 or b 3j+1, respectvely, the beams retrace ther paths. More specfcally for any k > 3j + 1, r k = r 6j+2 k and b k = b 6j+2 k. We call r 3j+1 and b 3j+1 the termnatng ponts for ther respectve beams. Pror to reachng ts termnatng pont, every thrd bounce of the blue beam concdes wth a bounce of the red beam. For all j, r = b 3 when j. The two beams are parallel to the bottom edge before the frst bounce, and both b 1 and r 1 are on the top edge. Proof. We proceed by nducton. For j = 1, the theorem holds; the arrangement generated from ths wedge contans 3(6j + 2) + 1 = 25 pseudolnes, each of whch ncdent to at most 8j + 2 = 10 vertces. See Fgure 3 for the wedge, and Fgure 4 for the assocated arrangement. b 1 b 3 = r 1 r 3 r 4 r 2 b 4 b 2 Fgure 3: The wedge for j = 1, the base case for our nducton. Assume that the theorem holds for j 1. We start wth the wedge produced from the (j 1) th case, and adjust ts angle to be π/(6j + 2). Let r (j 1) and b (j 1) be the ponts of the th bounce of the red and blue beams, respectvely, from the precedng case. We defne the r (j) and b (j) as follows: For < 3j 1, let r (j) For > 3j + 3, let r (j) = r (j 1) = r (j 1) 6 and b (j) and b (j) = b (j 1). = b (j 1) 6. We must specfy for the j th case how to construct the trples {r 3j 1, r 3j, r 3j+1 } and {b 3j 1, b 3j, b 3j+1 } for ther respectve beams. Note that r 3j+2 = r 3j and r 3j+3 = r 3j 1, and lkewse for the b. the electronc journal of combnatorcs 21(2) (2014), #P2.31 5

6 Fgure 4: The arrangement for j = 1, contanng 3(6j + 2) + 1 = 25 pseudolnes. Each pseudolne s ncdent to at most 10 vertces. We begn wth the smpler case of extendng the red beam. We place r 3j 1 on the sde opposte of the wedge from r 3j 2, and contnue to alternate sdes when placng r 3j and r 3j+1, each slghtly closer to the corner of the wedge than the prevous. To extend the blue beam we must cross the red beam once, placng b 3j 1 on the opposte sde of the wedge from b 3j 2. The subsequent pont, b 3j, concdes wth r j. As stated prevously, b 3 = r when j. Lastly, place b 3j+1 at an approprate locaton on the opposte sde of the edge (farther from the corner than r j+1 ). Wth ths, the constructon s complete. We must now consder the addtonal vertces (relatve to the (j 1) th case) formed on the lnes, resultng from ths constructon. For the blue lnes, a total of eght vertces were added, and lkewse, eght more were added for the red lnes. The edges of the wedge correspond to the lnes of the arrangement formng axes of symmetry. For one class of axes, we added eght vertces each. To the other, we added only sx each. (Whether the lnes gettng an addtonal sx vertces correspond to the top or bottom of the wedge depends on the party of j.) See Fgure 5 for the j = 2 case,.e., the frst complete extenson from the base case. In the resultng arrangement, there wll be 18j + 7 = (18(j 1) + 7) + (6 3) lnes wth none ncdent to more than 8j + 2 = (8(j 1) + 2) + 8 vertces, completng the nductve proof. the electronc journal of combnatorcs 21(2) (2014), #P2.31 6

7 b 1 b 3 =r 1 b 5 r 7 r 5 r3 b 7 r 6 r 4 b 6 =r 2 b 4 b 2 3 Open problems Fgure 5: The wedge for j = 2. The man nterest of the pseudolne arrangement presented here s that t shows that a natural conjecture that s wdely beleved to be true for straght lnes s defntely false for pseudolnes. Ths s relevant to a more general queston: how do the structural constrants on the ncdences between straght lnes and ponts dffer from those on ncdences between pseudolnes and ponts? In ths secton, we rase a number of specfc open problems on ths general theme. 3.1 Varatons on the Strong Drac There s no reason to expect that 4/9 s the best possble constant n the Weak Drac theorem for pseudolnes, and the gap between 4/9 and the best known lower bound s qute large. Problem 1. What s the supremum of values c for whch for all pseudolne arrangements L? r(l) cn + o(n) The next queston s whether (and by how much) the bound on r(l) for lne arrangements dffers from that for pseudolne arrangements. Problem 2. Is t possble to prove a lower bound on r(l) that holds for lne arrangements and not for pseudolne arrangements? One feature of the famly of pseudolne arrangements presented n Secton 2.2 s that (n 1)/3 lnes are all ncdent to a sngle vertex. A natural queston s whether ths s an essental feature of any pseudolne counterexample to the Strong Drac conjecture. Problem 3. Is there an nfnte famly of arrangements of n pseudolnes, such that no vertex of any arrangement n the famly s ncdent to Ω(n) pseudolnes, and no member of any arrangement s ncdent to more than n/(2 + ɛ) vertces for some ɛ > 0? the electronc journal of combnatorcs 21(2) (2014), #P2.31 7

