BOUNDS ON SOME VAN DER WAERDEN NUMBERS To Brow Departet of Matheatics, Sio Fraser Uiversity, Buraby, BC V5A S6 Bruce M Lada Departet of Matheatics, Uiversity of West Georgia, Carrollto, GA 308 Aaro Robertso Departet of Matheatics, Colgate Uiversity, Hailto, NY 3346 Abstract For positive itegers s ad k, k 2,, k s, the va der Waerde uber w(k, k 2,, k s ; s) is the iiu iteger such that for every s-colorig of {, 2,, }, with colors, 2,, s, there is a k i -ter arithetic progressio of color i for soe i We give a asyptotic lower boud for w(k, ; 2) for fixed We iclude a table of values of w(k, 3; 2) that are very close to this lower boud for = 3 We also give a lower boud for w(k, k,, k; s) that slightly iproves previously-kow bouds Upper bouds for w(k, 4; 2) ad w(4, 4,, 4; s) are also provided Itroductio Two fudaetal theores i cobiatorics are va der Waerde s Theore [8] ad Rasey s Theore [6] The theore of va der Waerde says that for all positive itegers s ad k, k 2,, k s, there exists a least positive iteger = w(k, k 2,, k s ; s) such that wheever [, ] = {, 2,, } is s-colored (ie, partitioed ito s sets), there is a k i -ter arithetic progressio with color i for soe i, i s Siilarly, Rasey s Theore has a associated threshold fuctio R(k, k 2,, k s ; s) (which we will ot defie here) The order of agitude of R(k, 3; 2) is kow to be k2 [], while log k the best kow upper boud o R(k, ; 2) is fairly close to the best kow lower boud I cotrast, the order of agitude of w(k, 3; 2) is ot kow, ad the best kow lower ad upper bouds o w(k, k; 2) are (k )2 (k ) w(k, k; 2) < 2 2222(k+9),
the lower boud kow oly whe k is prie The lower boud is due to Berlekap [] ad the upper boud is a celebrated result of Gowers [6], which aswered a log-stadig questio of Ro Graha Graha curretly offers 000 USD for a proof or disproof of w(k, k; 2) < 2 k2 [2] Several other ope probles are stated i [4] Recetly, there have bee soe further breakthroughs i the study of the va der Waerde fuctio w(k, ; 2) Oe was the aazig calculatio that w(6, 6; 2) = 32 by Kouril [2], extedig the list of previously kow values w(3, 3; 2) = 9, w(4, 4; 2) = 35, ad w(5, 5; 2) = 78 A list of other kow exact values of w(k, ; 2) appears i [5] Iproved lower bouds o several specific values of w(k, k; s) are give i [3] ad [0] I aother directio, Graha [7] gives a elegat proof that if oe defies w (k, 3) to be the least such that every 2-colorig of [, ] gives either k cosecutive itegers i the first color or a 3-ter arithetic progressio i the secod color, the k c log k < w (k, 3) < k dk2, for suitable costats c, d > 0 This iediately gives w(k, 3; 2) < k dk2 sice we trivially have w(k, 3; 2) w (k, 3) I view of Graha s bouds o w (k, 3), it would be desirable to obtai iproved bouds o w(k, 3; 2) Of particular iterest is the questio of whether or ot there is a o-polyoial lower boud for w(k, 3; 2) I this ote we give a lower boud of w(k, 3; 2) > k (2 o()) Although this ay see weak, we do kow that w(k, 3; 2) < k 2 for 5 k 6 (ie, for all kow values of w(k, 3; 2) with k 5; see Table ) More geerally, we give a lower boud o w(k, ; 2) for arbitrary fixed We also preset a lower boud for the classical va der Waerde ubers w(k, k,, k; s) that is a slight iproveet over previously published bouds I additio, we preset a upper boud for w(k, 4; 2) ad a upper boud for w(4, 4,, 4; s) 2 Upper ad Lower Bouds for Certai va der Waerde Fuctios We shall eed several defiitios, which we collect here For positive itegers k ad, r k () = ax { S : S cotais o k-ter arithetic progressio} S [,] For positive itegers k ad, deote by χ k () the iiu uber of colors required to color [, ] so that there is o oochroatic k-ter arithetic progressio The fuctio w (k, 3) has bee defied i Sectio Siilarly, we defie w (k, 4) to be the least such that every 2-colorig of [, ] has either k cosecutive itegers i the first color or a 4-ter arithetic progressio i the secod color 2
