Approximate Euclidean Ramsey theorems

Appro ximate Euclidean Ramsey theorems Adrian Dumitrescu ∗ No ve mber 13, 2018 Abstract According to a cla ssical result of Szemer´ edi, every dense subset of 1 , 2 , . . . , N cont a ins an arbitrar y long a rithmetic pr ogress io n, if N is large enough. Its analogue in higher dimensio ns due to F¨ urstenberg and Katznelson says that ev ery dens e subset of { 1 , 2 , . . . , N } d contains an arbitrar y large grid, if N is larg e enough. Here w e gener alize these r esults for s e parated p oint sets on the line and resp ectively in the Euclidean space: (i) every dense separated set of points in some int e r v al [0 , L ] on the line contains an a rbitrary long approximate arithmetic progr ession, if L is larg e eno ugh. (ii) every dense sepa rated set of p oints in the d -dimensio nal cub e [0 , L ] d in R d contains an ar bitrary lar g e approximate gr id, if L is larg e enough. A further gener alization for any finite pattern in R d is also e s tablished. The se pa ration condition is shown to b e necessa ry for such results to hold. In the end we show that e very sufficien tly large p oint se t in R d contains an a rbitrarily lar ge subset of almost collinea r p oints. No separatio n c ondition is needed in this case. Keyw ords : Euclidean Ramsey theory , approximat e arithmetic pr ogression, appro ximate h omoth- etic copy , almo st collinear p oin ts. 1 In tro du ction Let us start b y recalling the classical result of Ramsey from 1930: Theorem 1 (Ramsey [23]). L et p ≤ q , and r b e p ositive inte gers. Then ther e e xists a p ositive inte ger N = N ( p, q , r ) with the f ol low i ng pr op erty: If X is a set with N elements, f or any r - c oloring of the p -element subsets of X , ther e exists a sub se t Y of X with at le ast q elements such that al l p -e lement subsets of Y have the same c olor. As noted in [4], p erhaps the fir st Ramsey t yp e result of a geometric nature is V an d er W aerden’s theorem on arithmetic progressions: Theorem 2 (V an der W aerd en [26]). F or e v ery p ositive inte gers k and r , ther e exists a p ositive inte ger W = W ( k, r ) with the fol lowing pr op erty: F or ev e ry r -c oloring of the inte gers 1 , 2 , . . . , W ther e is a mono chr omatic arithmetic pr o gr e ssion of k terms. As early as 1936, Erd ˝ os and T ur ´ an hav e suggested that a stronger density statemen t must hold. Only in 197 5, Szemer ´ edi succeeded to co n firm this belief with his celebrated result: ∗ Department of Computer Science, Universi ty o f Wi sconsin–Milwa u kee, Email: ad@cs.u wm.edu . Supp orted in part by NSF CAREER grant CCF-0444188. 1 Theorem 3 (Szemer ´ edi [25]). F or every p ositive inte ger k and every c > 0 , ther e exists N = N ( k , c ) such that every subset X of { 1 , 2 , . . . , N } of size at le ast cN c ontains an arith metic pr o- gr ession with k terms. This is a f u ndamenta l result with relations to man y areas in mathematics. Szemer ´ edi’s pr o of is v ery complicated and is regarded as a mathematical tour de force in com binatorial r easoning [18, 22]. Another proof of t his result w as obtained by mea n s of ergo dic theory b y F¨ ursten b er g [8] in 1977. A homothetic copy of { 1 , 2 , . . . , k } d is also called a k - grid in R d . The follo wing generalization of V an der W aerden’s th eorem to higher dimensions is giv en by th e Gallai–Witt theorem [18, 22]: Theorem 4 (Gallai– Witt [22]). F or every p ositive inte gers d , k and r , ther e exists a p ositive inte