On the Set of Common Differences in van der Waerden s Theorem on Arithmetic Progressions

Transcription

1 Canad. Math. Bull. Vol. 42 (1), 1999 pp On the Set of Common Dfferences n van der Waerden s Theorem on Arthmetc Progressons Tom C. Brown, Ronald L. Graham and Bruce M. Landman Abstract. Analogues of van der Waerden s theorem on arthmetc progressons are consdered where the famly of all arthmetc progressons, AP, s replaced by some subfamly of AP. Specfcally, we want to know for whch sets A, of postve ntegers, the followng statement holds: for all postve ntegers r and k, there exsts a postve nteger n = w (k, r) such that for every r-colorng of [1, n] there exsts a monochromatc k-term arthmetc progresson whose common dfference belongs to A. We wll call any subset of the postve ntegers that has the above property large. A set havng ths property for a specfc fxed r wll be called r-large. Wegve some necessary condtons for a set to be large, ncludng the fact that every large set must contan an nfnte a number of multples of each postve nteger. Also, no large set {a n : n = 1, 2,...}can have lm nf n+1 > 1. n a n Suffcent condtons for a set to be large are also gven. We show that any set contanng n-cubes for arbtrarly large n, s a large set. Results nvolvng the connecton between the notons of large and 2-large are gven. Several open questons and a conjecture are presented. 1 Introducton An arthmetc progresson of length k s a set P = {x + d : = 0,...,k 1} where x and d are ntegers, d > 0. We call d the common dfference of P. Van der Waerden s classc theorem on arthmetc progressons [13] says that, for each postve nteger r, f the set of postve ntegers, Z +, s parttoned nto r classes, then at least one of the classes wll contan arbtrarly long arthmetc progressons. An alternate (and equvalent) form of van der Waerden s theorem says that for all postve ntegers k and r, there exsts a least postve nteger w(k, r) such that for every partton of the nterval [1, w(k, r)] = {1, 2,...,w(k,r)} nto r classes, at least one of the classes wll contan a k-term arthmetc progresson. A partton of a set nto r classes s often referred to as an r-colorng of the set. So van der Waerden s theorem can be stated: for all postve ntegers r and k, there exsts a postve nteger w(k, r) so that for every r-colorng of [1, w(k, r)] there exsts a monochromatc k- term arthmetc progresson. Analogues of van der Waerden s theorem may be consdered, where the famly of arthmetc progressons, AP, s replaced by some other famly of nteger sequences. That s, f r s a postve nteger and T s a famly of nteger sequences, we can ask whether for every r-colorng of the postve ntegers there are arbtrarly long monochromatc members of T. If T s a famly that does have ths property, we say that T has the r-ramsey property. IfT has the r-ramsey property for all r, we smply say that T has the Ramsey property. By van der Waerden s theorem, f T ncludes all the arthmetc progressons, then T has the Ramsey property. The Ramsey functons assocated wth T (.e., the functons w (k, r), Receved by the edtors Aprl 16, 1997; revsed Aprl 22, AMS subject classfcaton: Prmary: 11B25; secondary: 05D10. c Canadan Mathematcal Socety

2 26 Tom C. Brown, Ronald L. Graham and Bruce M. Landman analogous to w(k, r), butwheremembersoft are soughtratherthan membersofap) have been studed for a varety of such T (see [4], [8], [10]). In ths paper, we wsh to consder the Ramsey property for collectons T that are not supersets of AP, but rather subsets of AP. Ths s of nterest because f T s a proper subset of AP, and T has the Ramsey property, the concluson to van der Waerden s theorem s strengthened. Of course, f T AP s too small, t wll not have the Ramsey property. For example, t s well known that f F s any fnte set of postve ntegers, then t s possble to 2-color the postve ntegers n such a way that there do not exst arbtrarly long monochromatc arthmetc progressons wth common dfferences belongng to F (one proof can be found n [2]; ths fact s also a consequence of Theorem 2.1 below). On the other hand, one smple consequence (and strengthenng) of van der Waerden s theorem s that f F s a fxed fnte set of postve ntegers, then the famly of all arthmetc progressons havng common dfferences n Z + F has the Ramsey property. In fact, t s easy to see that f m s a fxed postve nteger, then the famly of all arthmetc progressons havng common