We begi with a upper boud for w (k, 4) The proof is essetially the sae as the proof give by Graha [7] of a upper boud for w (k, 3) For copleteess, we iclude the proof here We will ake use of a recet result of Gree ad Tao [9], who showed that for soe costat c > 0, r 4 () < e c log log () for all 3 Propositio 2 There exists a costat c > 0 such that w (k, 4) < e kc log k for all k 2 Proof Suppose we have a 2-colorig of [, ] (assue 4) with o 4-ter arithetic progressio of the secod color ad o k cosecutive itegers of the first color Let t < t 2 < < t be the itegers of the secod color Hece, < r 4 () Let us defie t 0 = 0 ad t + = The there ust be soe i, i, such that t i+ t i > 2r 4 () (Otherwise, usig r 4 () 3, we would have = i=0 (t i+ t i ) (+) 2r 4 (r 4()+) () 2r 4 + ) () 2 6 Usig (), we ow have a i with t i+ t i > 2r 4 () > log log 2 ec If e kd log k, d = c 2, the 2 ec log log k ad we have k cosecutive itegers of the first color, a cotradictio Hece, < e kd log k ad we are doe Clearly w(k, 4; 2) w (k, 4) Cosequetly, we have the followig result Corollary 22 There exists a costat d > 0 such that w(k, 4; 2) < e kd log k for all k 2 Usig Gree ad Tao s result, it is ot difficult to obtai a upper boud for w(4, 4,, 4; s) Propositio 23 There exists a costat d > 0 such that w(4, 4,, 4; s) < e sd log s for all s 2 Proof Cosider a χ 4 ()-colorig of [, ] for which there is o oochroatic 4-ter arithetic progressio Soe color ust be used at least χ 4 ties, ad hece r () χ 4 () 4() so that χ 4() Let c > 0 be such that () holds for all 3, ad let = e sd log s, where r 4 () d = c 2 The χ 4 () > r 4 () ec log log = s This eas that every s-colorig of [, ] has a oochroatic 4-ter arithetic progressio Sice = e sd log s, the proof is coplete It is iterestig that the bouds i Corollary 22 ad Propositio 23 have the sae for 3
The followig theore gives a lower boud o w(k, k,, k; s) It is deduced without too uch difficulty fro the Syetric Hypergraph Theore as it appears i [8], cobied with a old result of Raki [7] To the best of our kowledge it has ot appeared i prit before, eve though it is better, for large s, tha the stadard lower boud csk ( + o()) (see [8]), as k well as the lower bouds s k+ c(k + ) log(k + ) ad ksk due to Erdős ad Rado [4], ad e(k+) 2 Everts [5], respectively We give the proof i soe detail The proof akes use of the followig facts: 2 log χ k () < ( + o()), (2) r k () which appears i [8] as a cosequece of the Syetric Hypergraph Theore; ad r k () > e c(log ) log 2 k +, (3) which, for soe costat c > 0, holds for all 3 (this appears i [7]) Theore 24 Let k 3 be fixed, ad let z = log 2 k There exists a costat d > 0 such that w(k, k,, k; s) > s d(log s)z for all sufficietly large s Proof We ake use of the observatio that for positive itegers s ad, if s χ k (), the w(k, k,, k; s) >, which is clear fro the defiitios For large eough, (2) gives ( 2 log χ k () < + ) 3 log = r k () 2 r k () (4) Now let d = ( z+, 2c) where c is fro (3), ad let = s d(log s) z, where s is large eough so that (4) holds By (3), otig that log = d(log s) z+ = ( ) log s z+, 2c we have Therefore, r k () ) z+ < ec(log 3 log r k () for sufficietly large s Thus, for sufficietly large s, log s c = e 2c = s < 3d s(log s) z+ < s χ k () < 3 log r k () < s Accordig to the observatio at the begiig of the proof, this iplies that w(k, k,, k; s) > = s d(log s)z, as required We ow give a lower boud o w(k, ; 2) We ake use of the Lovász Local Lea (see [8] for a proof), which will be iplicitly stated i the proof 4
Theore 25 Let 3 be fixed The for all sufficietly large k, w(k, ; 2) > k log log k Proof Give, choose k > large eough so that ad k 2 log log k > ( ) log k (5) 6 < log k log log k (6) Next, let = k log log k To prove the theore, we will show that there exists a (red, blue)-colorig of [, ] for which there is o red k-ter arithetic progressio ad o blue -ter arithetic progressio For the purpose of usig the Lovász Local Lea, radoly color [, ] i the followig way For each i [, ], color i red with probability p = k α where ad color it blue with probability p α def = 2 log log k, Let P be ay k-ter arithetic progressio The, sice + x e x, the probability that P is red is p k = ( k α ) k (e k α ) k = e kα Hece, applyig (5), we have p k < ( ) ( )log k = e k Also, for ay -ter arithetic progressio Q, the probability that Q is blue is ( p) = (k α ) = k Now let P, P 2,, P t be all of the arithetic progressios i [, ] with legth k or So that we ay apply the Lovász Local Lea, we for the depedecy graph G by settig V (G) = {P, P 2,, P t } ad E(G) = {{P i, P j } : i j, P i P j } For each P i V (G), let d(p i ) deote the degree of the vertex P i i G, ie, {e E(G) : P i e} We ow estiate d(p i ) fro above Let x [, ] The uber of k-ter arithetic progressios P i [, ] that cotai x is bouded above by k, sice there are k positios that x ay occupy i k P ad sice the gap size of P caot exceed Siilarly, the uber of -ter arithetic k progressios Q i [, ] that cotai x is bouded above by 5
