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LINES INDUCED BY BICHROMATIC POINT SETS LOUISTHERAN Abstract. AnimportanttheoremofBecksaysthatanypointsetintheEuclideanplaneis either“nearlygeneralposition” or“nearlycollinear”: thereisaconstantC > 0suchthat, 1 givenn points inE2 with atmost rof them collinear, the number of lines induced bythe 1 pointsisatleastCr(n−r). 0 Recent work of Gutkin-Rams on billiards orbits requires the following elaboration of 2 Beck’sTheoremtobichromaticpointsets: thereisaconstantC > 0suchthat,givennred n pointsandnbluepointsinE2withatmostrofthemcollinear,thenumberoflinesspanning a atleastonepointofeachcolorisatleastCr(2n−r). J 7 ] O ntroduction C 1. I . h LetpbeasetofnpointsintheEuclideanplaneE2andletL(p)bethesetoflinesinduced t a by p. A line ℓ ∈ L(p) is k-rich if it is incident on at least k points of p. A well-known m theorem ofBeckrelates the size ofL(p)andthe maximum richness. [ Theorem 1(Beck’sInduced LinesTheorem [1]). Letp be asetof n points inE2,and letr be 1 v the maximumrichnessofany lineinL(p). Then L(p) ≫ r(n−r). 8 (cid:12) (cid:12) 48 Here f(n) ≫ g(n)meansthat f(n) ≥ Cg(n), f(cid:12)(cid:12)or an(cid:12)(cid:12)absolute constant C > 0. 1 In this note, we give an elaboration (using pretty much the same arguments) of Beck’s . 1 Theorem to bichromatic point sets, which arises in relation to the work of Gutkin and 0 Rams [2] on the dynamics of billiard orbits. Let p be a set of n red points and let q be a 1 set of n blue points with all points distinct (for a total of 2n). We define p ∪ q to be the 1 : bichromatic point set (p,q) and define the set of bichromatic induced lines B(p,q) to be the v i subset ofL(p,q)that isincidenton at leastone pointof each color. X r Theorem2(Beck-typetheoremforbichromaticpointsets). Let(p,q)beabichromaticpoint a set with n points in each color class (for a total of 2n). If the maximum richness of any line in L(p,q)isr, then B(p,q) ≫ n(2n−r). (cid:12) (cid:12) (cid:12) (cid:12) = In the particul(cid:12)ar case(cid:12)where r n, which is required by Gutkin and Rams, this shows that B(p,q) ≫ n2. Be(cid:12)ck’s Th(cid:12)eorem 1, and the present Theorem 2, may be deduced from the famous (cid:12) (cid:12) Szem(cid:12)eredi-T(cid:12)rotter Theorem on point-line incidences (Beck himself uses a weaker, but similar,statementashiskeylemma). Thefollowingformiswhatwerequireinthesequel. Theorem 3 (Szemeredi-Trotter Theorem [4]). Let p be a set of n points in E2 and let L be a finite setof linesinE2. Thenthenumber r of k-richlinesinL isr ≪ n2/k3 +n/k. Notations. Weuse p = (p)n and q = (q)n for point sets in E2. The notation f(n) ≫ Cg(n) i 1 i 1 meansthere isan absolute constant C > 0such that f(n) ≥ g(n) forall n ∈ N. 1 2 LOUISTHERAN roofs 2. P TheproofofthemaintheoremfollowsasimilarlinetoBeck’soriginalproof. Forapair ofpoints (p,q ), wedefine the richness ofthe pairto be the richnessof the linepq . i j i j Lemma 4. Let (p,q) be a bichromatic point setin E2. Then there is an absolute constant K > 0 1 suchthatthenumberofbichromaticpointpairsthatareeitheratmost1/K -richoratleastK n-rich 1 1 isat leastn2/2. Proof. Thereareexactlyn2pairsofpoints(p,q ),andeachoftheseinducesalineinB(p,q). i j DefinethesubsetB (p,q) ⊂ B(p,q)tobethesetofbichromaticlineswithrichnessbetween j 2j−1 and2j. Bythe Szemeredi-Trotter Theorem with k = 2j, (1) B (p,q) ≤ C(n2/23j +n/2j) j (cid:12) (cid:12) The number of bichromatic pair(cid:12)s induci(cid:12)ng any line ℓ ∈ B (p,q)is maximized when there (cid:12) (cid:12) j are 2j red pointsand 2j blueoneson ℓ,for a total of22j bichromatic pairs. Multiplying by the estimate of (1), the numberof bichromatic pairsinducing linesinB (p,q)isatmost j (2) C(n2/2j +n2j) for a large absolute constant C comingfrom the Szemeredi-Trotter Theorem. Now let K > 0 be a small constant to be selected later. We sum (2) over j such that 1 1/K ≤ j ≤ K n: 1 1 C n2 2−j +n 2j ≤ C n22−1/K1 +n·2K n   1  1/K1X≤j≤K1n 1/K1X≤j≤K1n  (cid:16) (cid:17)     PickingK smallenough(itdependsonC)ensuresthatatmostn2/2ofthemonochromatic 