dfferences n the set mz + has the Ramsey property (by van der Waerden s theorem, every r-colorng of {m, 2m,..., ( w(k,r) ) m}produces a k-term monochromatc arthmetc progresson, and ths progresson has common dfference n mz + ). The examples just mentoned lead us to ask the general queston: whch subcollectons of AP have the Ramsey property? For ths paper, we consder those subcollectons of AP whch consst of all arthmetc progressons havng common dfferences n a gven set A. In ths paper, we wll call a subset A of the postve ntegers large f the collecton T, consstng of all arthmetc progressons havng common dfferences n A, has the Ramsey property. A subset of the postve ntegers that s not large s called small. Smlarly, f T has the r-ramsey property, we wll say that A s r-large. Thus, we seek an answer to the queston: whch sets of postve ntegers are large? Furstenberg, usng dynamcal systems methods, showed n [5] that every nfnte cube (see defnton below) s large. Defnton Let {a 1, a 2,...,a n }be any set of postve ntegers. The n-cube Q(a 1,...,a n )s the set of all lnear combnatons c 1 a 1 + +c n a n such that c {0,1}for all, but where not all the c are 0. Smlarly, the nfnte cube Q(a 1, a 2,...) conssts of all fnte lnear combnatons of a 1, a 2,..., wth coeffcents n {0, 1}, except for 0. (Note: n the lterature, an n-cube often refers to any translate of our Q(a 1,...,a n ); also, 0 s often ncluded n the defnton, and then our Q(a 1,...,a n ) s called a punctured n-cube ). More recently Bergelson and Lebman [2, Corollary 1.9] showed, usng measurepreservng systems methods and ergodc theory, that f C s any nfnte cube, and p(x) s any polynomal wth nteger coeffcents, postve leadng coeffcent, and p(0) = 0, then {p(x) :x C} Z + s large (n partcular, {p(x) :x 1} Z + s large). In fact, they showed that f S s any subset of Z + wth postve upper densty, then S contans arbtrarly long arthmetc progressons wth common dfferences of the form p(x), x C. For an excellent exposton of some of these results and methods, see [1] and [7]. Here we gve some results on large sets usng completely elementary methods. In partcular, we strengthen Furstenberg s result mentoned above by showng (Theorem 3.2 below)

3 van der Waerden s Theorem 27 that f the set A contans an n-cube for arbtrarly large n, thenas large. Our Corollary 2.1 and Theorem 2.4 below are smlar to results of Furstenberg [6, Theorem 3.2 and Lemma 3.3] whch deal wth recurrence n measure-preservng systems. 2 Necessary Condtons We begn wth some condtons that are necesssary for a set to be large. Of course any condton that s necessary for 2-large sets s also necessary for large sets. For convenence, we wll often use the term d-a.p. to refer to an arthmetc progresson whose common dfference s d. Further, f A s a set of postve ntegers and f P s a d-a.p. where d A, we say that P s an A-a.p. Theorem 2.1 If A s 2-large, then for each postve nteger m, A contans an nfnte number of multples of m. Proof Itsuffces to showthat for everypostve ntegerm, A contans some multple of m. By way of contradcton, assume A contans no multples of the postve nteger n. Color the postve ntegers wth the colorng χ represented by the sequence , where each block of 0 s or 1 s has length n. Let d A and let X = {x 1,...,x n+1 } be a d-a.p. Snce d (mod 2n) for some 1 2n 1, we see that X must contan some element, a, of the form 2k 1 n + j 1,1 j 1 n, as well as some element, b, of the form 2k 2 n + j 2, n +1 j 2 2n.Thenχ(a) χ(b), so X s not monochromatc. Hence under χ there do not exst monochromatc (n +1)-term arthmetc progressons havng common dfference n A. Hence A s not 2-large. Before statng the next theorem we adopt some notaton and termnology. If I and J are ntervals of the same sze havng opposte color patterns (.e., whenever x s n poston of I, andys n poston of J, thenχ(x) χ(y)), then f C s a strng of 0 s and 1 s representng the color pattern of I, we say that J has color pattern C. Also, f χ s a 2- colorng and I and J are ntervals of the same sze wth color patterns C and D, respectvely, such that ether C = D orc = D,then we say that I and J mtate each other under χ. Theorem 2.2 (1) Let A = {a k } k=1 be a sequence of postve ntegers where ether a k 3a k 1 for k 2 or (2) a 1 = 1, a 2 = 2, and a k 3a k 1 when k 3. Then A s not 2-large. Proof (3) We wll prove that {a k } k=1 s not 2-large whenever a 1 = 1, a 2 2, and, for k 3, a k 3a k 1.

