Some Ramsey results for the n-cube

Transcription

1 Some Ramsey resuls for he n-cube Ron Graham Universiy of California, San Diego Jozsef Solymosi Universiy of Briish Columbia, Vancouver, Canada Absrac In his noe we esablish a Ramsey-ype resul for cerain subses of he n- dimensional cube. This can hen be applied o obain reasonable bounds on various relaed srucures, such as (parial) Hales-Jewe lines for alphabes of sizes 3 and 4, Hilber cubes in ses of real numbers wih small sumses, corners in he ineger laice in he plane, and 3-erm ineger geomeric progressions. 1 Preliminaries Le us define {0, 1} n = {(ɛ 1, ɛ 2,..., ɛ n ), ɛ i = 0 or 1, 1 i n}, D(n) := {0, 1} n {0, 1} n. We can hink of he poins of {0, 1} n as verices of an n-cube Q n, and D(n) as all he line segmens joining wo verices of Q n. We will ordinarily assume ha he wo verices are disinc. We can represen he poins (X, Y ) D(n) schemaically by he diagram shown in Figure 1. In he diagram, we place X o he lef of X if w(x) < w(x ) where w(z) denoes he number of 1 s in he binary n-uple Z. Similarly, we place Y below Y if w(y ) < w(y ). (If w(x) = w(x ) or w(y ) = w(y ), hen he order doesn maer). Wih [n] := {1, 2,..., n}, I [n] and I = [n]\i, a line L = L(I, C) consiss all he pairs ((x 1, x 2,..., x n ), (y 1, y 2,..., y n )) where C = (c j ) j I wih y i = 1 x i if i I, and x j = y j = c j if j I. This research was suppored in par by NSERC and OTKA grans and an Alfred P. Sloan Fellowship. 1

2 Figure 1: Represening poins in D(n). Thus, L(I, C) = 2 I. In his case we say ha I has dimension I. Fac. Every poin (X, Y ) D(n) lies on a unique line. Proof. Jus ake I = {i [n] : x i y i } and c j = x j = y j for j I. Figure 2: A corner in D(n). By a corner in D(n), we mean a se of hree poins of he form (X, Y ), (X, Y ), (X, Y ) where (X, Y ) and (X, Y ) are on a common line L (see Figure 2). 2

3 We can hink of a corner as a binary ree wih one level and roo (X, Y ). More generally, a binary ree B(m) wih m levels and roo (X, Y ) is defined by joining (X, Y ) o he roos of wo binary rees wih m 1 levels. All of he 2 k poins a level k are required o be a common line (see Figure 3). Figure 3: A binary ree wih 3 levels 2 The main resul. Our firs heorem is he following. Theorem 1. For all r and m, here is an n 0 = n 0 (r, m) such if n n 0 and he poins of D(n) are arbirarily r-colored, hen here is always a monochromaic binary ree B(m) wih m levels formed. In fac, we can ake n 0 (r, m) = c 6 rm for some absolue consan c. Proof. Le n be large (o be specified laer) and suppose he poins of D(n) are r-colored. Consider he 2 n poins on he line L 0 = L([n]). Le S 0 L 0 be he se of poins having he mos popular color c 0. Thus, S 0 2n r. Consider he grid G 1 (lower riangular par of a Caresian produc) defined by: (See Figure 4). G 1 = {(X, Y ) : (X, Y ) S 0, (X, Y ) S 0 wih X < X }. Thus, G 1 ( ) S0 > S r 2 4n := α 1 4 n. 3

4 Figure 4: A grid poin Le us call a line L of dimension small if < n 3 and deficien if L G 1 ( α1 4 )2. Thus, he oal number of poins on small or deficien lines is a mos ( 2 n ) 2 n + α ( 1 n ) n n < n 3 = α n ( n ) 3 + (1 α 1 4 ) < n 3 ( α 1 4 )4n + (1 α 1 4 )(3.8n ) ( α 1 2 )4n provided α 1 2 (.95) n. (since < n 3 2 n ( n ) ( n ) < 1.9 n follows easily by inducion) Thus, if we discard hese poins, we sill have a leas ( α1 2 )4n poins remaining in G 1, and all hese poins are on good lines, i.e., no small and no deficien. Le L 1 be such a good line, say of dimension I 1 = n 1 n 3. Le S 1 denoe he se of poins of L 1 G 1 wih he mos popular color c 1. Therefore Observe ha G 2 G 1 (see Figure 5). S 1 ( α 1 4r )2n1. Now le G 2 denoe he grid formed by S 1, i.e., G 2 = {(X, Y ) : (X, Y S 1, (X, Y ) S 1, wih X < X }. 4

5 Figure 5: G 2 G 1 Therefore, we have ( ) S1 G 2 2 ( α 1 8r )2 4 n1 := α 2 4 n1. As before, le us classify a line L of dimension as small if < n1 3, and as deficien if L G 2 ( α2 4 )2. A similar calculaion as before shows ha if we remove from G 2 all he poins on small or deficien lines, hen a leas ( α2 2 poins will remain in G )4n1 2, provided α 2 2 (.95) n1. Le S 2 L 2 G 2 have he mos popular color c 2, so ha S 2 ( α 2 4r )2n2. Then, wih G 3 defined o he he grid formed by S 2, we have G 3 ( α2 8r )2 4 n2, and so on. Noe ha G 3 G 2 G 1. We coninue his process for rm seps. In general, we define α i+1 = ( α i 8r )2, i i rm 1 wih α 1 = 1 4r. By consrucion, we have n 2 i+1 ni 3 for all i. In addiion, we will need o have α i 2 (.95) ni for all i for he argumen o be valid. In paricular, his implies ha in general α k = 1 2 2k+2 6 r 2k

