On the degree of regularity of generalized van der Waerden triples

Transcription

1 On the degree of regulrity of generlized vn der Werden triples Jcob Fox Msschusetts Institute of Technology, Cmbridge, MA 02139, USA Rdoš Rdoičić Deprtment of Mthemtics, Rutgers, The Stte University of New Jersey, Pisctwy, NJ 08854, USA Abstrct Let nd b be positive integers such tht b nd (, b) (1, 1). We prove tht there exists 6-coloring of the positive integers tht does not contin monochromtic (, b)-triple, tht is, triple (x, y, z) of positive integers such tht y = x + d nd z = bx + 2d for some positive integer d. This confirms conjecture of Lndmn nd Robertson. 1 Introduction In 1916, Schur [13] proved tht for every finite coloring of the positive integers there is monochromtic solution to x + y = z. In 1927, vn der Werden [15] proved tht every finite coloring of the positive integers contins rbitrrily long monochromtic rithmetic progressions. Rdo s 1933 thesis [12] ws seminl work in Rmsey theory, generlizing the erlier theorems of Schur nd vn der Werden. Rdo clled liner homogenous eqution 1 x n x n = 0 ( i s re nonzero integers) r-regulr if every r-coloring of N contins monochromtic solution to tht eqution. An eqution is regulr if it is r-regulr for ll positive integers r. Rdo s theorem for liner homogeneous eqution sttes tht n eqution is regulr if nd only if non-empty subset of i s sums to 0. Rdo lso mde conjecture Emil ddresses: licht@mit.edu (Jcob Fox), rdos@mth.rutgers.edu (Rdoš Rdoičić).

2 [12] tht further differentites between those liner homogeneous equtions tht re regulr nd those tht re not. Conjecture 1 (Rdo s Boundedness Conjecture, 1933) For every positive integer n, there exists n integer k := k(n) such tht every liner homogeneous eqution 1 x n x n = 0 tht is k-regulr must be regulr s well. This outstnding conjecture hs remined open except in the trivil cses (n = 1, 2) until recently, when the first uthor nd Kleitmn settled the first nontrivil cse n = 3 [4], [8]. They proved tht k(3) 24. Vn der Werden s theorem hs been strengthened nd generlized in numerous other wys [1], [2], [5], [6], [7], [11], [14]. In this note, we consider one of the generliztions, proposed by Lndmn nd Robertson in [10]. Let nd b be positive integers such tht b. A triple (x, y, z) of positive integers is clled n (, b)-triple if there exists positive integer d such tht y = x+d nd z = bx+2d. The degree of regulrity of n (, b)-triple, denoted by dor(, b), is the lrgest positive integer r, if it exists, such tht for every r-coloring of the positive integers there is monochromtic (, b)-triple. If no such r exists, tht is, for every finite coloring of the positive integers there is monochromtic (, b)-triple, then set dor(, b) =. Note tht vn der Werden s theorem for 3-term rithmetic progressions is equivlent to dor(1, 1) =. Lndmn nd Robertson proved tht dor(, b) = 1 if nd only if b = 2. They lso showed tht dor(, 2 1) = 2 for 2. For smll vlues of nd b, they provide few dditionl results: dor(1, 3) 3, dor(2, 2) 5, dor(2, 5) 3, dor(2, 6) 3, dor(3, 3) 5, dor(3, 4) 5, dor(3, 8) 3, dor(3, 9) 3. Finlly, they conjectured tht if (, b) (1, 1), then dor(, b) is finite [10], [11]. We confirm nd further strengthen their conjecture. Theorem 1 If (, b) (1, 1), then dor(, b) < 6. Our proof tht dor(, b) is finite uses Rdo s theorem for homogenous liner eqution. Proving specific upper bound of 6, regrdless of prmeters nd b, relies on the bove mentioned proof of Fox nd Kleitmn [4]. 2 Proof of Theorem 1 First, notice tht if (x, y, z) is n (, b)-triple, then (x, y, z) stisfies the eqution (b 2)x 2y + z = 0. 2

