J. Geom. c 016 Spriger Iteratioal Publishig DOI 10.1007/s000-016-033-5 Joural of Geometry Geeral positio subsets ad idepedet hyperplaes i d-space Jea Cardial, Csaba D. Tóth, ad David R. Wood Abstract. Erdős asked what is the maximum umber α( such that every set of poits i the plae with o four o a lie cotais α( poits i geeral positio. We cosider variats of this questio for d-dimesioal poit sets ad geeralize previously kow bouds. I particular, we prove the followig two results for fixed d: Every set H of hyperplaes i R d cotais a subset S Hof size at least c ( log 1/d, for some costat c = c(d > 0, such that o cell of the arragemet of H is bouded by hyperplaes of S oly. Every set of cq d log q poits i R d, for some costat c = c(d > 0, cotais a subset of q cohyperplaar poits or q poits i geeral positio. Two-dimesioal versios of the above results were respectively proved by Ackerma et al. [Electroic J. Combiatorics, 014] ad by Paye ad Wood [SIAM J. Discrete Math., 013]. Mathematics Subject Classificatio. Primary 5C35; Secodary 5C10. 1. Itroductio Poits i geeral positio A fiite set of poits i R d is said to be i geeral positio if o hyperplae cotais more tha d poits. Give a fiite set of poits P R d i which at most d + 1 poits lie o a hyperplae, let α(p be the size of a largest subset of P i geeral positio. Let α(, d = mi{α(p : P = }. For d =,Erdős [5] observed that α(, ad proposed the determiatio of α(, as a ope problem. 1 Füredi [6] proved log α(, o(, where the lower boud uses idepedet sets i Steier triple systems, Research of Wood is supported by the Australia Research Coucil. 1 We use the shorthad otatio to idicate iequality up to a costat factor for large. Hece f( g( isequivalettof( O(g(, ad f( g( isequivaletto f( Ω(g(.
J. Cardial et al. J. Geom. ad the upper boud relies o the desity versio of the Hales Jewett Theorem [7, 8]. Füredi s argumet combied with the quatitative boud for the desity Hales Jewett problem proved i the first polymath project [14] yields α(, / log (Theorem.. Our first goal is to derive upper ad lower bouds o α(, d for fixed d 3. We prove that the multi-dimesioal Hales Jewett theorem [8] yields α(, 3 o( (Theorem.4. But for d 4, oly the trivial upper boud α(, d O( is kow. We establish lower bouds α(, d ( log 1/d i a dual settig of hyperplae arragemets i R d as described below. Idepedet sets of hyperplaes For a fiite set H of hyperplaes i R d,bose et al. [] defied a hypergraph G(H with vertex set H such that the set of hyperplaes cotaiig the facets of each cell of the arragemet of H forms a hyperedge i G(H. A subset S Hof hyperplaes is called idepedet if it is a idepedet set of G(H; that is, if o cell of the arragemet of H is bouded by hyperplaes i S oly. Deote by β(h the maximum size of a idepedet set of H, ad let β(, d := mi{β(h : H = }. The followig relatio betwee α(, d adβ(, d was observed by Ackerma et al. [1] i the case d =. Lemma 1.1 (Ackerma et al. [1]. For d ad N, we have β(, d α(, d. Proof. For every set P of poits i R d i which at most d + 1 poits lie o a hyperplae, we costruct a set H of hyperplaes i R d such that β(h α(p. Cosider the set H 0 of hyperplaes obtaied from P by duality. Sice at most d + 1 poits of P lie o a hyperplae, at most d + 1 hyperplaes i H 0 have a commo itersectio poit. Perturb the hyperplaes i H 0 so that the d + 1 hyperplaes that itersect form a simplicial cell, ad deote by H the resultig set of hyperplaes. A idepedet subset of hyperplaes correspods to a subset i geeral positio i P.Thusα(P β(h. Ackerma et al. [1] proved that β(, log, usig a result by Kostochka et al. [11] o idepedet sets i bouded-degree hypergraphs. Lemma 1.1 implies that ay improvemet o this lower boud would immediately improve Füredi s lower boud for α(,. We geeralize the lower boud to higher dimesios by provig that β(, d ( log 1/d for fixed d (Theorem 3.3. Subsets either i Geeral Positio or i a Hyperplae We also cosider a geeralizatio of the first problem, ad defie α(, d, l, with a slight abuse of otatio, to be the largest iteger such that every set of poits i R d i which at most l poits lie i a hyperplae cotais a subset of α(, d, l poits i geeral positio. Note that α(, d =α(, d, d + 1 with this otatio, ad every set of poits i R d cotais α(, d, l poits i geeral positio or l+1 poits i a hyperplae.
