c 2013 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J. DISCRETE MATH. Vol. 7, No. 4, pp c 013 Society for Idustrial ad Applied Mathematics ON THE GENERAL POSITION SUBSET SELECTION PROBLEM MICHAEL S. PAYNE AND DAVID R. WOOD Abstract. Let f(, l) be the maximum iteger such that every set of poits i the plae with at most l colliear cotais a subset of f(, l) poits with o three colliear. First we prove that if l O( ), the f(, l) Ω( / l l). Secod we prove that if l O( (1 ɛ)/ ), the f(, l) Ω( log l ), which implies all previously kow lower bouds o f(, l) ad improves them whe l is ot fixed. A more geeral problem is to cosider subsets with at most k colliear poits i a poit set with at most l colliear. We also prove aalogous results i this settig. Key words. geeral positio, Erdős problems, discrete geometry AMS subject classificatio. 5C10 DOI / Itroductio. A set of poits i the plae is i geeral positio if it cotais o three colliear poits. The geeral positio subset selectio problem asks, give a fiite set of poits i the plae with at most l colliear, how big is the largest subset i geeral positio? That is, determie the maximum iteger f(, l) such that every set of poits i the plae with at most l colliear cotais a subset of f(, l) poits i geeral positio. Throughout this paper we assume l 3. Furthermore, as the results i this paper are all asymptotic i, the expressio fixed l is shorthad for l a costat ot depedet o. Otherwise l is allowed to grow as a fuctio of. The problem was origially posed by Erdős, first for the case l = 3 [8], ad later i a more geeral form [9]. Füredi [10] showed that the desity versio of the Hales Jewett theorem [11] implies that f(, l) o(), ad that a result of Phelps ad Rödl [0] o idepedet sets i partial Steier triple systems implies that f(, 3) Ω( l ). Util recetly, the best kow lower boud for l 4wasf(, l) /(l ), proved by a greedy selectio algorithm. Lefma [16] showed that for fixed l, f(, l) Ω( l ). (I fact, his results are more geeral; see sectio 3.) I relatio to the geeral positio subset selectio problem (ad its relatives), Brass, Moser, ad Pach [, p. 318] write, To make ay further progress, oe eeds to explore the geometric structure of the problem. We do this by usig the Szemerédi Trotter theorem [5]. We give improved lower bouds o f(, l) whel is ot fixed, with the improvemet beig most sigificat for values of l aroud. Our first result (Theorem.3) Received by the editors November, 01; accepted for publicatio (i revised form) July 10, 013; published electroically October 8, Departmet of Mathematics ad Statistics, The Uiversity of Melboure, Melboure, Australia (m.paye3@pgrad.uimelb.edu.au). This author s research was supported by a Australia Postgraduate Award from the Australia Govermet. School of Mathematical Scieces, Moash Uiversity, Melboure, Australia (david.wood@ moash.edu). This author s research was supported by the Australia Research Coucil. 177

2 178 MICHAEL S. PAYNE AND DAVID R. WOOD says that if l O( ), the f(, l) Ω( l l ). Our secod result (Theorem.5) says that if l O( (1 ɛ)/ ), the f(, l) Ω( log l ). For fixed l, this implies Lefma s lower boud o f(, l) metioed above. I sectio 3 we cosider a atural geeralizatio of the geeral positio subset selectio problem. Give k<l,erdős [9] asked for the maximum iteger f(, l, k) such that every set of poits i the plae with at most l colliear cotais a subset of f(, l, k) poits with at most k colliear. Thus f(, l) =f(, l, ). We prove results similar to Theorems.3 ad.5 i this settig too.. Results. Our mai tool is the followig lemma. Lemma.1. Let P be a set of poits i the plae with at most l colliear. The the umber of colliear triples i P is at most c( l l + l ) for some costat c. Proof. For i l, lets i be the umber of lies