3 By Rdo s theorem [12], this eqution is regulr if nd only if b {2 2, 2 1, 2 + 1}. Therefore, if b 2 { 2, 1, 1}, then dor(, b) is finite. Moreover, since k(3) 24 [4], if b 2 { 2, 1, 2}, then dor(, b) 23. As mentioned in the introduction, Lndmn nd Robertson [10] proved tht dor(, 2 1) = 2 for 2. For the remining cses, we use Lemm 1, which is stted nd proved next. Lemm 1 Let α nd β be rel numbers such tht 1 < α < β. Set r = log α β. Then every r-coloring of the positive integers contins integers x nd y of the sme color with αx y βx. Moreover, there is n (r + 1)-coloring of the positive integers tht contins no integers x nd y of the sme color with αx y βx. Proof: Consider coloring of N without x nd y of the sme color with αx y βx. Since r = log α β, then α r 1 < β. Let x 1 > r 2 k=0 αk /(β α r 1 ) be positive integer. For i > 1, set x i+1 = αx i. We hve αx i x i+1 < αx i + 1. Repetedly using the inequlity x i+1 < αx i + 1, we obtin x r < α r 1 x 1 + r 2 k=0 αk. Since we ppropritely chose x 1, the lst inequlity yields x r < βx 1. Hence, αx i x j βx i for 1 i < j r, so x 1,..., x r must ll hve different colors. Therefore, the number of colors is t lest r + 1. Next, we construct coloring of the positive integers by the elements of Z r+1 such tht there do not exist x nd y of the sme color with αx y βx. For every nonnegtive integer n, integers in the intervl [α n, α n+1 ) receive color n (mod r + 1). Within ech intervl, every pir of integers x nd y hve the sme color, but y < αx. For monochromtic x nd y from different intervls, with y > x, we hve y > α r x βx. Therefore, this (r + 1)-coloring of the integers hs no monochromtic x nd y such tht αx y βx. Now, we continue with the proof of Theorem 1. We hve two cses. Cse 1. b = In this cse, we hve y = x + d nd z = (2 + 1)x + 2d. Therefore, 2y < z < ( 2+1)y. Using Lemm 1 nd noting 1, we obtin dor(, 2 + 1) log 2 (2 + 1 ) = 2. Hence, for ll positive integers, we hve dor(, 2 + 1) = 2. Cse 2. b = 2 2. Since b must be positive integer, then 2. As mentioned in the introduction, Lndmn nd Robertson [10] proved tht dor(2, 2) 5. If > 2, then y = x + d nd z = (2 2)x + 2d. So, ( 2 2 )y < z < 2y. Using Lemm 1 nd 3, we obtin dor(, 2 2) log We hve 2 2 > 2 when > 3. Therefore, 2 dor(, 2 2) 3 for = 3 nd dor(, 2 2) = 2 for > 3. At this stge, we hve dor(, b) < 24, whenever (, b) (1, 1). Next, we improve the upper bound using some sophisticted tools from the pper of Fox nd Kleitmn [4]. For 3