Geeral positios subsets ad idepedet hyperplaes Motivated by a questio of Gowers [9], Paye ad Wood [13] studied α(,,l; that is, the miimum, take over all sets of poits i the plae with at most l colliear, of the maximum size a subset i geeral positio. They combie the Szemerédi Trotter Theorem [17] with lower bouds o maximal idepedet sets i bouded-degree hypergraphs to prove α(,,l log / log l for l 1/ ε. We geeralize some of their techiques, ad show that for fixed d ad all l,wehaveα(, d, l (/ log l 1/d (Theorem 4.1. It follows that every set of at least Cq d log q poits i R d, where C = C(d > 0 is a sufficietly large costat, cotais q cohyperplaar poits or q poits i geeral positio (Corollary 4... Subsets i geeral positio ad the Hales Jewett theorem Let [k] :={1,,...,k} for every positive iteger k. A subset S [k] m is a t-dimesioal combiatorial subspace of [k] m if there exists a partitio of [m] ito sets W 1,W,...,W t,x such that W 1,W,...,W t are oempty, ad S is exactly the set of elemets x [k] m for which x i = x j wheever i, j W l for some l [t], ad x i is costat if i X. A oe-dimesioal combiatorial subspace is called a combiatorial lie. To obtai a quatitative upper boud for α(,, we combie Füredi s argumet with the quatitative versio of the desity Hales Jewett theorem for k = 3 obtaied i the first polymath project (Note that the latter also cosider Moser umbers, ivolvig geometric lies ad ot oly combiatorial lies, but this is ot eeded here. Theorem.1 (Polymath [14]. The size of the largest subset of [3] m without a combiatorial lie is O(3 m / log m. Theorem.. α(, / log. Proof. Cosider the m-dimesioal grid [3] m i R m ad project it oto R usig a geeric projectio; that is, so that three poits i the projectio are colliear if ad oly if their preimages i [3] m are colliear. Deote by P the resultig plaar poit set ad let =3 m. Sice the projectio is geeric, the oly colliear subsets of P are projectios of colliear poits i the origial m-dimesioal grid, ad [3] m cotais at most three colliear poits. From Theorem.1, the largest subset of P with o three colliear poits has size at most the idicated upper boud. To boud α(, 3, we use the multidimesioal versio of the desity Hales Jewett Theorem. Theorem.3 (See [7, 14]. For every δ>0 ad every pair of positive itegers k ad t, there exists a positive iteger M := M(k, δ, t such that for every m>m, every subset of [k] m of desity at least δ cotais a t-dimesioal subspace. Theorem.4. α(, 3 o(.