cotaiig exactly i poits i P. A well-kow corollary of the Szemerédi Trotter theorem [5] states that for some costat c 1, for all i, ( s j c i 3 + ). i j i Thus the umber of colliear triples is i= ( ) i s i 3 c i i= j=i s j i= ( ci i 3 + ) i ( ) + i c( l l + l ). i i= Note that Lefma [15] proved Lemma.1 for the case of the grid via a direct coutig argumet. A statemet similar to Lemma.1 with l = also appears i the book by Tao ad Vu [6, Corollary 8.8]. To apply Lemma.1 it is useful to cosider the 3-uiform hypergraph H(P ) determied by a set of poits P, with vertex set P, ad a edge for each colliear triple i P. A subset of P is i geeral positio if ad oly if it is a idepedet set i H(P ). The size of the largest idepedet set i a hypergraph H is deoted α(h). Specer [3] proved the followig lower boud o α(h). Lemma. (Specer [3]). Let H be a r-uiform hypergraph with vertices ad m edges. If m</r,theα(h) >/. Ifm /r, the α(h) > r 1 r r/(r 1) (m/) 1/(r 1). Lemmas.1 ad. imply our first result. Theorem.3. Let P be a set of poits with at most l colliear. The P cotais a set of Ω(/ l l + l ) poits i geeral positio. I particular, if l O( ), thep cotais a set of Ω( l l ) poits i geeral positio. Proof. Letm be the umber of edges i H(P ). By Lemma.1, m/ c( l l+l ) for some costat c. Now apply Lemma. with r =3. Ifm</3, the α(h(p )) > /, as required. Otherwise, α(h(p )) > 3 3/ (m/) 1/ 3 3/ c( l l + l ) = 3 3c. l l + l

3 ON THE GENERAL POSITION SUBSET SELECTION PROBLEM 179 Note that Theorem.3 also shows that if l / l l, thef(, l) Ω(/l). This improves upo the greedy boud metioed i the itroductio, ad is withi a costat factor of optimal, sice there are poit sets with at most l colliear that ca be covered by /l lies. Theorem.3 aswers, up to a logarithmic factor, a symmetric Ramsey-style versio of the geeral positio subset selectio problem posed by Gowers [13]. He asked for the miimum iteger GP(q) such that every set of at least GP(q) poitsithe plae cotais q colliear poits or q poits i geeral positio. Gowers oted that Ω(q ) GP(q) O(q 3 ). Theorem.3 with l = q 1ad =GP(q) implies that Ω( GP(q)/ l(q 1)) q ad so GP(q) O(q l q). The boud GP(q) Ω(q ) comes from the q q grid, which cotais o q +1 colliear poits, ad o more tha q + 1 i geeral positio, sice each row ca have at most poits. Determiig the maximum umber of poits i geeral positio i the q q grid is kow as the o-three-i-lie problem, first posed by Dudeey i 1917 [4]. See [14] for the best kow boud ad for more o its history. As a aside, ote that Pach ad Sharir [18] proved a result somewhat similar to Lemma.1 for the umber of triples i P determiig a fixed agle α (0,π). Their proof is similar to that of Lemma.1 i its use of the Szemerédi Trotter theorem. Also, Elekes [6] employed Lemma. to prove a similar result to Theorem.3 for the problem of fidig large subsets with o triple determiig a give agle α (0,π). Pach ad Sharir ad Elekes did ot allow the case α = 0, that is, colliear triples. This may be because their work did ot cosider the parameter l, without which the case α = 0 is exceptioal sice P could be etirely colliear, ad all triples could determie the same agle. The followig lemma of Sudakov [4, Propositio.3] is a corollary of a result by Duke, Lefma, ad Rödl [5]. Lemma.4 (Sudakov [4]). Let H be a 3-uiform hypergraph o vertices with m edges. Let t m/ ad suppose there exists a costat ɛ>0 such that the umber of edges cotaiig ay fixed pair of vertices of H is at most t 1 ɛ.the α(h) Ω( t l t). Lemmas.1 ad.4 ca be used to prove our secod result. Theorem.5. Fix costats ɛ>0 ad d>0. LetP be a set of poits i the plae with at most l colliear poits, where l (d) (1 ɛ)/. The P cotais a set of Ω( log l ) poits i geeral positio. Proof. Letm be the umber of edges i H(P ). By Lemma.1, for some costat c 1, m cl + c l l<cd + c l l (d +1)c l l. Defie t := (d +1)c l l. Thus t m/. Each pair of vertices i H is i less tha l edges of H, ad l (d) (1 ɛ)/ < ((d +1)c l l) (1 ɛ)/ = t 1 ɛ. Thus the assumptios i Lemma.4 are satisfied. So H cotais a idepedet set of size Ω( t l t). Moreover, l t = l (d +1)c l l t (d +1)cl l (d +1)c l l 1 l

