A solution to a problem of Grünbaum and Motzkin and of Erdős and Purdy about bichromatic configurations of points in the plane

Transcription

1 A solution to a poblem of Günbaum and Motzkin and of Edős and Pudy about bichomatic configuations of points in the plane Rom Pinchasi July 29, 2012 Abstact Let P be a set of n blue points in the plane, not all on a line. Let R be a set of m ed points such that P R = and evey line detemined by P contains a point fom R. We povide an answe to an old poblem by Günbaum and Motzkin [9] and independently by Edős and Pudy [6] who asked how lage must m bn tems of n in such a case? Moe specifically, both [9] and [6] wee looking fo the best absolute constant c such that m cn. We povide an answe to this poblem and show that m n Intoduction A beautiful esult of Motzkin [14], Rabin, and Chakeian [4] states that any set of non-collinea ed and blue points in the plane detemines a monochomatic line. Günbaum and Motzkin [9] initiated the study of biased coloing, that is, coloing of the points such that no puely blue line is detemined. Thntuition behind this study is that if the numbe of blue points is much lage than the numbe of ed points, then unless the set of blue points is collinea the set of blue and ed points should detemine a monochomatic blue line. The same poblem was independently consideed by Edős and Pudy [6] who stated it in a slightly diffeent way. Poblem A. Given a non-collinea set P of n points in the plane we wish to stab all the lines detemined by this set by anothe set R of m ed points such that P R =. Give a lowe bound fo m in tems of n. It will be moe convenient fo us to conside the equivalent dual poblem (by point-line duality in the plane) fo lines in the plane: Poblem B. Given a set L of n non-concuent blue lines in the eal pojective plane we wish to find a set R of m ed lines diffeent fom the blue ones such that evey intesection point of blue lines is incident to a ed line. Give a lowe bound fo m in tems of n. An Ω(n) bound fo the cadinality of R in Poblem B follows fom the so called weak Diac s conjectue. In 1951 Diac [5] conjectued that in any set of n non-concuent lines thee exists a line incident to at least n 2 O(1) intesection points with othe lines in the set. Szemeédi and Totte Mathematics Dept., Technion Isael Institute of Technology, Haifa 32000, Isael. oom@math.technion.ac.il. 1

2 [20] and Beck [1] poved a weake esult which is that unde the above conditions thee exists a line incident to Ω(n) intesection points. This esult is known as the weak Diac s conjectue. The poofs in [1, 20] used the uppe bound of Szemeédi and Totte [20] fo the numbe of incidences between points and lines in the plane. Notice that any lowe bound fo the poblem of Diac implies immediately the same lowe bound fo Poblem B. Indeed, if thee exists a blue line l incident to m intesection points with othe blue lines, then clealy m distinct ed lines ae equied just to ensue that evey intesection point on l is incident to a ed line. In the oiginal papes [20] and [1] lowe bounds of n and n, espectively, ae shown fo the weak Diac s conjectue. Today much bette bounds in tems of the multiplicative constant ae known (see [17] fo the best bound) fo the numbe of incidences between points and lines in the plane. Consequently also the constant in the weak Diac s conjectue is impoved. Vey ecently, Payne and Wood [18] did cay out this calculation of the best constant in the weak Diac s conjectue. They combined the above mentioned pogess on the numbe of point-line incidences in the plane and with some modeas showed that the lowe bound in the weak Diac s conjectue can bmpoved to n On the othe hand and also vey ecently, Lund, Pudy, and Smith [13] showed that Diac s conjectus falsf we eplace lines by pseudolines. They constucted examples of n pseudolines whee each ons incident to at most 4 9n intesection points. This means in paticula that one cannot hope fo a bette lowe bound than 4 9n fo (the pseudo-line vesion of) Poblem B though finding a blue line with many intesection points on it. An impotant special case of Poblem A was solved in [21] and extended in [15]: In 1970 Scott [19] conjectued that any set of n non-collinea points in the plane detemines at least 2 n 2 lines with distinct diections. In the same pape [19], Scott also includes an analogous conjectun thee dimensions. Scott s conjectun the plane was poved by Unga [21]. Notice that this is equivalent to saying that given a set of n blue points in the plane and a set of m points on the line at infinity (theefoe, in fact, on any given line) such that thes no monochomatic blue line, then m 2 n 2. This bound is best possible. In [15] this esult is extended as follows: Suppose that P is a set of n non-collinea blue points in the plane and R is a set of m ed points such that P R = and evey line detemined by P contains a ed point that is exteme on that line (with espect to its incident blue points), then m 2 n 2. (This esult is late used in [15] and [16] to solve Scott s conjectun thee dimensions.) It is evident, howeve, that the answe to Poblems A,B is diffeent. Constuctions found by Günbaum show that m can be as small as n 4 in Poblems A,B and thee ae spoadic constuctions (that is, fo small values of n) in which R is equal to P 6 (see [10]). In this pape we povide the following patial answe to Poblems A,B which impoves significantly on the bound of n 3 76 that can be deduced by using the bound on the weak Diac s conjectue in [18]. Ou poof uses a puely combinatoial agument that does not ely on the asymptotic bounds on the numbe of point-linncidences in the plane: Theoem 1. Let L be a set of n non-concuent blue lines and let R be a set of m ed lines in the eal pojective plane. If L R = and thes a line fom R though evey intesection point of lines in L, then m n