ger N = N ( d, k , r ) with the fol lowing pr op erty: F or every r -c oloring of th e inte ger lat tic e p oints in { 1 , 2 , . . . , N } d , ther e exists a mono chr omatic homothetic c opy of { 1 , 2 , . . . , k } d . Mor e pr e cisely, ther e exist ( a 1 , a 2 , . . . , a d ) ∈ { 1 , 2 , . . . , N } d , and a p ositive inte ger x such tha t al l p oints of th e form ( a 1 + i 1 x, a 2 + i 2 x, . . . , a d + i d x ) , i 1 , i 2 , . . . , i d ∈ { 0 , 1 , . . . , k − 1 } ar e of the same c olor. A h igher dimensional generalizati on of Szemer ´ edi’s density theorem was obtained b y F ¨ urstenberg and Katznelson [9]; see also [22]. Theorem 5 (F ¨ urstenberg–Katznelson [9]). F or e very p ositive inte gers d , k and every c > 0 , ther e exists a p ositive inte ger N = N ( d, k , c ) with the f ol lowing pr op erty: every subset X of { 1 , 2 , . . . , N } d of size at le ast cN d c ontains a homothetic c opy of { 1 , 2 , . . . , k } d . The pro of o f F ¨ ur sten b erg a n d Katz n elson uses infinitary met h o ds in ergodic theory . As noted in [22], no com binatorial pro of is kno w n. In the first part of our pap er (Section 2), w e present analogues of Th eorems 2, 3 , 4, and 5, for p oin t sets in the E u clidean sp ace. Sp ecifically , w e obtain (restricted) Ramsey theorems for sep ar ate d p oin t s ets, for fin ding appro ximate homothetic copies of an arithmetic p rogression on the line and resp ectiv ely of a grid in R d . The latter result carries o ver for an y finite pattern p oint set and ev ery d ense and su fficien tly large separated p oin t set in R d . It is w orth noting that the separation condition is necessary for such results to hold (Prop osition 1 in Section 2). While for Theorems 2, 3, 4, an d 5, the separation condition comes f or f ree for any set of intege r s, it h as to b e explicitly enforced for p oin t sets. The exact statemen ts of our results (Theorems 6, 7 and 8) are to b e found in Section 2 f ollo w ing the definitions. F ortunately , th e pro ofs of these theorems are m uch simpler than of their exact coun terparts previously mentio n ed. Moreo v er, the resulting upp er b ound s are m uc h b etter than those one w ould get from the inte ger theorems. The pro ofs are constructive and y ield v ery s imple and efficient algorithms for compu tin g the resp ectiv e ap p ro ximate h omothetic copies give n input p oint sets satisfying the requiremen ts. In the second part (Section 3), we present an unrestricted theorem (T h eorem 9) w h ic h shows the existence of an arbitrary large subset of almost collinear p oin ts in ev ery su fficien tly large p oint set in R d . No separation condition is n eeded in th is r esult. 2 Applications. Man y other Ramsey type p roblems in the E u clidean s p ace ha v e b een inv estigated in a series of pap ers b y Er d˝ os et al. [4, 5, 6] in the early 1970s, and later by Gr aham [10, 11, 12, 13, 14]. V an der W aerden’s theorem on arithmetic progressions h as inspired n ew conn ections an d n u merous results in n umb er theory , combinatorics, and com b inatorial geomet ry [1, 2, 7, 10, 15, 16, 17, 18, 19, 20 , 22, 24], where w e only named a few here. Our analogues of Theorems 2, 3, 4, an d 5, for p oint sets in the Euclidean sp ace ma y also find fruitful app lications in com binatorial and computational geometry . It is obvio u s that general p oin t sets are m uch more common in these areas th an the r ather sp ecial in teger or lattice p oint sets that o ccur in num b er theory and in teger com binatorics. A first application needs to b e men tioned: A result similar to our Theorem 6 h as b een p r o v ed instrum en tal in settling a conjecture of Mitc hell [21] on illumination for maximal un it disk pac kin gs: It is shown [2] that an y dens e (circular) forest with congruen t unit trees that is deep enough has a hid den p oint. The result th at is n eeded there is an appro ximate equidistribution lemma for separated p oin ts on th e line, w hic h is a relaxed v ersion of our Th eorem 6. 