4 28 Tom C. Brown, Ronald L. Graham and Bruce M. Landman Ths s suffcent for the theorem snce (2) s a specal case of (3), and any A satsfyng (1) s a subset of some A satsfyng (3). Let A = {a k } satsfy (3). Then for each k 3, a k can be expressed unquely n the form (4) k 1 a k = c (k) j a j, j=1 where the c (k) j are defned recursvely as follows: () c (k) k 1 s the largest postve nteger such that a k (c (k) k 1 +1)a k 1; () f k 4, then for each j = 2,...,k 2, once c (k) k 1, c(k) k 2,...,c(k) j+1 have been defned, defne c(k) j a k ( k 1 = j+1 to be the largest nteger such that c (k) a ) +(c (k) j +1)a j ; () fnally, c (k) 1 s defned so that a k = k 1 =1 c(k) a.fork=2, set c (k) 1 = c (2) 1 = a 2.Itthen follows from (3) that c (k) 2forallk 2andall1 k 1. Thus, for each k 2, we can partton [1, a k ] nto subntervals: B (k) (k 1, 1)...B (k) (k 1,c (k) k 1 )B(k) (k 2,1)...B (k) (k 2,c (k) k 2 )......B (k) (1, 1)...B (k) (1, c (k) 1 ), where B (k) (, j) = a for 1 k 1, 1 j c (k), and where the subntervals are lsted such that subnterval B precedes subnterval B when the members of B are less than the members of B. Let χ be the 2-colorng of the postve ntegers defned by: () χ(1) = 1; () once [1, a 1 ],...,[1, a k 1 ] have been colored, color [1, a k ] by colorng the B (k) (, j) as follows: χ ( B (k) (, j) ) { C f j s odd = f j s even where C denotes the color pattern for [1, a ]. We show that under χ there s no 5-term monochromatc a m -a.p. for any m. Letm>0 be fxed and let {x 1,...,x 5 }be a 5-term a m -a.p. Let k be the smallest postve nteger such that k > m and x 1 [1, a k ]. We consder three cases. C Case 1 x 1 B (k) (, j)where1 m 1and1 j c (k).let S= {B (k) (,j):1 m 1,1 j c (k) }. Then S = m 1 =1 c(k) a. By the way c (k) m a k ( k 1 =m+1 s defned, we have c (k) a ) +(c (k) m +2)a m.