3 2 Poof of Theoem 1 Thdea of the poof is to estimatn two diffeent ways the cadinality of the following set T of special tiples (, e, c) such that: s an edgn the aangement A(L) delimited by two vetices, say W and Z, c is a linn L passing though the vetex W, and is a linn R passing though the vetex Z. See Figue 1 fo an example of a tipln T. We note that thoughout all of ou dawings below, lines in L ae dawn solid while lines in R ae dawn dashed. A simple lowe bound fo T is agued as follows. Conside any two lines b and c in L. Let W be thntesection point of b and c. Then thee ae pecisely two edges e of A(L) on the line b that ancident to W (hee we use the fact that not all the lines in L pass though the same point. Notice also that the two edges incident to W may have the same othe vetex Z, in the case whee all the lines in L, but one, ae concuent). Fo each of these two edges thes at least one ed line in R passing though the vetex of the edge distinct fom W. Theefoe, fo evey (odeed) pai of blue lines b and c we obtain two distinct tiples in T, so that no tiple aises moe than oncn this manne. This implies that T 2n(n 1) (see Figue 1). Z c e W e Z b Figue 1: Evey b, c L givse to two tiples in T. To obtain a good lowe bound fo m in tems of n it will theefoe be helpful to bound fom above the numbe of tiples in T to which a given ed line belongs. We denote by T() the set of tiples in T in which the ed lins. The following lemma povides an uppe bound fo the cadinality of T(). As the eade may notice this is closely elated to the so called Zone Theoem (see [2, 3]). Ou poof is indeed inspied by the poof in [3]. As we shall comment late, it is vey well possible that one may be able to use a moe elaboated agument, as used in [2] fo the zone theoem, to povide an impoved bound fo the lemma: Lemma 1. Fo evey line R we have T() 6n. Poof. Let be the given ed line and assume without loss of geneality that is hoizontal. That is, we conside an affine pictue of the pojective plann which is hoizontal. We can also assume, by applying a suitable pojective tansfomation, that no two lines in L R ae paallel in this affine pictue. 3