2 Appro ximate homothetic copies of an y pattern Definitions. Let δ > 0. A p oint set S in R d is said to b e δ - sep ar ate d if th e minimum pairwise distance among p oin ts in S is a t least δ . F or t w o p oints p, q ∈ R d , let d ( p, q ) denote the Euclidean distance b et w een them. The closed ball of rad iu s r in R d cen tered at p oint z = ( z 1 , . . . , z d ) is B d ( z , r ) = { x ∈ R d | d ( z , x ) ≤ r } = { ( x 1 , . . . , x d ) | d X i =1 ( x i − z i ) 2 ≤ r 2 } . Giv en a p oin t set (or “pattern”) P = { p 1 , . . . , p k } of k p oin ts in R d and another p oin t set Q with k p oints: (i) Q is similar to P , if it is a magnified/shr unk and p ossibly rotated cop y of P . (ii) Q is homoth etic to P , if it is a magnified/shr unk cop y of P in the same p osition (with no rotatio n s). Appro ximate similar copies and app ro ximate homothetic copies are defined as follo w s. See also Fig. 1 for an illustration. Giv en p oin t sets P and Q as ab o ve and 0 < ε ≤ 1 / 3: • Q is an ε - appr oximate similar copy of o f P , if there exists Q ′ so that Q ′ is sim ilar to P , and eac h p oint q ′ i ∈ Q ′ con tains a (distinct) p oin t q i ∈ Q in the ball of radius εd cen tered at q ′ i , where d is the m in im um p airwise d istance among points in Q ′ . • Q is an ε - appr oximate homoth etic cop y of of P , if there exists Q ′ so that Q ′ is homothetic to P , and eac h p oin t q ′ i ∈ Q ′ con tains a (distinct) point q i ∈ Q in the ball of radius εd cen tered at q ′ i , wh ere d is the minim um p airwise distance among p oin ts in Q ′ . The condition ε ≤ 1 / 3 is imp osed to en s ure that any tw o balls of radius εd aroun d p oin ts in Q ′ are disjoint, and m oreov er, that any tw o distin ct p oin ts of Q are separate d b y a constant times d , in this case b y at least d/ 3. In our theorems, ε -approxima te m eans ε -appro ximate homothetic cop y . W e start with ε -approxima te arithmetic progressions on the line by p ro ving the f ollo wing analogue of Theorem 3 for points on the line: Theorem 6 F or every p ositive inte ger k , c, δ > 0 , and 0 < ε ≤ 1 / 3 , ther e exists a p ositive numb er Z 0 = Z 0 ( k , c, δ, ε ) with the fol lowing pr op e rty: L et S b e a δ - sep ar ate d p oint set in an interval I of length | I | = L with at le ast cL p oints, wher e L ≥ Z 0 . Then S c ontains a k -p oint su bset that forms an ε -appr oximate arithmetic pr o gr e ssion of k terms. Mor e over, one c an set Z 0 ( k , c, δ, ε ) = 2 δ · ( k s ) j , where s =  1 ε  , r = k k − 1 , j = & log 2 cδ log r ' . 