5 van der Waerden s Theorem 29 Hence, usng (4), 2a m S. Therefore x 3 / S. We consder two subcases. In case x 2 S, thenx 3 and x 4 both belong to [a k +1,a k + 2a m ]. Hence, by the defnton of k t follows that x 3 and x 4 both belong to an nterval that mtates B (m+1) (m, 1) B (m+1) (m, 2). Then from the defnton of χ, χ(x 3 ) χ(x 4 ). In case x 2 / S, then by the same reasonng we have that χ(x 2 ) χ(x 3 ). Thus, n ether subcase, {x 1, x 2, x 3, x 4 } s not a 4-term monochromatc a m -a.p. Case 2 x 1 B (k) (m, j) for some j, 1 j c m (k).nowa m a k k 1 =m c(k) a,sothata m S. Therefore x 2 S,sobyCase1{x 2,x 3,x 4,x 5 }cannot be monochromatc. Case 3 x 1 B (k) (, j) wherem+1 k 1and1 j c (k). The block B (k) (, j) mtates [1, a ] under χ,.e., t mtates B () ( 1, 1)...B () ( 1,c () 1 ) B() ( 2,1)...B () ( 2,c () 2 )......B () (1, 1)...B () (1, c () 1 ). Hence by Cases 1 and 2 we may assume x 1 belongs to a sub-block of B (k) (, j) thatmtates B () (r, j) wherem<r 1. If r > m + 1, then we can repeat ths argument, so that by a smple nducton proof we may assume that x 1 belongs to a block B that mtates [1, a m+1 ] under χ. Ths means that we can assume k = m +1,andwe redonebycase1 and Case 2. We have shown that n all cases, there s no 5-term monochromatc a m -a.p. for any m, so that A s not 2-large. The next theorem shows that we can weaken the hypothess of Theorem 2.2 from a 3a 1 to a 2a 1, f we make the addtonal assumpton that a 1 dvdes a for all. Theorem 2.3 If A = {a } =1 s an ncreasng sequence of postve ntegers where a dvdes a +1 for all, then A s not 2-large. Proof Defne a 2-colorng χ on the set of postve ntegers recursvely, as follows. Frst let χ(x) = 1forallx [1, a 1 ]. Once χ has been defned on [1, a ], we defne χ on [1, a +1 ]by χ(x) χ(x a )foreachx [a +1,a +1 ]. Frst note that, from the way χ s defned on [a +1,a +1 ], for each 1theresno 2-term monochromatc a -a.p. contaned n [1, a +1 ]. Now assume j > +1andthat{x 1,x 2,x 3 }s a monochromatc a -a.p. that s contaned n [1, a j ]. Snce a +1 dvdes a j, we see that every subnterval of [1, a j ] of the form [ka , (k +1)a +1 ], k 1, mtates [1, a +1 ]. Hence, snce there s no 2-term monochromatc a -a.p. n [1, a +1 ], nether of the pars {x 1, x 2 } and {x 2, x 3 } could be n any one nterval [ka +1 +1,(k+1)a +1 ]. Ths mples that x 3 x 1 > a +1 2a, a contradcton. We have shown that for each 1 and each j 1, there s no 3-term monochromatc a -a.p. wth respect to χ, whch s contaned n [1, a j ], whch proves the theorem.

6 30 Tom C. Brown, Ronald L. Graham and Bruce M. Landman The next theorem, mportant n provng many subsequent results, shows that fnte unons of small sets cannot be large. Theorem 2.4 If A = A 1 A n and A s large, then some A s large. Proof By an obvous nducton argument, t suffces to prove the result for n = 2. So let A = B C, and assume nether B nor C s large. Snce B s not large, there exst postve ntegers r and k 1, and some r-colorng ρ of Z +, under whch there s no monochromatc k 1 -term B-a.p. Lkewse, there exst postve ntegers s and k 2,andans-colorng σ of Z +, under whch there s no monochromatc k 2 -term C-a.p. Defne χ to be the rs-colorng of Z + gven by χ(n) = ( ρ(n),σ(n) ).Letk=max{k 1, k 2 }. Let X be any k-term a.p. that s monochromatc wth respect to χ. Then X must also be monochromatc wth respect to each of the the colorngs ρ and σ. Hence X must have common dfference lyng outsde B C. Hence B C s not large. Corollary 2.1 Let c > 1 be a fxed real number. Let A = {a } =1 be an ncreasng sequence of postve ntegers such that a ca 1 for all but a fnte number of 2. Then A s not large. Proof Consder frst the case n whch a ca 1 for all 2. Let n be such that c n 3. For each = 1,...,n,letA ={a jn+ : j = 0,1,2,...