8 The authors are not even aware of an nfnte famly of pseudolne arrangements such that no vertex s ncdent to Ω(n) pseudolnes and no pseudolne s ncdent to more than cn vertces for some c < 1. Both Felsner s example, and the example presented n Secton 2.2 have dhedral symmetry. Assumng that the Strong Drac conjecture holds for lne arrangements, t may be easer to prove for the specal case of dhedrally symmetrc lne arrangements. Problem 4. Does the Strong Drac conjecture hold for lne arrangements wth dhedral symmetry? The example presented here shows that any method used to gve an affrmatve answer to Problem 4 would need to be able to dstngush between lne arrangements and pseudolne arrangements. 3.2 Drac-Motzkn for Pseudolnes Another classc queston from ncdence geometry concerns the mnmum number of ordnary vertces n an arrangement of lnes. An ordnary vertex s one that s ncdent to exactly 2 lnes of the arrangement. The famous conjecture on ths queston was known, untl ts recent proof by Green and Tao [11], as the Drac-Motzkn conjecture. Theorem 4. Let L be an arrangement of n lnes n the plane, not all through one pont. Suppose that n > n 0 for a suffcently large constant n 0. Then, L determnes at least n/2 ordnary vertces. The best result on the Drac-Motzkn problem pror to Green and Tao s proof of Theorem 4 was by Csma and Sawyer [5]. They showed that, f L s an arrangement of n > 7 lnes n the plane, not all through one pont, then L determnes at least 6n/13 ordnary vertces. A key dfference between the proof of Csma and Sawyer and that of Green and Tao s that the result of Csma and Sawyer can be generalzed to apply to arrangements of pseudolnes n a straghtforward manner [15], but the result of Green and Tao reles on algebrac statements, ncludng the Cayley-Bacharach theorem, that do not apply to pseudolnes. Ths rases the queston: s the generalzaton of Theorem 4 for arrangements of pseudolnes true? Problem 5. Is there an arrangement of n > 13 pseudolnes, not all through one pont, that determnes fewer than n/2 ordnary vertces? References [1] J. Akyama, H. Ito, M. Kobayash, and G. Nakamura, Arrangements of n ponts whose ncdent-lne-numbers are at most n/2, Graphs and Combnatorcs, vol. 27, pp , the electronc journal of combnatorcs 21(2) (2014), #P2.31 8

9 [2] J. Beck, On the lattce property of the plane and some problems of Drac, Motzkn and Erdős n combnatoral geometry, Combnatorca, vol. 3(3), pp , [3] L. Berman, Symmetrc smplcal pseudolne arrangements, Electronc Journal of Combnatorcs, vol. 15, #R13, [4] P. Brass, W. O. J. Moser, and J. Pach, Research problems n dscrete geometry. New York: Sprnger, [5] J. Csma and E. Sawyer, There exst 6n/13 ordnary ponts, Dscrete & Computatonal Geometry, vol. 9, no. 1, pp , [6] G. Drac, Collnearty propertes of sets of ponts, Quarterly Journal of Mathematcs, vol. 2, no. 1, pp , [7] Z. Dvr, Incdence theorems and ther applcatons, Foundatons and Trends n Theoretcal Computer Scence, vol. 6, no. 4, pp , [8] D. Eppsten, A kaledoscope of smplcal arrangements, [Onlne]. Avalable: [9] P. Erdős, Some unsolved problems, Magyar Tud. Akad. Mat. Kutató Int. Közl., vol. 6, pp , [10] S. Felsner, Geometrc graphs and arrangements : some chapters from combnatonal geometry, 1st ed. Wesbaden: Veweg, [11] B. Green and T. Tao, On sets defnng few ordnary lnes, Dscrete Comput. Geom., vol. 50, no. 2, pp , [12] B. Grünbaum, Arrangements and Spreads, ser. Regonal Conference Seres n Mathematcs. Amercan Mathematcal Socety, 1972, vol. 10. [13] B. Grünbaum, A catalogue of smplcal arrangements n the real projectve plane, Ars Mathematca Contemporanea, vol. 2, no. 1, [14] F. Hrzebruch, Sngulartes of algebrac surfaces and characterstc numbers, Contemp. Math, vol. 58, pp , [15] J. Lenchner, Sylvester-Galla Results and Other Contrbutons to Combnatoral and Computatonal Geometry. ProQuest, [16] J. Matoušek, The dawn of an algebrac era n dscrete geometry? n 27th European Workshop on Computatonal Geometry (EuroCG 2011), extended abstract, [17] M. Payne and D. R. Wood, Progress on Drac s conjecture, Electronc Journal of Combnatorcs, vol. 21(2), #P2.12, [18] E. Szemeréd and W. Trotter, Extremal problems n dscrete geometry, Combnatorca, vol. 3, pp , [19] T. Tao, Algebrac combnatoral geometry: the polynomal method n arthmetc combnatorcs, ncdence combnatorcs, and number theory, EMS Surveys n Mathematcal Scences, vol. 1, pp. 1 46, the electronc journal of combnatorcs 21(2) (2014), #P2.31 9

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