3 Figure 1: Left: a 4- term arithmetic pr o gressio n (thick vertical ba rs) and a 1 / 3-approximate 4-ter m arith- metic pr ogressio n (filled cir cles) o n the line. Right: a 3-grid (empt y circles) and a 1 / 4-approximate 3 -grid (filled circ le s ) in R 2 . Pro of. Without loss of generalit y , I = [0 , L ]. Pu t s = ⌈ 1 ε ⌉ . Condu ct an iterativ e pr o cess as f ollo w s. In step 0: Let I 0 = I , and sub divide the in terv al I 0 in to k s half-closed int erv als 1 of equal length. Let x = L/ ( k s ) b e the common length o f the sub-inte r v als. F or t = 0 , . . . , s − 1 consider the system I t of k disj oint sub-inte r v als with left endp oints of coord in ates tx, ( t + s ) x, ( t + 2 s ) x, . . . , ( t + ( k − 1) s ) x . Observe that the s systems of inte r v als I t partition the in terv al I 0 . If for some t , 0 ≤ t ≤ s − 1, eac h of the k in terv als c ontains at least one p oin t in S , s top. Otherwise in eac h of the s systems of k interv als, at least one of the k in terv als is empty , s o all the p oints are con tained in at most k s − s in terv als from the total of k s . No w pick one of the remaining ( k − 1) s in terv als, which con tains the most p oints of S , sa y I 1 . I n step i , i ≥ 1: Sub divide I i in to k s half-closed int erv als of equal length and pr o ceed as b efore. In the curren t step i , the pro cess either (i) terminates su ccessfully by finding an in terv al I i sub d ivided in to k s sub-in terv als m aking s sy s tems of in terv als, and in at least one of the systems, eac h su b-in terv al con tains at least one p oin t in S , or (ii) it cont inues with another su b division in step i + 1. W e show that if L is large enough, and the num b er of su b division steps is large enough, the iterativ e pro cess term in ates s u ccessfully . Let L 0 = | I | = L b e the initia l int erv al length, a n d m 0 ≥ c | I | b e the (initial) num b er of p oints in I 0 . A t step i , i ≥ 0, let m i b e the n umb er of p oin ts in I i , and let L i = | I i | b e the length of in terv al I i . Clearly L i = L ( k s ) i , and m i ≥ m 0 ( k − 1) i s i ≥ cL ( k − 1) i s i . (1) Let j b e a p ositiv e in teger so that c · δ ·  k k − 1  j ≥ 2 , e . g ., set j = & log 2 cδ log r ' , where r = k k − 1 . (2) No w set Z 0 ( k , c, δ, ε ) = 2 δ · ( k s ) j . If L ≥ Z 0 , as assumed, then b y our choice of parameters we ha v e L j = L ( k s ) j ≥ Z 0 ( k s ) j = 2 δ · ( k s ) j ( k s ) j = 2 δ, (3) 1 When sub divid in g a closed interv al, the first k − 1 resulting sub-interv als are half-closed, and the k th sub-interv al is closed. When sub dividing a half-closed interv al, all resulting sub-interv als are h alf-closed. 4 and m j · δ ≥ cL · δ ( k − 1) j s j = c · δ ·  k k − 1  j L ( k s ) j ≥ 2 L ( k s ) j = 2 L j . (4) Since the p oin t set is δ -separated, an in terv al pac king argumen t on the line us in g (3 ) giv es m j δ ≤ L j + δ 2 + δ 2 = L j + δ ≤ 3 2 L j . (5) Observe that (5) is in con tradiction to (4 ), which means th at the iterativ e p ro cess cannot reac h step j . W e conclude that for s ome 0 ≤ i ≤ j − 1, step i is successfu l: w e foun d a system of k in terv als of length x with left endp oin ts at coordinates a 0 = tx, a 1 = ( t + s ) x, a 2 = ( t + 2 s ) x, . . . , a k − 1 = ( t + ( k − 1) s ) x , eac h conta ining a distinct point, say b p ∈ S , p = 0 , 1 , . . . , k − 1. Ob serv e that the k p oints { a p : p = 0 , 1 , . . . , k − 1 } form an (exact) arithmetic progression of k terms with common difference equal to sx . It is now easy to verify that the k p oints b p form an ε -approximat e arithmetic progression of k terms, since for p = 0 , 1 , . . . , k − 1 a p ≤ b p ≤ a p + x and εs ≥ 1 , th us x ≤ εsx and b p ∈ [ a p , a p + εsx ] . This completes the pro of. ✷ The next prop osition sh o ws that the separation condition in the th eorem is necessary , fo r otherwise, ev en a 3-term approxima te a r ithmetic progression cannot b e guaran teed, irresp ectiv e of