1 pairsinduce lineswith richness between 1/K and K n. (cid:3) 1 1 Thefollowing lemmaisa bichromatic variant of Beck’sTwo-Extremes Theorem. Lemma5. Let(p,q)beabichromaticpointsetinE2. Theneither A Thenumber of bichromaticlines B(p,q) ≫ n2. B There is a line ℓ ∈ B(p,q) incide(cid:12)nt on at(cid:12)least K n red points and K n blue points for an (cid:12) (cid:12) 2 2 absolute constant K > 0. (cid:12) (cid:12) 2 Proof. We partition the bichromatic pairs (p,q ) into three sets: L is the set of pairs with i j richness less than 1/K ; M is the set of pairs with richness in the interval [1/K ,K n]; H is 1 1 1 the setof pairswith richnessgreater than K n. 1 ByLemma4, |L∪H| ≥ n2/2. There are now three cases: Case I: (Alternative A) If |L| ≥ n2/4, then we are in alternative A, since quadratically manypairscan be covered onlyby quadraticallymanylinesof constant richness. Case II: (Alternative B) If we are not in Case I, then, |H| ≥ n2/4. In particular, since H is not empty, there are lines incident on at least one point of each color and at least K n 1 pointsintotal. Ifoneoftheselinesislineincidenttoatleast K1npointsofeachcolor, then 6 we are in alternative B. CaseIII:(AlternativeA)IfwearenotinCaseIorCaseII,theneverylineinducedbya bichromaticpairinHhasatleast 5K nredpointsincidentonitor 5K nblueonesincident 6 1 6 1 on it. LINESINDUCEDBYBICHROMATICPOINT SETS 3 Sincethereare|H| ≥ n2/4bichromaticpointpairsincidentonaveryrichline,theremust be atleastn/4 differentpoints of eachcolor participating in some pointpairin H. Each line induced by a pair in H generates at least K n incidences, so the number of 1 these lines is at most 1 n. But then if all the lines induced by H span at most 1K n blue K1 6 1 points, the total number of blue incidences is less than n/4, which is a contradiction. We can make asimilar argument forred points. Thusthere isalineℓ spanningatleast 5K nbluepointsanda distinctlineℓ spanning 1 6 1 2 at least 5K n red points. From this configuration we get at least (5K n − 1)2 distinct 6 1 6 1 (cid:3) bichromatic lines,putting usagain in alternative A. Proof of Theorem2. Ifalternative Aof Lemma5 holds, then we are alreadydone. If we are in alternative B, then there must be a line ℓ of richness r ≥ 2K n incident to 2 at least K n points of each color. Now pick any subset X of K (2n−r) points not incident 2 2 to ℓ. There are at least 1K2n(2n−r)bichromatic point pairsdetermined byone point in X 2 2 and one pointincidentto ℓ. Thus weget atleast 1 K (2n−r) 1 K2n(2n−r)− 2 ≥ K3n(2n−r) 2 2 2 ! 2 2 (cid:3) bichromatic lines. onclusion 3. C We proved an extension of Beck’s Theorem [1] to bichromatic point sets using a fairly standard argument, completingthe combinatorial stepinGutkin-Rams’srecentpaperon billiards. Thiskindofbichromaticresultcan,duetothegeneralnatureoftheproofs,beextended to any setting where a Szemeredi-Trotter-type result is available (see, e.g., [3] for many examples). Moreover, by“forgetting” colors and repeatedlysquaring the constants, The- orem 2 holds for bichromatic lines in multi-chromatic point sets. It would, however, be interesting to know whether this is the correct order of growth for the constants in a multi-chromatic version of Theorem 2. eferences R [1] Jo´zsefBeck. OnthelatticepropertyoftheplaneandsomeproblemsofDirac,Motzkin andErdo˝s incombinatorial geometry. Combinatorica, 3(3-4):281–297,1983. ISSN 0209- 9683. doi: 10.1007/BF02579184. [2] Eugene Gutkin and Michael Rams. Complexities for cartesian products of homoge- neousLagrangian systems, with applicationsto billiards. Preprint, 2010. [3] La´szlo´ A. Sze´kely. Crossing numbers and hard Erdo˝s problems in discrete geometry. Combin. Probab. Comput., 6(3):353–358, 1997. ISSN 0963-5483. doi: 10.1017/S0963548397002976. [4] Endre Szemere´di and William T. Trotter, Jr. Extremal problems in discrete geometry. Combinatorica, 3(3-4):381–392,1983. ISSN 0209-9683. doi: 10.1007/BF02579194. DepartmentofMathematics,TempleUniversity E-mailaddress: [email protected] URL:http://math.temple.edu/˜theran/