Let P i be ay k-ter arithetic progressio cotaied i [, ] The total uber of k- ter arithetic progressios P ad -ter arithetic progressios Q i [, ] that ay have o-epty itersectio with P i is bouded above by ( ) ( k k k + < k 2 + 2 ), (7) sice k > Thus, d(p i ) < k ( ) 2 + 2 whe Pi = k Likewise, d(p i ) < ( ) 2 + 2 whe P i = Thus, for all vertices P i of G, we have d(p i ) < k ( 2 + 2 ) To fiish settig up the hypotheses for the Lovász Local Lea, we let X i deote the evet that the arithetic progressio P i is red if P i = k, or blue if P i =, ad we let d = ax d(p i) We have see above that for all i, i t, the probability that X i occurs is i t at ost q def =, while fro (7) we have d < 2k ( + ) k We are ow ready to apply the Lovász Local Lea, which says that i these circustaces, if the coditio eq(d + ) < is satisfied, the there is a (red, blue)-colorig of [, ] such that o evet X i occurs, ie, such that there is o red k-ter arithetic progressio ad o blue -ter arithetic progressio This will iply w(k, ; 2) > = k log log k, as desired Thus, the proof will be coplete whe we verify that eq(d + ) < Usig 3, we have d < 3k, so that d + < 3k + < e 2 k Hece, it is sufficiet to verify that e 3 qk < (8) Sice q = k is ow coplete ad k log log k, iequality (8) ay be reduced to (6), ad the proof Reark As log as k > e 6, the iequality of Theore 25 holds To show this, we eed oly show that coditios (5) ad (6) hold if k > e 6 That (6) holds is obvious For (5), it suffices to have k 2 log log k > log k; that is log k > 2 log log k(log + log log k) Whe k e 6, we have 2 log log k(log + log log k) 2(log k) /6 log log k( log log k + log log k) = 6 7 (log 3 k)/6 (log log k) 2 Sice (log log k) < (log k) 7/20 for k e 6 we have 2 log log k(log + log log k) 7(log 3 k)3/5 Fially, sice (log k) 2/5 7 for k 3 e6, coditio (5) is satisfied We ed with a table of coputed values These were all coputed with a stadard backtrack algorith except for w(4, 3; 2), w(5, 3; 2), ad w(6, 3; 2), which are due to Michal Kouril [3] The values w(k, 3; 2), k 2, appeared previously i [5] 6
k 2 3 4 5 6 7 8 9 0 2 3 4 5 6 w(k, 3; 2) 6 9 8 22 32 46 58 77 97 4 35 60 86 28 238 w (k, 3) 9 23 34 73 3 93????????? Table : Sall values of w(k, 3) ad w (k, 3) Refereces E Berlekap, A costructio for partitios which avoid log arithetic progressios, Caad Math Bull (968), 409-44 2 F Chug, P Erdős, ad R Graha, O sparse sets hittig liear fors, Nuber theory for the illeiu, I (Urbaa, IL, 2000), 257-272, A K Peters, Natick, MA, 2002 3 MR Drasfield, L Liu, VW Marek, ad M Truszczyski, Satisfiability ad coputig va der Waerde ubers, Elec J Cobiatorics (2004), R4 4 P Erdős ad R Rado, Cobiatorial theores o classificatios of subsets of a give set, Proc Lodo Math Soc 3 (952), 47-439 5 F Everts, Colorigs of sets, PhD thesis, Uiversity of Colorado, 977 6 W T Gowers, A ew proof of Szeerédi s theore, Geo Fuct Aal (200), o 3, 465-588 7 R Graha, O the growth of a va der Waerde-like fuctio, Itegers: Elec J Cobiatorial Nuber Theory 6 (2006), A29 8 R Graha, B Rothschild, ad J Specer, Rasey Theory, Secod Editio, Wiley, New York, 990 9 B Gree ad T Tao, New bouds for Szeerédi s theore II: A ew boud for r 4 (N), preprit: arxiv:ath//060604v 0 PR Herwig, MJH Heule, PM va Labalge, ad H va Maare, A ew ethod to costruct lower bouds for va der Waerde ubers, Elec J Cobiatorics 4 (), 2007, R6 J H Ki, The Rasey uber R(3, t) has order of agitude t 2 / log t, Rado Structures ad Algoriths 7 (995), o 3, 73-207 2 M Kouril, PhD thesis, Uiversity of Ciciati, 2007 3 M Kouril, private couicatio, 2007 4 B Lada ad A Robertso, Rasey Theory o the Itegers, Aerica Matheatical Society, Providece, RI, 2004 5 B Lada, A Robertso, ad C Culver, Soe ew exact va der Waerde ubers, Itegers: Elec J Cobiatorial Nuber Theory 5(2) (2005), A0 6 F Rasey, O a proble of foral logic, Proc Lodo Math Soc 30 (930), 264-286 7 R Raki, Sets of itegers cotaiig ot ore tha a give ubers of ters i arithetical progressio, Proc Roy Soc Ediburgh Sect A 65 (960/96), 332-344 8 B L Va der Waerde, Beweis eier baudetsche Verutug, Nieuw Archief voor Wiskude 5 (927), 22-26 7