the size of the p oin t set. Prop osition 1 F or any n , and for any 0 ≤ ε < 1 / 3 , ther e exists a set of n p oints in [0 , 1] , without an ε -appr oximate arithmetic pr o gr e ssion of 3 terms. Pro of. Let ξ = 1 3 − ε . Let S = { ξ i | i = 0 , . . . , n − 1 } . Assume f or cont r adiction that { q 1 , q 2 , q 3 } is an ε -appro ximate arithmetic progression of 3 terms, where q 1 < q 2 < q 3 , and q 1 , q 2 , q 3 ∈ S . Then there exist a and r > 0, so that a − r , a and a + r form a 3-term arithmetic p rogression, and we ha ve: a − r − εr ≤ q 1 ≤ a − r + εr , a − εr ≤ q 2 ≤ a + εr, a + r − εr ≤ q 3 ≤ a + r + εr . F rom the first an d the third inequalit ies w e obtain a − εr ≤ q 1 + q 3 2 ≤ a + εr, therefore     q 1 + q 3 2 − q 2     ≤ 2 εr . (6) F ur ther note that q 3 − q 1 ≥ a + r − εr − ( a − r + εr ) = 2(1 − ε ) r , hence r ≤ q 3 − q 1 2(1 − ε ) . 5 By su bstituting this boun d into (6), w e ha ve     q 1 + q 3 2 − q 2     ≤ ε 1 − ε · ( q 3 − q 1 ) ≤ ε 1 − ε · q 3 . (7) On the ot her hand     q 1 + q 3 2 − q 2     ≥ q 1 + q 3 2 − q 2 ≥ q 3 2 − q 2 . (8) Putting in equalities (7) and (8) together and dividing by q 3 yields 1 2 − q 2 q 3 ≤ ε 1 − ε . Ob viously , q 2 q 3 ≤ ξ , hence 1 6 + ε = 1 2 − 1 3 + ε = 1 2 − ξ ≤ 1 2 − q 2 q 3 ≤ ε 1 − ε . Equiv alen tly , 1 6 ≤ ε 2 1 − ε , whic h is imp ossible for ε < 1 / 3. Ind eed, th e quadratic function f ( x ) = 6 x 2 + x − 1 is strictly negativ e for 0 < x < 1 / 3. W e h a v e thereb y reac hed a contradictio n . W e conclude that S has no ε -appro ximate arithmetic pr ogression of 3 te r ms. ✷ Remark. T h e follo win g sligh tly different form of P r op osition 1 ma y b e conv en ien t: F or an y n there exists a set o f n p oin ts in [ 0 , 1], withou t an ε -approximat e arithmetic p r ogression of 3 terms, for any 0 ≤ ε ≤ 1 / 4. F or the pr o of, tak e S = { 1 / 8 i | i = 0 , . . . , n − 1 } , and pro ceed in th e same w ay . F or a d -dimensional cub e Π d i =1 [ a i , b i ], let us refer to ( a 1 , . . . , a d ) as the first vertex of the d - dimensional cu b e. W e no w con tin u e with ε -appro ximate grids in R d b y proving the follo w in g analogue of Theorem 5 for points in R d : Theorem 7 F or every p ositive inte g ers d, k , and c, δ > 0 , and 0 < ε ≤ 1 / 3 , ther e exists a p ositive numb er Z 0 = Z 0 ( d, k , c, δ , ε ) with the fol lowing pr op erty: L et S b e a δ - se p ar ate d p oint set in the d -dimensional cub e Q = [0 , L ] d , with at le ast cL d p oints, wher e L ≥ Z 0 . Then S c ontains a subset that forms an ε -app r oximate k -grid in R d . Mor e over, one c an set Z 0 ( d, k , c, δ , ε ) = 2 δ · ( k s ) j , where s = & √ d ε ' , r = k d k d − 1 , j =  log κ d cδ log r  . Her e κ d (in the expr ession of j ) is a c onstant dep ending on d : κ d =  3 d · ( d/ 2)! π d/ 2  , if d is ev en , and κ d =  3 d · (1 · 3 · · · d ) 2 · (2 π ) ( d − 1) / 2  , if d is odd . (9) Pro of. F or simplicit y of calculat ions, we fi r st presen t th e pro of for d = 2 b y outlining the differences from the one-dimensional case ; the argumen t for d ≥ 3 is analogous, with the sp ecific calculations in the second part of the pro of. Recall that we h a v e set κ 2 = ⌈ 9 π ⌉ = 3. Put s = ⌈ √ 2 ε ⌉ . Conduct an iterativ e pro cess as follo ws. In step 0: Let Q 0 = Q , and sub divide the square Q 0 in to ( k s ) 2 smaller congruen t squares. Let 6 x = L/ ( k s ) b e the common side length of these squares. F or t 1 , t 2 ∈ { 0 , . . . , s − 1 } consider th e system Q t 1 ,t 2 of k 2 disjoin t sq u ares with