4 the ske of completeness nd clrity, we repet some of their nlysis tht pplies in our context. We need the following bit of nottion. Definition: Let p be prime number. For every integer n, let v p (n) denote the lrgest power of p tht divides n. If n = 0, let v p (n) = +. Notice tht v p (m 1 m 2 ) = v p (m 1 ) + v p (m 2 ) for every prime p, nd integers m 1 nd m 2. The following strightforwrd lemm (Lemm 3 in [4]) gives bsic properties of the function v p, which we will repetedly use. Lemm 2 If m 1, m 2, m 3 re integers with v p (m 1 ) v p (m 2 ) v p (m 3 ) nd v p (m 1 ) < v p (m 1 + m 2 + m 3 ), then v p (m 1 ) = v p (m 2 ). Furthermore, if v p (m 3 ) < v p (m 1 + m 2 + m 3 ), then lso v p (m 1 + m 2 ) = v p (m 3 ). Recll tht if (x, y, z) is n (, b)-triple, then (x, y, z) stisfies the eqution (b 2)x + 2y z = 0. Let t x = b 2, t y = 2, nd t z = 1 denote the coefficients of x, y, nd z in this eqution, respectively. We hve three cses to consider, depending on t x. Cse A. t x is multiple of 4. Clerly, we hve v 2 (t x ) > v 2 (t y ) = 1 > v 2 (t z ) = 0. Let S = {v 2 (t x ), v 2 (t y ), v 2 (t x ) v 2 (t y )}, nd let Γ(S) be the undirected Cyley grph of the group (Z, +) with genertors being the elements of S. Since every vertex of Γ(S) hs degree 2 S, there exists proper (greedy) ( S + 1)-coloring χ of its vertices. This result is folklore nd we refer the reder to Lemm 2 in [4] for detils. Now, define χ(n) = χ (v 2 (n)), for every n N. We clim tht in the 4-coloring χ of N there re no x, y, nd z, ll of the sme color nd v 2 (t x x+t y y+t z z) > min{v 2 (t x x), v 2 (t y y), v 2 (t z z)}. Indeed, otherwise (by Lemm 2) we hve v 2 (t x x) = v 2 (t y y); or v 2 (t x x) = v 2 (t z z); or v 2 (t y y) = v 2 (t z z). This implies v 2 (y) v 2 (x) = v 2 (t x ) v 2 (t y ); or v 2 (z) v 2 (x) = v 2 (t x ); or v 2 (z) v 2 (y) = v 2 (t y ). However, this contrdicts tht χ is proper coloring of Γ(S) nd v 2 (x), v 2 (y), nd v 2 (z) re ll of the sme color. Since v 2 (0) = +, by definition, nd since there re no x, y, z, ll of the sme color nd v 2 (t x x + t y y + t z z) > min{v 2 (t x x), v 2 (t y y), v 2 (t z z)}, then, in prticulr, there re no monochromtic solutions to t x x + t y y + t z z = 0 (i.e. (b 2)x + 2y z = 0) in χ. Cse A is equivlent to Lemm 4 (with p = 2) in [4]. Cse B. t x hs n odd prime fctor p. 1 In this cse we hve v p (t x ) > v p (t y ) = v p (t z ) = 0. Let d = v p (t x ). We construct 6-coloring χ tht is product of 2-coloring χ 1 nd 3-coloring χ 2. For n N define χ 1 (n) v p(n) (mod 2). The coloring χ d 1 (n) colors intervls of v p vlues of length d, open on one side, periodiclly in 2 colors with period 2. Let Γ be the undirected Cyley grph on Z p \ {0} such tht (u, v) is n edge of Γ if nd only if u 2v 0 (mod p) or 2u v 0 (mod p). Since every vertex of Γ hs degree 2, there exists proper 3-coloring χ 2 : V (Γ) {0, 1, 2}. For n N define χ 2 (n) = χ 2(m mod p), where n = mp vp(n). Finlly, for n N define χ(n) = (χ 1 (n), χ 2 (n)). We clim tht in the 6-coloring χ of N there re no x, y, nd z, ll of the sme color nd 1 Note tht Cses A nd B overlp. However, it is only importnt tht Cses A, B, nd C cover ll the possibilities. 4

