General position subsets and independent hyperplanes in d-space

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1 J. Geom. c 016 Spriger Iteratioal Publishig DOI /s 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 5C 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(.

2 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.

3 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(.

4 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.

5 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.

6 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.

7 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.

8 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

9 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.

10 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, (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 (005 [5] Erdős, P.: O some metric ad combiatorial geometric problems. Discret. Math. 60, (1986 [6] Füredi, Z.: Maximal idepedet subsets i Steier systems ad i plaar sets. SIAM J. Discret. Math. 4(, (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, (1991 [9] Gowers, T.: A geometric Ramsey problem /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: [13] Paye, M.S., Wood, D.R.: O the geeral positio subset selectio problem. SIAM J. Discret. Math. 7(4, (013 [14] Polymath, D.H.J.: A ew proof of the desity Hales Jewett theorem. A. Math. 175(, (01 [15] Solymosi, J., Stojaković, M.: May colliear k-tuples with o k +1 colliear poits. Discret. Comput. Geom. 50, (013 [16] Specer, J.: Turá s theorem for k-graphs. Discret. Math. (, (197 [17] Szemerédi, E., Trotter, W.T.: Extremal problems i discrete geometry. Combiatorica 3(3, (1983

11 Geeral positios subsets ad idepedet hyperplaes Jea Cardial Uiversité libre de Bruxelles (ULB Brussels Belgium Csaba D. Tóth Califoria State Uiversity Northridge Northridge, CA USA David R. Wood Moash Uiversity Melboure Australia Received: October 17, 015.