first v ertices of coordinates ( t 1 + i 1 s, t 2 + i 2 s ) x , where i 1 , i 2 ∈ { 0 , 1 , . . . , k − 1 } . Observe that the s 2 systems of squares Q t 1 ,t 2 partition the squ are Q 0 . If for some ( t 1 , t 2 ) , 0 ≤ t 1 , t 2 ≤ s − 1, eac h of the k 2 squares in the resp ectiv e s ystem con tains at least one p oint in S , stop. O therwise in eac h of the s 2 systems of k 2 squares, at least one of the k 2 squares is empty , so all the points are con tained in a t most k 2 s 2 − s 2 squares from the total of k 2 s 2 . No w pic k one of th e remaining s 2 ( k 2 − 1) squares, whic h con tains the most p oints of S , sa y Q 1 . I n step i , i ≥ 1: Sub divide Q i in to ( ks ) 2 smaller congruent squares a n d pro ceed as b efore. In the curr ent step i , the pro cess either (i) terminates successfully by fin ding a square Q i sub d ivided in to ( ks ) 2 smaller sq u ares making s 2 systems of squares, and in at least one of the systems, eac h smaller square con tains at least one p oint in S , or (ii) it con tin u es with another sub d ivision in step i + 1. W e sho w that similar to the one-dimensional case, if L is large enough, and the n umber of sub division steps is large enough, the ite rativ e pro cess term in ates successfully . Let L 0 = L b e the initial squ are side of Q 0 , and m 0 ≥ cL 2 b e the (initial) n umber of p oin ts in Q 0 . A t step i , i ≥ 0, let m i b e the n umber of p oints in Q i , and let L i b e the side length of Q i . Clearly L i = L ( k s ) i , and m i ≥ m 0 ( k 2 − 1) i s 2 i ≥ cL 2 ( k 2 − 1) i s 2 i . Let j b e a p ositiv e in teger s o that c · δ 2 ·  k 2 k 2 − 1  j ≥ κ 2 = 3 , e . g ., set j = & log 3 cδ 2 log r ' , where r = k 2 k 2 − 1 . (10) No w set Z 0 (2 , k , c, δ , ε ) = 2 δ · ( k s ) j . If L ≥ Z 0 , as assumed, then by our c hoice of parameters we ha ve L j = L ( k s ) j ≥ Z 0 ( k s ) j = 2 δ · ( k s ) j ( k s ) j = 2 δ, (11) and m j · δ 2 ≥ cL 2 · δ 2 ( k 2 − 1) j s 2 j = c · δ 2 ·  k 2 k 2 − 1  j L 2 ( k s ) 2 j ≥ 3 L 2 ( k s ) 2 j = 3 L 2 j . (12) Note that (11) is identica l with (3) from the one-dimensional case. Since S is δ -separated, the disks of radius δ / 2 cen tered at the p oints of S are in terior-disjoin t. A straigh tforward p ac king argumen t yields m j π δ 2 4 ≤ ( L j + δ ) 2 ≤  3 2 L j  2 = 9 4 L 2 j , (13) where the la s t inequalit y is implied b y (11). Inequalit y (13) is e quiv alent to m j · δ 2 ≤ 9 π L 2 j . (14) Ho we ver th is is contradict ion with inequalit y (12) (by the setting κ 2 = ⌈ 9 π ⌉ = 3). T his means that the iterativ e pro cess cann ot reac h step j . W e conclude that for s ome 0 ≤ i ≤ j − 1, step i is successful: we found a system of k 2 disjoin t squares of side x with first vertic es a i 1 ,i 2 = ( t 1 + i 1 s, t 2 + i 2 s ) x , where i 1 , i 2 ∈ { 0 , 1 , . . . , k − 1 } , eac h con taining a distinct p oin t, sa y b i 1 ,i 2 ∈ S , for i 1 , i 2 ∈ { 0 , 1 , . . . , k − 1 } . Observ e that the k 2 p oint s a i 1 ,i 2 form an (exact ) grid Q ′ of k 2 p oint s with side length equal to sx . As in the one-dimensional 7 case, it is now easy to v erify that th e k 2 p oint s b i 1 ,i 2 form an ε -appro ximate grid of k 2 p oint s , sin ce for i 1 , i 2 ∈ { 0 , 1 , . . . , k − 1 } εs ≥ √ 2 , th us d ( a i 1 ,i 2 , b i 1 ,i 2 ) ≤ x √ 2 ≤ εxs. (15) Note that the min im um d istance among the p oints in Q ′ is sx , and this completes the pro of for th e planar case ( d = 2). The argumen t for the general case d ≥ 3 is analogous and th e calculations in der ivin g the up p er b