4 We denote by T 1 () the set of tiples (, e, c) T() such that e lies above. T 2 () will denote the complementay set of tiples in T(), namely those tiples (, e, c) T() such that e lies below. We show that T 1 () 3n; a symmetic bound holds fo T 2 (). We may assume without loss of geneality that fo evey tiple (, e, c) in T 1 () the edge s bounded. This is because we can apply a pojective tansfomation that takes to the line at infinity a line paallel to and located slightly below it. Let e 1,...,e s denote all the edges such that (,, c) is in T 1 () fo some c L (notice that also unbounded edges have two endpoints as they wap aound infinity in the pojective plane). Fo evey i let b i L denote the line containing and let Z i be the vetex of that is the intesection point of b i and. We assume that thndexing of the edges e 1,...,e s is accoding to the location of Z 1,...,Z s on fom left to ight. Note that evey vetex Z i aises by at least two distinct blue lines (see fo example thntesection point of a unique blue line with in Figue 2, which is not a vetex in A(L)). If two lines b i and b j meet at the same point Z i = Z j, then we assume that if i < j, then above b i is to the left of b j. Fo evey 1 i s we denote by the vetex of diffeent fom Z i (see Figue 2). b 1 b 2 b3 W 1 W 2 b 4 b 5 b 6 e 1 W 3 W 4 W 5 W 6 W7 b 7 e 2 e 3 e 4 e 5 e 6 e 7 Z 1 = Z 2 Z 3 = Z 4 = Z 5 Z 6 = Z 7 Figue 2: Notation used in the poof. Fix an index 1 i s. A line c L though, diffeent fom b i, will be called a left line with espect to if its intesection point with lies to the left of Z i. In a simila way we define a ight line with espect to. c will be called exteme line with espect to, if its intesection point with is exteme (leftmost o ightmost) on among all thntesection points of with lines in L passing though. If c is not exteme with espect to it will be called tame. Obseve that fo evey 1 i s thee ae at most two exteme lines with espect to (see Figue 3). Claim 1. If = W j and i < j, then j = i + 1. Poof. Assume not then one of the (at least two) lines in L passing though Z i+1 must intesect eithe the elativnteio of o the elativnteio of e j, contay to the assumption that these ae two edges in the aangement A(L) (see Figue 4). 4

5 tame lines exteme left line extemght line Z i left lines b i ight lines Figue 3: Right, left, and exteme lines with espect to. = W j e j Z i Z i+1 Z j b i bi+1 b j Figue 4: Illustating the poof of Claim 1. As a simple consequence of Claim 1, thee ae no thee distinct indices i 1, i 2, i 3 such that 1 = 2 = 3. Claim 2. If a line c in L is tame with espect to two edges, e j, then = W j. Consequently, because of Claim 1, a linn L can be tame with espect to at most two edges. Poof. Assume without loss of geneality that i < j and that W j. If c is a ight line with espect to and a left line with espect to e j, then eithe b i cosses the elativnteio of the edge e j, o b j cosses the elativnteio of the edge, a contadiction (see Figue 5 (a) and (b), espectively). Note that becaus < j, it is not possible that c is a left line with espect to and a ight line with espect to e j. Assume that c is a left line with espect to both and e j, then is close to along c than W j, and then the exteme left line with espect to cosses the elativnteio of the edge e j, a contadiction (see Figue 5 (c)). A symmetic agument applies if c is a ight line with espect to both and e j. Claim 3. Suppose an edge has two exteme lines with espect to it (a left line and a ight line). Then b i may be tame with espect to at most one edge e j. In the latte case b i and b j meet at. 5

6 W j W j W j e j ej e j b j b i b j bi b j b i (a) c (b) c c (c) Figue 5: Illustating the poof of Claim 2. Poof. Suppose that b i is tame with espect to two edges e j, e k. By Claim 2, W j = W k and theefoe, by Claim 1, we may assume k = j + 1. It cannot be that = W j = W k, as a consequence of Claim 1. Theefoe, W j = W k, and then lies in the elativnteio of the segment between W j and Z i on b i. If i < j, then the left exteme line with espect to must coss the edge e j (see Figue 6) and if j < i, then thght exteme line with espect to must coss the edge e j. These contadictions complete the poof. W j e j Z i b j b i Figue 6: Illustating the poof of Claim 3. Given any tiple (, e, c) T 1 () eithe c is an exteme line with espect to e o it is tame with espect to e. Fo evey i = 1,...,s we denote by x i the numbe of exteme lines with espect to. We denote by t i the numbe of times b i is tame with espect to anothe edge e j. The numbe of tiples in T 1 () is theefoe s i=1 (x i + t i ). Clealy x i 2 fo evey i. By Claim 3, if x i = 2, then t i = 1. Recall that by Claim 2, t i 2 fo evey i. Theefoe, we have x i + t i 3 fo evey 1 i s. This poves the uppe bound of 3n fo the cadinality of T 1 (). Symmetic aguments show that T 2 () 3n, thus poving Lemma 1. Lemma 1 implies an uppe bound of 6mn fo the cadinality of T. Togethe with the lowe bound of 2n(n 1) we get a lowe bound fo m, namely m n 1 3, thus poving Theoem 1. 6