4 1730 MICHAEL S. PAYNE AND DAVID R. WOOD = 1 (d +1)c =Ω( log l ). l l l Thus P cotais a subset of Ω( log l ) poits i geeral positio. 3. Geeralizatios. I this sectio we cosider the fuctio f(, l, k) defied to be the maximum iteger such that every set of poits i the plae with at most l colliear cotais a subset of f(, l, k) poitswithatmostk colliear, where k<l. Brass [1] cosidered this questio for fixed l = k + 1 ad showed that o() f(, k +1,k) Ω( (k 1)/k (l ) 1/k ). This ca be see as a geeralizatio of the results of Füredi [10] for f(, 3, ). As i Füredi s work, the lower boud comes from a result o partial Steier systems [], ad the upper boud comes from the desity Hales Jewett theorem [1]. Lefma [16] further geeralized these results for fixed l ad k by showig that f(, l, k) Ω( (k 1)/k (l ) 1/k ). The desity Hales Jewett theorem also implies the geeral boud f(, l, k) o(). The result of Lefma may be geeralized to iclude the depedece of f(, l, k) o l for fixed k 3, aalogously to Theorems.3 ad.5 for k =. The first result we eed is a geeralizatio of Lemma.1. It is proved i the same way. Lemma 3.1. Let P be a set of poits i the plae with at most l colliear. The, for k 4, the umber of colliear k-tuples i P is at most c(l k 3 + l k 1 ) for some absolute costat c. Lemmas. ad 3.1 imply the followig theorem, which is proved i the same way as Theorem.3. Theorem 3.. If k 3 is fixed ad l O( ), thef(, l, k) Ω ( ) (k 1)/k l. (k )/k For l = ad fixed k 3, Theorem 3. implies f(,, k) Ω ( ) (k 1)/k = (k )/k Ω ( (k k+)/k) =Ω( ). This aswers completely a geeralized versio of Gowers questio [13], amely, to determie the miimum iteger GP k (q) such that every set of at least GP k (q) poits i the plae cotais q colliear poits or q poits with at most k colliear, for k 3. Thus GP k (q) O(q ). The boud GP k (q) Ω(q ) comes from the followig costructio. Let m := (q 1)/k ad let P be the m m grid. The P has at most m poits colliear, ad m<q. If S is a subset of P with at most k colliear, the S has at most k poits i each row. So S km q 1. Theorem.5 ca be geeralized usig Lemma 3.1 ad a theorem of Duke, Lefma, ad Rödl [5] (the oe that implies Lemma.4). Theorem 3.3 (Duke, Lefma, ad Rödl [5]). Let H be a k-uiform hypergraph with maximum degree Δ(H) t k 1 where t k. Let p j (H) be the umber of pairs of edges of H sharig exactly j vertices. If p j (H) t k j 1 γ for j =,...,k 1 ad some γ>0, theα(h) C(k, γ) t (l )1/(k 1) for some costat C(k, γ) > 0. Theorem 3.4. Fix costats d > 0 ad ɛ (0, 1). If k 3 is fixed ad 4 l d (1 ɛ)/,the ( ) (k 1)/k f(, l, k) Ω (l )1/k. l (k )/k Proof. Give a set P of poits with at most l colliear, a subset with at most k colliear poits correspods to a idepedet set i the (k + 1)-uiform hypergraph

5 ON THE GENERAL POSITION SUBSET SELECTION PROBLEM 1731 H k+1 (P ) of colliear (k + 1)-tuples i P. By Lemma 3.1, the umber of edges i H k+1 (P )ism c( l k + l k )forsomecostatc. The first term domiates sice l o( ). For large eough, m/ cl k. To limit the maximum degree of H k+1 (P ), discard vertices of degree greater tha (k +1)m/. Letñ be the umber of such vertices. Cosiderig the sum of degrees, (k +1)m ñ(k +1)m/, adsoñ /. Thus discardig these vertices yields a ew poit set P such that P / adδ(h k+1 (P )) 4(k +1)cl k.notethat a idepedet set i H k+1 (P ) is also idepedet i H k+1 (P ). Set t := (4(k+1)cl k ) 1/k,som 1 (k+1) tk ad Δ(H k+1 (P )) t k,asrequired for Theorem 3.3. By assumptio, l