}. For each and each j 1, a jn+ 3a (j 1)n+. Hence by Theorem 2.2 each A s not 2-large. Snce A = A 1 A n, by Theorem 2.4 A s not large (by the proof of Theorem 2.4, A s not 2 n -large). To complete the proof, let m 2besuchthata ca 1 for all m. Bytheabove case, {a m 1, a m,...}s not large. By Theorem 2.1, {a 1,...,a m 2 }s not large. Hence, by Theorem 2.4, A s not large (by the proof of Theorem 2.4, t s not 2 n+1 -large). Remarks By Theorem 2.4 t s obvous that Corollary 2.1 can be extended to any set A = B 0 B m havng the property that for each k = 0,...,m,thereexstsac k >1such that B k = {b k, : = 1, 2,...}wth b k, c k b k, 1 for 2. For example, let { f n : n 1} be the set of Fbonacc numbers. It s easy to show that for each k 0thereexstsac k >1 such that for all n 2, f n + k c k ( f n 1 + k) (whenk=0 we can take c k = 3/2). Hence f m s a fxed nonnegatve nteger, the set n=1 [ f n, f n + m] s not large. Although the complement of a small set s large (by Theorem 2.4), the complement of a large set need not be small. For example, by the work of Bergelson and Lebman, A = {n 2 } and B = {2n 2 } are both large sets, but snce B Z + A, wehavethatz + As large. We note that f A and B are small sets, then the proof of Theorem 2.4 does not necessarly gve the best (.e., the least) value of n such that A B s not n-large. For example, let m 3beoddandletA={a }be an ncreasng sequence of postve multples of m such that a 1 dvdes a for all 2. By Theorem 2.2 (or Theorem 2.3), A s not 2-large. Now let B be the set of all postve ntegers n such that m n. By Theorem 2.1, B s not 2-large. Hence, accordng to Theorem 2.4, A B s not 4-large. However, by the followng result, we can make the stonger statement that A B s not 3-large. Before proceedng we defne, for m > 1, a k-term a.p. (mod m) tobeanncreasng sequence of postve ntegers {x } k =1 such that for some d {1,2,...,m 1},x x 1 d

7 van der Waerden s Theorem 31 (mod m) forall =2,...,k. Denote by (AP) m the famly of all a.p. s (mod m). In [11] t was shown that f m s odd and A s a fnte set of multples of m, then the famly A (AP) m does not have the 3-Ramsey property, where A s the famly of all A-a.p. s. The next proposton extends ths result to certan cases n whch A contans an nfnte number of multples of m. Proposton 2.1 Let m 3 be odd and let A = {a } be a sequence of postve ntegers such that m a 1 and a 1 a for 2. LetA be the famly of all A-a.p. s. Then A (AP) m does not have the 3-Ramsey property. Proof We gve a 3-colorng χ of the postve ntegers, and show that under χ there s no monochromatc m-term a.p. (mod m) and no monochromatc 3-term a -a.p. for all. Let S 1 = [ 1, m/3 ], S 2 = [ m/3 +1, 2m/3 ],ands 3 = [ 2m/3 +1,m ]. Denote by C 1 the strng of length m representng the colorng defned by 1 fx S 1 C 1 (x) = 2 fx S 2 3 fx S 3. Denote by C 2 the strng of length m representng the colorng defned by 3 fx S 1 C 2 (x) = 1 fx S 2 2 fx S 3. If I s an nterval of length m and χ s a colorng such that χ(i) = C 1, defne χ(i) tobethe colorng C 2 ;andfχ(i)=c 2, defne χ(i) to be the colorng C 1. We now defne the colorng χ recursvely as follows: () for each x [1, a 1 ], let x be the element of [1, m] suchthatx x (mod m), and let χ(x) = C 1 (x ); () once [1, a 1 ], 2, has been colored, we color [a 1 +1,a ] as follows: χ ( [ka 1 +(j 1)m +1,ka 1 + jm] ) = χ ( [(k 1)a 1 +(j 1)m +1,(k 1)a 1 + jm] ), for 1 k (a /a 1 ) 1and1 j a 1 /m. Note that, from elementary group theory, snce m 3, any m-term a.p. (mod m) contans at least one element from each of S 1, S 2,andS 3. Hence there s no m-term monochromatc a.p. (mod m). Now assume that {x, y} s monochromatc wth y x = a. Consder the partton of the postve ntegers Z + = j=1 B j,whereb j = [( j 1)a +1 +1,ja +1 ]. Note that, by the way χ s defned, x and y cannot be monochromatc and le n the same B j. Thus, y and y + a dolenthesameb j, and hence χ(y) χ(y + a ). That s, there s no 3-term monochromatc a -a.p., and the proof s complete.