J. Cardial et al. J. Geom. Proof. Cosider the m-dimesioal hypercube [] m i R m ad project it oto R 3 usig a geeric projectio. Let P be the resultig poit set i R 3 ad let := m. Sice the projectio is geeric, the oly coplaar subsets of P are projectios of poits of the m-dimesioal grid [] m lyig i a two-dimesioal subspace. Therefore P does ot cotai more tha four coplaar poits. From Theorem.3 with k = t =, for every δ>0ad sufficietly large m, every subset of P with at least δ elemets cotais k t = 4 coplaar poits. Hece every idepedet subset of P has o( elemets. We would like to prove α(, d o( for fixed d. However, we caot apply the same techique, because a m-cube has too may co-hyperplaar poits, which remai co-hyperpaar i projectio. By the multidimesioal Hales Jewett theorem, every costat fractio of vertices of a high-dimesioal hypercube has this property. It is a coicidece that a projectio of a hypercube to R d works for d = 3, because d 1 = d + 1 i that case. 3. Lower bouds for idepedet hyperplaes We also give a lower boud o β(, dford. By a simple chargig argumet (see Cardial ad Felser [3], oe ca establish that β(, d 1/d. Ispired by the recet result of Ackerma et al. [1], we improve this boud by a factor of (log 1/d. Lemma 3.1. Let H be a fiite set of hyperplaes i R d. For every subset of d hyperplaes i H, there are at most d simplicial cells i the arragemet of H such that all d hyperplaes cotai some facets of the cell. Proof. A simplicial cell σ i the arragemet of H has exactly d + 1 vertices, ad exactly d + 1 facets. Ay d hyperplaes alog the facets of σ itersect i a sigle poit, amely at a vertex of σ. Everysetofd hyperplaes i H that itersect i a sigle poit ca cotai d facets of at most d simplicial cells (sice o two such cells ca lie o the same side of all d hyperplaes. The followig is a reformulatio of a result of Kostochka et al. [11], that is similar to the reformulatio of Ackerma et al. [1] i the case d =. I what follows, d = O(1, ad asymptotic otatios refer to. Theorem 3. (Kostochka et al. [11]. Cosider a -vertex (d + 1-uiform hypergraph H such that every d-tuple of vertices is cotaied i at most t = O(1 edges, ad apply the followig procedure: 1. let X be the subset of vertices obtaied by choosig each vertex idepedetly at radom with probability p, such that p = (/(t log log log 3/(3d 1,. remove the miimum umber of vertices of X so that the resultig subset Y iduces a triagle-free liear hypergraph H[Y ]. A hypergraph is liear if it has o pair of distict edges sharig two or more vertices.
Geeral positios subsets ad idepedet hyperplaes The with high probability H[Y ] has a idepedet set of size at least ( t log t 1 d. Theorem 3.3. For fixed d, we have β(, d ( log 1/d. Proof. LetH be a set of hyperplaes i R d ad cosider the (d + 1-uiform hypergraph H havig oe vertex for each hyperplae i H, ad a hyperedge of size d + 1 for each set of d + 1 hyperplaes formig a simplicial cell i the arragemet of H. From Lemma 3.1, everyd-tuple of vertices of H is cotaied i at most t := d edges. We ca apply Theorem 3. ad obtai a subset S of hyperplaes of size Ω ( (( log( 1/d such that o simplicial cell is bouded d d by hyperplaes of S oly. However, there might be osimplicial cells of the arragemet that are bouded by hyperplaes of S oly. Let p be the probability used to defie X i Theorem 3.. It is kow [10] that the total umber of cells i a arragemet of d-dimesioal hyperplaes is less tha d d. Hece for a iteger c d + 1, the expected umber of cells of size c that are bouded by hyperplaes of X oly is at most p c d d (4 3dc/(3d 1 ( d log log log 3/(3d 1 dd d (4 3dc/(3d 1+d. Note that for c d +, the expoet of satisfies (4 3dc 3d 1 + d<0. Therefore the expected umber of such cells of size at least d + is vaishig. O the other had we ca boud the expected umber of cells that are of size at most d, ad that are bouded by hyperplaes of X oly, where the expectatio is agai with respect to the choice of X. Note that cells of size d are ecessarily ubouded, ad i a simple arragemet, o cell has size less tha d. The umber of ubouded cells i a d-dimesioal arragemet is O(d d 1 [10]. Therefore, the umber we eed to boud is at most p d O(d d 1 (4 3dd/(3d 1+d 1 1/(3d 1 = o( 1/d. Cosider ow a maximum idepedet set S i the hypergraph H[Y ], where Y is defied as i Theorem 3., ad for each cell that is bouded by hyperplaes of S oly, remove from S oe of the hyperplaes boudig the cell. Sice S X, the expected umber of such cells is o( 1/d, hece there exists a X for which the umber of remaiig hyperplaes i S X is still Ω ( ( log 1/d,ad they ow form a idepedet set. We have the followig colorig variat of Theorem 3.3. Corollary 3.4. Hyperplaes of a simple arragemet of size i R d for fixed d ca be colored with O ( 1 1/d /(log 1/d colors so that o cell is bouded by hyperplaes of a sigle color.