5 v p (t x x+t y y +t z z) > mx{v p (t x x), v p (t y y), v p (t z z)}. Indeed, otherwise (by Lemm 2) we hve v p (t x x) = v p (t y y) v p (t z z); or v p (t x x) = v p (t z z) v p (t y y); or v p (t y y) = v p (t z z) v p (t x x). If v p (t x x) = v p (t y y), then d + v p (x) = v p (y), hence, χ 1 (x) χ 1 (y), which contrdicts χ(x) = χ(y). If v p (t x x) = v p (t z z), then d + v p (x) = v p (z), hence, χ 1 (x) χ 1 (z), which contrdicts χ(x) = χ(z). So, ssume tht v p (t y y) = v p (t z z) v p (t x x). Reclling our coefficients, we obtin v p (y) = v p (z) d+v p (x). By (the second prt of) Lemm 2, we lso hve v p (2y z) = d+v p (x). Let e denote the common vlue of v p (y) nd v p (z). Let y = y p e nd z = z p e. Since χ 2 (y) = χ 2 (z), then χ 2(y mod p) = χ 2(z mod p), hence, 2y z 0 (mod p). However, this implies v p (2y z) = e, so v p (y) = v p (z) = e = d + v p (x). It follows from here tht χ 1 (x) is different from χ 1 (y) nd χ 1 (z), which contrdicts χ(x) = χ(y) = χ(z). Since v p (0) = + nd there re no x, y, z, ll of the sme color nd v p (t x x+t y y+t z z) > mx{v p (t x x), v p (t y y), v p (t z z)}, then, in prticulr, there re no monochromtic solutions to t x x + t y y + t z z = 0 (i.e. (b 2)x + 2y z = 0) in χ. Cse B is essentilly equivlent to Lemm 6 (with s = 1) in [4]. Notice tht one cn define χ 2 to be 2-coloring in the proof bove, s long s the order of 2 mod p is even. Cse C. t x { 2, 1, 1, 2} Cse t x = 1 is tken cre of in [10], s mentioned before, while cses t x = 1 nd t x = 2 correspond to Cses 1 nd 2, respectively. The only remining cse is t x = 2. 2 In this cse, we hve y = x + d nd z = (2 + 2)x + 2d. Therefore, 2y < z < 4y. Using Lemm 1, we obtin dor(, 2 + 2) log 2 4 = 2. Hence, for ll positive integers, we hve dor(, 2 + 2) = 2. 3 Concluding remrks As consequence of our proof, we obtined dor(1, 3) = 2, dor(2, 5) = 2, dor(2, 6) = 2, dor(3, 3) 3, dor(3, 4) 3, dor(3, 8) = 2. These results improve the corresponding entries in the tble provided by Lndmn nd Robertson [10] for smll vlues of nd b. After submission, we lerned tht Frntzikinkis, Lndmn, nd Robertson [9] independently showed tht dor(, b) is finite unless (, b) = (1, 1). References [1] V. Bergelson nd A. Leibmn, Polynomil extensions of vn der Werden s nd Szemerédi s theorems, J. Amer. Mth. Soc. 9 (1996) The eqution is not regulr in this cse, however this possibility is not covered by Cses A nd B of the improved upper bound nlysis. 5

6 [2] T. C. Brown, R. L. Grhm, nd B. M. Lndmn, On the set of common differences in vn der Werden s theorem on rithmetic progressions, Cnd. Mth. Bull. 42 (1999), [3] T. C. Brown, B. M. Lndmn, nd M. Mishn, Monochromtic homothetic copies of {1, 1 + s, 1 + s + t}, Cndin Mth. Bull. 40 (1997), [4] J. Fox nd D. Kleitmn, On Rdo s Boundedness Conjecture, Journl of Combintoril Theory, Series A, ccepted. [5] H. Furstenberg, Recurrences in Ergodic Theory nd Combintoril Number Theory. Princeton University Press, Princeton, [6] T. Gowers, A new proof of Szemerédi s theorem, Geometric nd Functionl Anlysis, 11 (2001) [7] R. L. Grhm, B. L. Rothschild, nd J. Spencer, Rmsey Theory. John Wiley & Sons Inc., New York, [8] N. Hindmn, I. Leder, nd D. Struss, Open problems in prtition regulrity, Combintorics, Probbility, nd Computing, 12 (2003) [9] B. M. Lndmn, personl communiction. [10] B. M. Lndmn nd A. Robertson, On generlized vn der Werden triples, Discrete Mthemtics 256 (2002) [11] B. M. Lndmn nd A. Robertson, Rmsey theory on the integers. Student Mthemticl Librry, 24. Americn Mthemticl Society, Providence, RI, (Reserch Problem 5.4, p. 159.) [12] R. Rdo, Studien zur Kombintorik, Mth. Zeit. 36 (1933) [13] I. Schur, Uber die Kongruenze x m +y m z m (mod p), Jber. Deutsch. Mth.-Verein. 25 (1916), [14] E. Szemerédi, On sets of integers contining no k elements in rithmetic progression, Act Arith. 27 (1975), [15] B. L. vn der Werden, Beweis einer Budetschen Vermutung, Nieuw Arch. Wisk. 15 (1927),