8 32 Tom C. Brown, Ronald L. Graham and Bruce M. Landman Corollary 2.2 Let m 3 be odd and let B be the set of all postve ntegers not dvsble by m. Let A = {a } be an ncreasng sequence of postve multples of m such that a 1 a for 2. ThenA B s not 3-large. Proof Ths s mmedate from Proposton 2.1 snce each B-a.p. s a member of (AP) m. Remark Corollary 2.2 follows from Proposton 2.1 by the fact that A (m), the set of all arthmetc progressons havng common dfferences whch are not multples of m,sasubset of the set (AP) m. On the other hand, there are examples showng that we cannot always replace the collecton (AP) m by the collecton A (m) and expect the Ramsey propertes to be unaffected. Forexample, n [12] t was shownthat f D s the set of all arthmetc progressons wth common dfference 2, then the famly (AP) 2 D has the 3-Ramsey property. In contrast, by Theorem 2.1, we see that A (2) D does not even have the 2-Ramsey property,.e., A = {2n 1 :n Z + } {2} s not 2-large. In fact, B = {2n 1 :n Z + } {2 n : n 0} s not 2-large, snce B contans no multples of sx. 3 Some Postve Results In ths secton we gve some suffcent condtons for a set to be large. Lemma 1 For each postve nteger r, f A s r-large and F s fnte, then A F s r-large. Proof It suffces to show that A {a}s r-large for each a A. Assume ths s not the case. Then there s an a 0 A, anr-colorng f of Z +,andak Z + such that there s no monochromatc k-term (A {a 0 })-a.p. Hence, under f, there are arbtrarly long monochromatc a 0 -a.p. s. By Theorem 2.1, ma 0 A for some m > 1. Under f,thereare arbtrarly long monochromatc ma 0 -a.p. s, a contradcton. Theorem 3.1 Let p(x) be a polynomal wth nteger coeffcents and leadng coeffcent postve. If x + a dvdes p(x) for some nteger a, then ( range(p) = {p(x) :x=1,2,...} ) Z + s large. Proof Let p(x) = (x + a)s(x). Let q(x) = p(x a). So q(x) = xs(x a). By the result of Bergelson and Lebman mentoned n Secton 1, range(q) Z + s large. If a 0, then range(q) range(p), so range(p) Z + s large. If a > 0, then range(p) = range(q) F, where F s fnte, so by Lemma 3.1 range(p) Z + s large. Remarks Does the converse of Theorem 3.1 hold? That s, f no x+a dvdes the polynomal p(x), does t follow that {p(n) :n Z + }s not large? It s easy to see (by Theorem 2.1) that the answer s yes f p(x) has degree one, for f p(x) = ax + b, wherebs not a multple of a, then the range of p(x) contans no multples of a. We do not know the answer for the general case. Another queston: s t true that whenever the range of a polynomal p(x) s not large, then the range fals to contan multples of some postve nteger m? It would be

9 van der Waerden s Theorem 33 nterestng to know whether the range of the polynomal f (x) = (x 2 13)(x 2 17)(x 2 221) s large, because although f (x) has no lnear factors, ts range contans an nfnte number of multples of every postve nteger m (usng the propertes of the Legendre symbol, one can show that the congruence f (x) 0 (mod m) s solvable for all m). Theorem 3.2 Aslarge. If A s a set of postve ntegers contanng n-cubes for arbtrarly large n, then Proof The proof makes use of the Hales-Jewett theorem [9], for whch we need some notaton. Let k be any fxed postve nteger, and let B = {0, 1,...,k 1}. For a postve nteger n, we consder the set B n of n-tuples wth entres from B, and the set (B λ) n of n-tuples wth entres from B λ, whereλs an ndetermnate. Let w(λ) denote an element of (B λ) n n whch at least one λ occurs. For each, 0 k 1, let w() denote the result of replacng each occurrence of λ n w(λ) by. Acombnatoral lne n B n s any set L havng the form L = {w(0), w(1),...,w(k 1)} for some n-tuple w(λ) n(b λ) n,wherew(λ) has at least one occurrence of λ. The Hales-Jewett theorem asserts that for any gven postve ntegers k and r, wthb= {0,1,...k 1}, there exsts a postve nteger n such that