J. Cardial et al. J. Geom. Proof. From Theorem 3.3, there always exists a idepedet set of hyperplaes of size at least c ( log 1/d for some costat c. We assume here that all logarithms are base. We defie a ew costat c such that c = ( 1 c + c /d 1 c = /d 1 c(1 /d 1. We ow prove that hyperplaes formig a simple arragemet i R d ca be colored with c ( 1 1/d /(log 1/d colors so that o cell is bouded by hyperplaes of a sigle color. We proceed by iductio ad suppose this holds for / hyperplaes. We apply the greedy algorithm ad iteratively pick a maximum idepedet set util there are at most / hyperplaes left. We assig a ew color to each idepedet set, the use the iductio hypothesis for the remaiig hyperplaes. This clearly yields a proper colorig. Sice every idepedet set has size at least c ( log 1/d, the umber of iteratios before we are left with at most / hyperplaes is at most t c ( log The umber of colors is therefore at most 1 1/d ( ( t + c ( log 1/d c ( = log ( 1 c + c 1/d. ( ( 1 1/d 1/d + c ( log 1/d ( ( 1 1/d ( log 1/d ( ( 1 c + c /d 1 1 1/d (log 1/d ( = c 1 1/d, (log 1/d as claimed. I the peultimate lie, we used the fact that log > 1 log for >4. 4. Large subsets i geeral positio or i a hyperplae We wish to prove the followig. Theorem 4.1. Fix d. Every set of poits i R d ( with at most l cohyperplaar poits, where l 1/, cotais a subset of Ω (/ log l 1/d poits i geeral positio. That is, α(, d, l (/ log l 1/d for l.
Geeral positios subsets ad idepedet hyperplaes This is a higher-dimesioal versio of the result by Paye ad Wood [13]. The followig Ramsey-type statemet is a immediate corollary. Corollary 4.. For fixed d there is a costat c such that every set of at least cq d log q poits i R d cotais q cohyperplaar poits or q poits i geeral positio. I order to give some ituitio about Corollary 4., it is worth metioig a easy proof whe cq d log q is replaced by q (q ( d. Cosider a set of = q q d poits i R d, ad let S be a maximal subset i geeral positio. Either S q ad we are doe, or S spas ( ( S d q d hyperplaes, ad, by maximality, every poit lies o at least oe of these hyperplaes. Hece by the pigeohole priciple, oe of the hyperplaes i S must cotai at least / ( q d = q poits. We ow use kow icidece bouds to estimate the maximum umber of cohyperplaar (d+1-tuples i a poit set. I what follows we cosider a fiite set P of poits i R d such that at most l poits of P are cohyperplaar, where l := l( 1/ is a fixed fuctio of. Ford 3, a hyperplae h is said to be γ-degeerate if at most γ P h poits i P h lie o a (d -flat. A flat is said to be k-rich wheever it cotais at least k poits of P.The followig is a stadard reformulatio of the classic Szemerédi Trotter theorem o poit-lie icideces i the plae [17]. Theorem 4.3 (Szemerédi ad Trotter [17]. For every set of poits i R, the umber of k-rich lies is at most ( O k 3 + k This boud is the best possible. Elekes ad Tóth proved the followig higher-dimesioal versio, ivolvig a additioal o-degeeracy coditio. Theorem 4.4 (Elekes ad Tóth [4]. For every iteger d 3, there exist costats C d > 0 ad γ d > 0 such that for every set of poits i R d, the umber. of k-rich γ d -degeerate plaes is at most ( d d 1 C d + kd+1 k d 1 This boud is the best possible apart from costat factors. We prove the followig upper boud o the umber of cohyperplaar (d + 1- tuples i a poit set. Lemma 4.5. Fix d. Let P be a set of poits i R d with at most l cohyperplaar poits, where l 1/. The the umber of cohyperplaar (d + 1-tuples i P is O ( d log l. Proof. We proceed by iductio o d. The base case d = was established by Paye ad Wood [13], usig the Szemerédi Trotter boud (Theorem 4.3. We reproduce it here for completeess. We wish to boud the umber of colliear triples i a set P of poits i the plae. Let h k be the umber.