d (1 ɛ)/.thus ( t k l k ) 1 ɛ l d. 4(k +1)c Hece l 1 ɛ +k d/(1 ɛ) t k 4(k+1)c, implyig l C 1 ɛ 1(k)t +k 1 ɛ+ = C 1 (k)t ɛ k for some costat C 1 (k). Defie ɛ := 1 1 ɛ,soɛ > 0(siceɛ<1) ad l C 1 (k)t 1 ɛ.to 1 ɛ+ ɛ k boud p j (H k+1 (P )) for j =,...,k, first choose oe edge (which determies a lie), the choose the subset to be shared, the choose poits from the lie to complete the secod edge of the pair. Thus for γ := ɛ / ad sufficietly large, ( )( ) k +1 l k 1 p j (H k+1 (P )) m j k +1 j C (k)t k l k+1 j C (k)(c 1 (k)) k+1 j t k t (1 ɛ )(k+1 j) t (k+1) j 1 γ. Hece the secod requiremet of Theorem 3.3 is satisfied. Thus ( α(h k+1 (P )) Ω (l t)1/k) t ( (k 1)/k ( ) ) 1/k Ω l((l k ) 1/k ) l (k )/k ( ) (k 1)/k Ω (l )1/k. l (k )/k 4. Cojectures. Theorem 3. suggests the followig cojecture, which would completely aswer Gowers s questio [13], showig that GP(q) =Θ(q ). It is true for the grid [14], [7, Appedix]. Cojecture 4.1. f(, ) Ω( ). A atural variatio of the geeral positio subset selectio problem is to color the poits of P with as few colors as possible, such that each color class is i geeral positio. A easy applicatio of the Lovász local lemma shows that uder this requiremet, poits with at most l colliear are colorable with O( l) colors. The followig cojecture would imply Cojecture 4.1. It is also true for the grid [7]. Cojecture 4.. Every set P of poits i the plae with at most colliear ca be colored with O( ) colors such that each color class is i geeral positio. The followig propositio is somewhat weaker tha Cojecture 4.. k 1 ɛ

6 173 MICHAEL S. PAYNE AND DAVID R. WOOD Propositio 4.3. Every set P of poits i the plae with at most colliear ca be colored with O( l 3/ ) colors such that each color class is i geeral positio. Proof. Color P by iteratively selectig a largest subset i geeral positio ad givig it a ew color. Let P 0 := P. Let C i be a largest subset of P i i geeral positio ad let P i+1 := P i \ C i. Defie i := P i. Applyig Lemma.1 to P i shows that H(P i )haso( i l l + l i ) edges. Thus the average degree of H(P i )isatmost O( i l l + l ), which is O( l ) sice i ad l. Applyig Lemma. gives C i = α(h(p i )) >c i / l for some costat c>0. Thus i (1 c/ l ) i. It is well kow (ad ot difficult to show) that if a sequece of umbers m i satisfies m i m(1 1/x) i for some x>1adif j>xl m, them j 1. Hece if k l l /c, the k 1, so the umber of colors used is O( l 3/ ). The problem of determiig the correct asymptotics of f(, l) (adf(, l, k)) for fixed l remais wide ope. The Szemerédi Trotter theorem is essetially tight for the grid [19], but says othig for poit sets with bouded colliearities. For this reaso, the lower bouds o f(, l) forfixedl remai essetially combiatorial. Fidig a way to brig geometric iformatio to bear i this situatio is a iterestig challege. Cojecture 4.4. If l is fixed, the f(, l) Ω(/ polylog()). The poit set that gives the upper boud f(, l) o() (from the desity Hales Jewett theorem) is the geeric projectio to the plae of the log l -dimesioal l l l iteger lattice (heceforth [l] d,whered := log l () ). The problem of fidig large geeral positio subsets i this poit set for l = 3 is kow as Moser s cube problem [17, 1], ad the best kow asymptotic lower boud is Ω(/ l ) [3, 1]. I the colorig settig, the followig cojecture is equivalet to Cojecture 4.4 by a argumet similar to that of Propositio 4.3. Cojecture 4.5. For all fixed l 3, every set of poits i the plae with at most l colliear ca be colored with O(polylog()) colors such that each color class is i geeral positio. Cojecture 4.5 is true for [l] d, which ca be colored with O(d l 1 ) colors as follows. For each x [l] d, defie a sigature vector i Z l whose etries are the umber of etries i x equal to 1,,...,l. The umber of such sigatures is