ound are as follo ws. Th e inequ alit y (11) remains v alid. By the choice of parameters r and j , we ha ve c · δ d ·  k d k d − 1  j ≥ κ d . (16) The analogue of (12) is m j · δ d ≥ cL d · δ d ( k d − 1) j s d j = c · δ d ·  k d k d − 1  j L d ( k s ) d j ≥ κ d · L d ( k s ) d j = κ d ·  L ( k s ) j  d = κ d · L d j . (17) The packing argumen t in R d yields m j · V ol d  δ 2  ≤  3 2  d L d j , (18) where V ol d ( r ) is the v olume of the sph ere of radius r in R d . I t is we ll-known that V ol d ( r ) =          π d/ 2 ( d/ 2)! · r d if d is ev en , 2 · (2 π ) ( d − 1) / 2 1 · 3 · · · d · r d if d is o dd . (19) T o obtain a cont radiction in the argument, as in the pr evious cases, one sets κ d as in (9) taking in to accoun t (19). The setting of s is suc h that the analogue of (15) is ensur ed . This completes the pro of of Theorem 7. ✷ By selecting a sufficiently fin e grid in Th eorem 7, one obtains b y similar means the follo w ing general statemen t for an y pattern in R d : Theorem 8 F or every p ositive inte ger d , finite p attern P ⊂ R d , | P | = k , and c, δ > 0 , and 0 < ε ≤ 1 / 3 , ther e exists a p ositive numb er Z 0 = Z 0 ( d, P , c, δ , ε ) with the fol lowing pr op erty: L et S b e a δ - se p ar ate d p oint set in the d -dimensional cub e Q = [0 , L ] d , with at le ast cL d p oints, wher e L ≥ Z 0 . Then S c ontains a subset that is an ε - appr oximate homothetic c opy of P . Observe that the iterativ e pr o cedures u sed in the pr o ofs of Th eorems 6, 7 and 8, yield very simple a n d effici en t algorithms for computin g the resp ective app r o ximate homothetic copies giv en input p oin t sets satisfying the imp osed r equ iremen ts. F or instance in Th eorems 6 and 7, the num b er of iterations, j , is giv en b y (2) and resp. (10), and eac h iteration tak es linear time (in the num b er of p oin ts). 8 Remark. The follo win g conn ection b et ween our r esult Th eorem 6 and Szemer ´ edi’s T heorem 3 is worth making. If one mak es abstraction of the b ounds obtained, the qu alitativ e state men t in Theorem 6 can b e obtained as a corollary from Theorem 3. Here is a p ro of. F or simp licit y let n = L b e inte ger. T ak e any set of cn p oin ts. Since the s et is δ -separated every in terv al [ i, i + 1] has at most 1 /δ p oints. Therefore, there are at lea st cδ n interv als with at least o n e p oin t. By Theorem 3, w e kno w that if n is large enough then we can find k /ε interv als whic h form an arithmetic progression of length k /ε (just think of eac h in terv al [ i, i + 1] as the in teger i ). T o b e more precise, if Theorem 3 w orks for n ≥ N 0 ( k , c ) then w e apply it with N 0 ( k /ε, cδ ). Let i 0 , . . . , i k /ε b e the in terv als of this arithmetic progression. Then, b y d efinition eac h of these k /ε in terv als has a p oint from the set. Pic k an arb itrary elemen t of the set from the k in terv als i 0 , i 1 /ε , i 2 /ε , . . . , i k /ε . Then we get an ε -appro ximate k -term arithmetic progression since the distance b et we en th ese in terv als is at least 1 /ε , so the error f rom pic king an arbitrary p oin t in eac h interv al is at most ε relativ e to the distance b et ween the p oints. It is also w orth n oting that our p ro of of T heorem 6 is self conta in ed and m u c h simpler (from first prin ciples) than the pro of one gets fr om Szemer ´ edi’s theorem as describ ed ab o ve. Moreo v er , the upp er b ound resulting from our pro of is m u ch b etter than that one gets from the intege r theorem. T hat is, w ith th e quantita tive b ound s included, the t wo theorems (6 and 7) cannot b e deriv ed as corollaries of the classical integ er theorems. Indeed, as mentioned in the intro d uction no combinatorial p ro of is kno wn for the higher dimensional generalization of Szemer ´ edi’s theorem due to F¨ ur sten b erg and Katznelson. 