J. Cardial et al. J. Geom. of lies cotaiig exactly k poits of P. The umber of colliear 3-tuples is ( k h k k h i 3 k=3 k=3 k=3 i=k ( k k 3 + k log l + l log l. We ow cosider the geeral case d 3. Let P be a set of poits i R d, o l i a hyperplae, where d + ad l.letγ := γ d > 0bea costat specified i Theorem 4.4. We distiguish the followig three types of (d + 1-tuples: Type 1: (d +1-tuples cotaied i some (d -flat spaed by P Deote by s k the umber of (d -flats spaed by P that cotai exactly k poits of P. Project P oto a (d 1-flat i a geeric directio to obtai a set of poits P i R d 1.Nows k is the umber of hyperplaes of P cotaiig exactly k poits of P. By applyig the iductio hypothesis o P, the umber of cohyperplaar d-tuples is ( k s k d 1/ log l. d k=d Hece the umber of (d + 1-tuples of P lyig i a (d -flat spaed by P satisfies ( k s k l d 1 log l d log l. d +1 k=d+1 Type : (d +1-tuples of P that spa a γ-degeerate hyperplae Let h k be the umber of γ-degeerate hyperplaes cotaiig exactly k poits of P.By Theorem 4.4, ( k h k k d h i d +1 k=d+1 k=d+1 k=d+1 i=k ( k d d d 1 + kd+1 k d 1 d log l + l d 1 d log l. Type 3: (d +1-tuples of P that spa a hyperplae that is ot γ-degeerate Recall that if a hyperplae H paed by P is ot γ-degeerate, the more tha a γ fractio of its poits lie i a (d -flat L(H. We may assume that L(H is also spaed by P. Cosider a (d -flat L spaed by P ad cotaiig exactly k poits of P. The hyperplaes spaed by P that cotai L
Geeral positios subsets ad idepedet hyperplaes partitio P \L. Let r be the umber of hyperplaes cotaiig L ad exactly r poits of P \L. Wehave l r=1 rr. If a hyperplae H is ot γ-degeerate, cotais a (d -flat L = L(H with exactly k poits, ad r other poits of P, the k>γ(r+k, hece r<( 1 γ 1k. Furthermore, all (d + 1-tuples that spa H must cotai at least oe poit that is ot i L. Hece the umber of (d + 1-tuples that spa H is at most O(rk d. The total umber of (d + 1-tuples of type 3 that spa a hyperplae H with a commo (d -flat L = L(H is therefore at most r rk d k d. r=1 Recall that s k deotes the umber of (d -flats cotaiig exactly k poits. Summig over all such (d -flats ad applyig the iductio hypothesis yields the followig upper boud o the total umber of (d+1-tuples spaig hyperplaes that are ot γ-degeerate: s k k d d log l. k=d+1 Summig over all three cases, the total umber of cohyperplaar (d+1-tuples is O( d log l as claimed. I the plae, Lemma 4.5 gives a O( log l boud for the umber of colliear triples i a -elemet poit set with o l o a lie, where l O(. This boud is tight for l =Θ( fora sectio of the iteger lattice. It is almost tight for l Θ(1, Solymosi ad Stojaković [15] recetly costructed -elemet poit sets for every costat l ad ε>0 that cotais at most l poits o a lie ad Ω( ε colliear l-tuples, hece Ω( ε( l 3 Ω( ε colliear triples. Armed with Lemma 4.5, we ow apply the followig stadard result from hypergraph theory due to Specer [16]. Theorem 4.6 (Specer [16]. Every r-uiform hypergraph with vertices ad m edges cotais a idepedet set of size at least r 1 r r/(r 1 ( m 1/(r 1. (1 Proof of Theorem 4.1. We apply Theorem 4.6 to the hypergraph formed by cosiderig all cohyperplaar (d + 1-tuples i a give set of poits i R d, with o l cohyperplaar. Substitutig m d log l ad r = d +1 i (1, we get a lower boud ( 1/d ( d 1 log l =, 1/d log l for the maximum size of a subset i geeral positio, as desired.