7 The best constuction we ae awae of whee T is lags such that T equals oughly 5n. It is highly possible that this is the best uppe bound one can takn Lemma 1 and consequently impove the lowe bound in Theoem 1 to m 2 5n. So fa we havndications that the bound Lemma 1 is not best possible. Howeve, ou aguments to showing this ae a lot moe technically involved than those pesented hee and would damage the pesentation quite a bit (compae fo this matte the agument in [3] with the monvolved onn [2] fo an uppe bound of the complexity of a zonn the zone theoem). We theefoe choose to leave this question open at the moment. Refeences [1] J. Beck, On the lattice popety of the plane and some poblems of Diac, Motzkin and Edős in Combinatoial geomety, Combinatoica 3 (1983), [2] M. Ben, D. Eppstein, F. Yao, Hoizon theoems fo lines and polygons. Discete and computational geomety (New Bunswick, NJ, 1989/1990), 45 66, DIMACS Se. Discete Math. Theoet. Comput. Sci., 6, Ame. Math. Soc., Povidence, RI, [3] B. Chazelle, L. J. Guibas, D.T. Lee, The powe of geometic duality. BIT 25 (1985), no. 1, [4] G. D. Chakeian, Sylveste s poblem on collinea points and a elative, Ame. Math. Monthly 77 (1970), [5] G. A. Diac, Collineaity popeties of sets of points, Quately J. Math. 2 (1951), [6] P. Edős and G. Pudy, Some combinatoial poblems in the plane, J. Combinatoial Theoy, Se. A 25 (1978), [7] J. E. Goodman and R. Pollack, A combinatoial pespective on some poblems in geomety, Cong. Nume. 32 (1981), [8] J. E. Goodman and R. Pollack, Allowable sequences and ode types in discete and computational geomety, in: New Tends in Discete and Computational Geomety (J. Pach, ed.), Algoithms Combin. 10, Spinge, Belin, 1993, [9] B. Günbaum, Aangements of coloed lines, Abstact , Notices Ame. Math. Soc. 22 (1975), A-200. [10] B. Günbaum, Monochomatic intesection points in families of coloed lines, Geombinatoics 9 (1999), 3 9. [11] R. E. Jamison, Suvey of the slope poblem, in: Discete Geomety and Convexity, Ann. New Yok Acad. Sci. 440, New Yok Acad. Sci., New Yok, 1985, [12] R. E. Jamison and D. Hill, A catalogue of spoadic slope-citical configuations. in: Poceedings of the Fouteenth Southeasten Confeence on Combinatoics, Gaph Theoy and Computing (Boca Raton, Fla., 1983), Cong. Nume. 40 (1983), [13] B. D. Lund, G. B. Pudy, J. W. Smith, Some Results Related to a Conjectue of Diac s axiv: v1. 7

8 [14] T. S. Motzkin, Nonmixed connecting lines, Abstact 67T 605, Notices Ame. Math. Soc. 14 (1967), 837. [15] J. Pach, R. Pinchasi, and M. Shai, On the numbe of diections detemined by a theedimensional point set, J. Combinat. Theoy, Se. A, 108 (2004), Also in Poc. 19th Annu. ACM Sympos. Comput. Geom. (2003), [16] J. Pach, R. Pinchasi, and M. Shai, Solution of Scott s poblem on the numbe of diections detemined by a point set in 3-space, Discete Comput. Geom. 38 (2007), [17] J. Pach, R. Radoičić, G. Tados, and G. Tóth, Impoving the Cossing Lemma by finding moe cossings in spase gaphs, Discete and Computational Geomety 36 (2006), [18] M. Payne and D.R. Wood, Pogess on Diac s Conjectue, axiv: v1. [19] P. R. Scott, On the sets of diections detemined by n points, Ame. Math. Monthly 77 (1970), [20] E. Szemeédi and W.T. Totte, J., Extemal poblems in discete geomety, Combinatoica 3 (1983), [21] P. Unga, 2N noncollinea points detemine at least 2N diections, J. Combin. Theoy, Se. A 33 (1982),