3 Almost collinear p oin ts Let 0 < ε < 1, and let S b e a fi nite p oin t set in R d . S is sai d to b e ε - c ol line ar , if in ev ery triangle determined by S , t w o of its (in terior) angles are at most ε . Note that in particular, this co n dition implies that an ε -collinear point set is conta in ed in a section of a cylinder w hose axis is a diameter pair o f the p oin t set, a n d with radius εD , where D is the diamete r ; the cylinder radius is at most D 2 tan ε ≤ εD , for ε < 1. Theorem 9 F or an y dimension d , p ositive inte ger k , and ε > 0 , ther e exists N = N ( d, k , ε ) , such that any p oint set S i n R d with at le ast N p oints has a subset of k p oints that is ε -c ol line ar. Pro of. F or simplicit y , w e presen t the pro of for d = 2; the argument for d ≥ 3 is analogous. Finitely color all the segmen ts determined b y S as follo w s . Cho ose a co ordinate system, so that no t wo p oint s hav e the same x -coordin ate. Put r = ⌈ π /ε ⌉ + 1, and let I b e a uniform sub division of the in terv al [ − π / 2 , π / 2] in to r half-c losed subinte r v als of length at most ε . Let pq b e an y segmen t, w h ere x ( p ) < x ( q ). Color pq b y i if the angle made b y pq w ith the x -axis b elongs to the i th subinterv al. O b viously this is an r -coloring of the segment s d etermined b y S . Let N = N (2 , k , r ), where N ( · ) is as in Theorem 1. By Ramsey’s theorem (Theorem 1), for ev ery r -coloring of the segmen ts of an N -elemen t p oint set, there exists a mon o c hromatic set K of k p oin ts, that is, all segments h a ve the same color, sa y i . Let ∆ pq r b e an y triangle determined b y K , and assume that x ( p ) < x ( q ) < x ( r ). Then b y construction, w e ha ve ∠ q pr , ∠ pr q ≤ ε . T his means that K is ε -collinear, as required.. ✷ W e conclude w ith an inform al remark. Observe that the limit of an ε -col linear set, when ε → 0, is a collinear set of p oint s . It should b e noted that one cannot hop e to find any other non-c ol line ar pattern which is the limit of some approxima te patterns o ccur r ing in an y sufficient ly large p oin t set, no mat ter ho w large. Indeed, by taking all points in our g round set on a common l ine, all its subsets w ill b e collinear. 9 Ac kno w ledgmen ts. Th e author thanks the anon ymous review er of an earlier version for the observ ation and the pr o of in th e remark at the end o f Sect ion 2. References [1] P . Braß and J. P ac h : Problems and results on geometric patterns, in Gr aph The ory and Combinatoria l Optimization (D. Avis e t al., editors), Spr inger, 2005 , p p. 17–36 . [2] A. Du m itrescu and M. Jiang: Mono chromatic simplices of an y v olume, Discr ete Mathem atics , 310 (2010), 95 6–960. [3] A. Dumitrescu and M. J iang: The forest hid in g problem, Pr o c e e dings of the 21st A nnual ACM-SIAM Symp osium on Discr ete Algorithms , (S OD A 2010), Au stin, T exas, Jan. 2010, pp. 1566–15 79. [4] P . Erd˝ os, R. Graham, P . Montgo m ery , B. Rothschild, J. S p encer, and E. Straus: E u clidean Ramsey theorems, I, J. Combinator ial The ory Ser. A 14 (19 73), 341–36 3. [5] P . Erd˝ os, R. 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