J. Cardial et al. J. Geom. Note added i proof Recall that i Sect. we asked whether α(, d o( for every d 3. Subsequetly, Milićević [1] aouced a aswer to this questio i the affirmative. Refereces [1] Ackerma, E., Pach, J., Pichasi, R., Radoičić, R., Tóth, G.:A ote o colorig lie arragemets. Electro. J. Comb. 1(, #P.3 (014 [] Bose, P., Cardial, J., Collette, S., Hurtado, F., Korma, M., Lagerma, S., Taslakia, P.: Colorig ad guardig arragemets. Discret. Math. Theor. Comput. Sci. 15(3, 139 154 (013 [3] Cardial, J., Felser, S.: Coverig partial cubes with zoes. I: Proceedigs of the 16th Japa coferece o discrete ad computatioal geometry ad graphs (JCDCG 013, Lecture otes i computer sciece. Spriger, Berli (014 [4] Elekes, G., Tóth, C.D.: Icideces of ot-too-degeerate hyperplaes. I: Proceedigs of the ACM symposium o computatioal geometry (SoCG, pp. 16 1 (005 [5] Erdős, P.: O some metric ad combiatorial geometric problems. Discret. Math. 60, 147 153 (1986 [6] Füredi, Z.: Maximal idepedet subsets i Steier systems ad i plaar sets. SIAM J. Discret. Math. 4(, 196 199 (1991 [7] Fursteberg, H., Katzelso, Y.: A desity versio of the Hales Jewett theorem for k = 3. Discret. Math. 75(1-3, 7 41 (1989 [8] Fursteberg, H., Katzelso, Y.: A desity versio of the Hales Jewett theorem. J. d aalyse Math. 57, 64 119 (1991 [9] Gowers, T.: A geometric Ramsey problem. http://mathoverflow.et/questios/ 5098/a-geometric-ramsey-problem (01 [10] Halperi, D.: Arragemets. I: Goodma, J.E., O Rourke, J. (eds Hadbook of Discrete ad Computatioal Geometry, chapter 4., d ed., CRC Press, Boca Rato (004 [11] Kostochka, A.V., Mubayi, D., Verstraëte, J.: O idepedet sets i hypergraphs. Radom Struct. Algorithms 44(, 4 39 (014 [1] Milićević, L.: Sets i almost geeral positio (015. arxiv:1601.0706 [13] Paye, M.S., Wood, D.R.: O the geeral positio subset selectio problem. SIAM J. Discret. Math. 7(4, 177 1733 (013 [14] Polymath, D.H.J.: A ew proof of the desity Hales Jewett theorem. A. Math. 175(, 183 137 (01 [15] Solymosi, J., Stojaković, M.: May colliear k-tuples with o k +1 colliear poits. Discret. Comput. Geom. 50, 811 80 (013 [16] Specer, J.: Turá s theorem for k-graphs. Discret. Math. (, 183 186 (197 [17] Szemerédi, E., Trotter, W.T.: Extremal problems i discrete geometry. Combiatorica 3(3, 381 39 (1983
Geeral positios subsets ad idepedet hyperplaes Jea Cardial Uiversité libre de Bruxelles (ULB Brussels Belgium e-mail: jcardi@ulb.ac.be Csaba D. Tóth Califoria State Uiversity Northridge Northridge, CA USA e-mail: csaba.toth@csu.edu David R. Wood Moash Uiversity Melboure Australia e-mail: david.wood@moash.edu Received: October 17, 015.