the umber of partitios of d ito at most l parts, which is O(d l 1 ). Give each set of poits with the same sigature its ow color. To see that this is a proper colorig, suppose that {a, b, c} [l] d is a moochromatic colliear triple, with b betwee a ad c. Permute the coordiates so that the etries of b are odecreasig. Cosider the first coordiate i i which a i, b i,adc i are ot all equal. The without loss of geerality a i <b i. But this implies that a has more etries equal to a i tha b does, cotradictig the assumptio that the sigatures are equal. Ackowledgmets. Thaks to Timothy Gowers for postig his motivatig problem o MathOverflow [13]; to Moritz Schmitt ad Louis Thera for iterestig discussios, ad for makig us aware of referece [13]; to Jáos Pach for poitig out refereces [16] ad [18]; ad to the referee for helpful feedback ad for poitig out referece [6]. REFERENCES [1] P. Brass, O poit sets without k colliear poits, i Discrete Geometry, Moogr. Textbooks Pure Appl. Math. 53, Dekker, New York, 003, pp

7 ON THE GENERAL POSITION SUBSET SELECTION PROBLEM 1733 [] P. Brass, W. Moser, ad J. Pach, Research Problems i Discrete Geometry, Spriger, New York, 005. [3] V. Chvátal, Remarks o a problem of Moser, Caad. Math. Bull., 15 (197), pp [4] H. Dudeey, Amusemets i Mathematics, Nelso, Ediburgh, [5] R. A. Duke, H. Lefma, ad V. Rödl, O ucrowded hypergraphs, Radom Structures Algorithms, 6 (1995), pp [6] G. Elekes, A ote o a problem of Erdős o right agles, Discrete Math., 309 (009), pp [7] K. F. Roth, O a problem of Heilbro, J. Lodo Math. Soc., 6 (1951), pp (appedix by P. Erdős). [8] P. Erdős, O some metric ad combiatorial geometric problems, Discrete Math., 60 (1986), pp [9] P. Erdős, Some old ad ew problems i combiatorial geometry, i Applicatios of Discrete Mathematics, R. D. Rigeise ad F. S. Roberts, eds., SIAM, Philadelphia, 1988, pp [10] Z. Füredi, Maximal idepedet subsets i Steier systems ad i plaar sets, SIAM J. Discrete Math., 4 (1991), pp [11] H. Fursteberg ad Y. Katzelso, A desity versio of the Hales-Jewett theorem for k = 3, Discrete Math., 75 (1989), pp [1] H. Fursteberg ad Y. Katzelso, A desity versio of the Hales-Jewett theorem, J.Aal. Math., 57 (1991), pp [13] T. Gowers, A Geometric Ramsey Problem, accessed July 01. [14] R. R. Hall, T. H. Jackso, A. Sudbery, ad K. Wild, Some advaces i the o-three-i-lie problem, J. Combi. Theory Ser. A, 18 (1975), pp [15] H. Lefma, Distributios of poits i the uit square ad large k-gos, Europea J. Combi., 9 (008), pp [16] H. Lefma, Extesios of the No-Three-i-Lie Problem, preprit, 01; available olie from o three submitted.pdf. [17] L. Moser, Problem P.170, Caad. Math. Bull., 13 (1970), p. 68. [18] J. Pach ad M. Sharir, Repeated agles i the plae ad related problems, J. Combi. Theory Ser. A, 59 (199), pp. 1. [19] J. Pach ad G. Tóth, Graphs draw with few crossigs per edge, Combiatorica, 17 (1997), pp [0] K. T. Phelps ad V. Rödl, Steier triple systems with miimum idepedece umber, Ars Combi., 1 (1986), pp [1] D. H. J. Polymath, Desity Hales-Jewett ad Moser umbers, i A Irregular Mid, Bolyai Soc. Math. Stud. 1, Jáos Bolyai Mathematical Society, Budapest, 010, pp [] V. Rödl ad E. Šiňajová, Note o idepedet sets i Steier systems, Radom Structures Algorithms, 5 (1994), pp [3] J. Specer, Turá s theorem for k-graphs, Discrete Math., (197), pp [4] B. Sudakov, Large K r-free subgraphs i K s-free graphs ad some other Ramsey-type problems, Radom Structures Algorithms, 6 (005), pp [5] E. Szemerédi ad W. T. Trotter, Jr., Extremal problems i discrete geometry, Combiatorica, 3 (1983), pp [6] T. Tao ad V. Vu, Additive Combiatorics, Cambridge Stud. Adv. Math. 105, Cambridge Uiversity Press, Cambridge, UK, 006. [7] D. R. Wood, A ote o colourig the plae grid, Geombiatorics, 13 (004), pp