for every r-colorng of B n there s a monochromatc combnatoral lne. Now let r be a postve nteger and let χ be any r-colorng of the postve ntegers. We assume that A contans arbtrarly large cubes, and we wsh to show that for each k there are monochromatc k-term A-a.p. s. Let k be gven, and choose n accordng to the Hales-Jewett theorem so that every r-colorng of B n,whereb={0,...,k 1}, has a monochromatc combnatoral lne. Snce A contans arbtrarly large cubes, A contans an n-cube, say Q(a 1,...,a n ). Now defne an r-colorng σ, ofb n, as follows: for each (x 1,...,x n )nb n,let σ(x 1,...,x n )=χ(x 1 a 1 +x 2 a 2 + +x n a n ). By the choce of n, there exsts a combnatoral lne L = {w(0),...,w(k 1)} that s monochromatc wth respect to σ. To smplfy our notaton, we may assume that w(λ) = (x 1, x 2,...,x s,λ,λ,...,λ), where s n 1, all the x s belong to B, and there are n s occurrences of λ. Thenfor0 k 1, σ ( w() ) = σ(x 1, x 2,...,x s,,,...,)=χ ( x 1 a 1 + x s a s +(a s+1 + a n ) ). Wrtng a = x 1 a 1 + x s a s and d = a s+1 + +a n,wehavethatd Q(a 1,...,a n ) A, and χ s constant on the k-term arthmetc progresson P = {a + d :0 k 1}. Hence we have shown that for each r > 0 and each k > 0, every r-colorng of the postve ntegers contans a monochromatc k-term A-a.p. In the next corollary, the symbol {r} denotes the fractonal part of the real number r,.e., {r} = r r. Corollary 3.1 (a) Let α>0be rratonal and let ɛ>0.thena={ Z + :{α}<ɛ}s large.

10 34 Tom C. Brown, Ronald L. Graham and Bruce M. Landman (b) Let α>0.thena={ nα : n Z + } s large. (c) If A s a set of postve ntegers contanng arbtrarly long ntervals, then A s large. Proof (a) Snce {{α} : Z + } s dense n the unt nterval, for each k 1we may choose a k A such that {a k α} < ɛ/2 k. Let n be a gven postve nteger. Then {(a 1 + +a n )α}<ɛ,soacontans the nfnte cube Q(a 1, a 2,...). By Theorem 3.2, A s large. (b) The proof s essentally the same as the proof of (a). (c) The set must contan an nfnte cube (see [5, p. 171]). By Theorem 3.1, the range of any polynomal wth nteger coeffcents, that s dvsble by x + a for some a, s large. However, t s easy to fnd a large set A wth the property that for each polynomal p(x) wth nteger coeffcents, A does not contan all but fntely many elements of range(p) (we say all but fntely many because of Lemma 3.1). Ths follows easly from the followng more general result. Proposton 3.1 If R 1, R 2,... s a sequence of nfnte subsets of Z +, then there exsts a large set A such that the complement of A contans nfntely many elements of every R. Proof We defne B = {b 1, b 2,...}as follows. Let b 1 and b 2 be arbtrary elements of R 1 such that b 2 > b 1. Now assume j 3. Once b 1,...,b j 1 have been defned, choose b j such that b j R and b j b j 1 > b j 1 b j 2. Then the sequence {b j : j = 1, 2,...}contans nfntely many members of each R.Also,b j b j 1 goes to nfnty wth j. Hence, f A = Z + {b j : j =1,2,...},Acontans arbtrarly long ntervals, hence s large by Corollary 3.1(c). The followng theorem provdes some smple ways of obtanng large sets from other large sets. The proofs are relatvely straghtforward and are omtted. Theorem 3.3 (a) If A s large and m s a postve nteger, then ma s large. (b) If A s large and m s a postve nteger, then A {x:m x}s large. (c) If A s r-large, and f all elements of A are multples of the postve nteger m, then 1 m As r-large (hence, A large mples 1 m Alarge). We have yet to see an example of a set A that s r-large for some r 2, but that s not large. We make the followng conjecture. Conjecture If A s 2-large, then A s large. We have some partal results concernng the above conjecture. In the followng theorem, we use the symbol A n,whereas a set of postve ntegers, to denote the set of products {x 1 x 2...x n :x A}. Theorem 3.4 If A s 2-large, then A n s 2 n -large.

11 van der Waerden s Theorem 35 Sketch of Proof We prove ths by nducton on n. For the nducton step, let χ be a 2 n+1 - colorng of the postve ntegers, usng the colors 1, 2, 3,...,2 n+1. Defne the 2-colorng λ on Z + by { a f χ(x) [1, 2 n ] λ(x) = b f χ(x) [2 n +1,2 n+1 ]. Corollary 3.2 If A s 2-large and s closed under multplcaton, then A s large. 4 Remarks and Questons WedonotknowfTheorem2.4sstlltruefwereplace large wth r-large. In partcular, f A B s 2-large, must t follow that at least one of A or B s 2-large? We remarked n Secton 2 that, by Theorem 2.1, the set {2n 1:n 1} {2 n : n 0}s not 2-large. Wewouldlketoknowf{2n 1:n 1} {n!:n 1}(an example not covered by Theorem 2.1) s 2-large (t s not 4-large by Theorem 2.2 and the proof of Theorem 2.4). We also ask: whch sets A have the property that some translaton of A s large,.e., for whch A does there exst an nteger x such that x + A = {x + a : a A} s large? By the result of Bergelson and Lebman, the range of any polynomal p(x) has ths property, snce p(x) p(0) sends 0 nto 0. Also, t follows from Theorem 2.4 that any set A wth bounded gaps has ths property snce then Z + = s =0 (A + ) for some s. It would be nterestng to know f some translaton of the set of prmes s large (by Theorem 2.1, t would have to be an odd translaton). Let p(x) be any polynomal wth nteger coeffcents, postve leadng coeffcent, and p(0) = 0, and let A be a large set. Must {p(x) : x A} be large? In partcular, must {x 2 : x A} be large? References Acknowledgements The authors are grateful for suggestons from the referee, whch led to sgnfcant mprovements of ths paper. [1] V. Bergelson, Ergodc Ramsey theory an update. In: Ergodc theory of Z d -actons (Eds. Pollcott and Schmdt), London Math. Soc. Lecture Note Ser. 228(1996), [2] V. Bergelson and A. Lebman, Polynomal extensons of van der Waerden s and Szemeréd s theorems.j.amer. Math. Soc. 9(1996), [3] T. C. Brown, On van der Waerden s theorem and a theorem of Pars and Harrngton. J. Combn. Theory Ser. A 30(1981), [4] T. C. Brown, P. Erdős and A. R. Freedman, Quas-progressons and descendng waves. J. Combn. Theory Ser. A 53(1990), [5] H. Furstenberg, Recurrence n Ergodc Theory and Combnatoral Number Theory. Prnceton Unversty Press, Prnceton, [6], Poncaré Recurrence and Number Theory. Bull. Amercan Math. Soc. 5(1981), [7], A Polynomal Szemeréd Theorem. In: Combnatorcs, Paul Erdős s Eghty, Janos Bolya Mathematcal Socety, Budapest, [8] R.N.GreenwellandB.M.Landman,On the exstence of a reasonable upper bound for the van der Waerden numbers. J. Combn. Theory Ser. A 50(1989), [9] A.W.HalesandR.I.Jewett,Regularty and postonal games.trans.amer.math.soc.106(1963), [10] B. M. Landman, Ramsey functons for quas-progressons. Graphs and Combnatorcs, to appear. [11], Avodng arthmetc progressons (mod m) and arthmetc progressons. Utltas Math., to appear.

12 36 Tom C. Brown, Ronald L. Graham and Bruce M. Landman [12] B. M. Landman and B. Wysocka, Collectons of sequences havng the Ramsey property only for few colors.bull. Australan Math. Soc. 55(1997), [13] B. L. van der Waerden, Bewes ener Baudetschen Vermutung. Neuw Arch. Wsk. 15(1927), Department of Mathematcs and Statstcs Smon Fraser Unversty Burnaby, BC V5A 1S6 AT&T Labs 180 Park Avenue Florham Park, NJ USA Department of Mathematcal Scences Unversty of North Carolna at Greensboro Greensboro, NC USA