6 I is now sraighforward o check ha all he required inequaliies are saisfied by choosing n n 0 (r, m) = c 6 rm for a suiable absolue consan c. Hence, here mus be m indices i 1 < i 2 <... < i m such ha all he ses S ik have he same color. These m ses S ik conain he desired monochromaic binary ree B(m). 3 Some inerpreaions 3.1 Self-crossing pahs As we saed a he beginning, we can hink of D(n) as he se of all he diagonals of he n-cube Q n. Le us call a pair {x, x} = {(x 1,..., x n ), ( x 1,..., x n )} a main diagonal of Q n where x i = 1 x i. An affine k-subcube of Q n is defined o be a subse of 2 k poins of he form {(y 1,..., y n ) : y i = 0 or 1 if and only if i I} for some k-subse I [n] = {1, 2,..., n}. We will say ha hree conneced diagonals of he form {x, y}, {y, z}, {z, w} form a self-crossing pah, denoed by, if {x, y} and {z, w} are boh main diagonals of he same subcube. Corollary 1. In any r-coloring of he edges in D(n), here is always a monochromaic self-crossing pah, provided n > c 6 r (where c is a suiable absolue consan). The same argumen works for any subgraph G of D(n), provided ha G has enough edges and for any pair of crossing main diagonals, G has all he edges beween he pair s endpoins. 3.2 Corners The preceding echniques can be used o prove he following. Theorem 3. For every r, here exiss δ = δ(r) and n 0 = n 0 (r) wih he following propery: If A and B are ses of real numbers wih A = B = n n 0 and A+B n 1+δ, hen any r-coloring of A B conains a monochromaic corner, i.e., a se of 3 poins of he form (a, b), (a + d, b), (a, b + d) for some posiive number d. In fac, he argumen shows ha we can choose δ = 1 2 r+1. The calculaion goes as follows; The Caresian produc A B can be covered 6

7 by n 1+δ lines of slope -1. Choose he line wih he mos poins from A B, denoed by L 0. There are a leas n 2 /n 1+δ poins in L 0 A B. Choose he se of poins S 1 wih he mos popular color in L 0 A B. ( S 1 n 1 δ /r) As before, consider he grid G 1 defined by S 1, and choose he slope -1 line, L 2, which has he larges inersecion wih G 1. Choose he se of poins, S 2, having he mos popular color and repea he process wih G 2, he grid defined by S 2. We can have more han r ieraions wihou having a monochromaic corner. Solving he simple recurrence a n+1 = 2a n + 1 in he exponen, one can see ha afer r seps he size of S r is a leas c r n 1 δ(2r+1 1). If his quaniy is a leas 2, hen we have a leas one more sep and he monochromaic corner is unavoidable. The inequaliy can be rearranged ino c r n 1 δ(2r+1 1) 2 n 1 δ2r+1 2 c r n δ. From his we see ha choosing δ = 2 r 1 guaranees ha for large enough n he inequaliy above is rue, proving our saemen. By ieraing hese echniques, one can show ha he same hypoheses on A and B (wih appropriae δ = δ(r, m) and n 0 = n 0 (r, m), imply ha if A B is r-colored hen each se conains a monochromaic ranslae of a large Hilber cube, i.e., a se of he form H m (a, a 1,..., a m ) = {a + ɛ i a i } A, where ɛ i = 0 or 1, 1 i m. H m (b, a 1,..., a m ) = {b + 1 i m 1 i m ɛ i a i } B 3.3 Parial Hales-Jewe lines Corollary 2. For every r here is an n = n 0 (r) c6 r, wih he following propery. For every r-coloring of {0, 1, 2, 3} n wih n > n 0, here is always a monochromaic se of 3 poins of he form: (..., a,..., 0,..., b,..., 3,..., 0,..., c,..., 3,..., d,...) (..., a,..., 1,..., b,..., 2,..., 1,..., c,..., 2,..., d,...) (..., a,..., 2,..., b,..., 1,..., 2,..., c,..., 1,..., d,...) In oher words, every column is eiher consan, increasing from 0, or decreasing from 3. 7

8 Proof. ( To each poin (x 1, x 2,..., x n ) in {0, 1, 2, 3} n, we associae he poin (a1, a 2,..., a n ), (b 1, b 2,..., b n ) ) in {0, 1} n {0, 1} n by he following rule: x k a k b k Then i no hard o verify ha a monochromaic corner in D(n) corresponds o a monochromaic se of 3 poins as described above, a srucure which we migh call a parial Hales-Jewe line. Corollary 3. For every r here is an n = n 0 (r) c6 r, wih he following propery. For every r-coloring of {0, 1, 2} n wih n > n 0, here is always a monochromaic se of 3 poins of he form: (..., a,..., 0,..., b,..., 0,..., 0,..., c,..., 0,..., d,...) (..., a,..., 1,..., b,..., 2,..., 1,..., c,..., 2,..., d,...) (..., a,..., 2,..., b,..., 1,..., 2,..., c,..., 1,..., d,...) Proof. Map he poins (a 1, a 2,..., a n ) {0, 1, 2, 3} o poins (b 1, b 2,..., b n ) {0, 1, 2} n by: a i = 0 or 3 b i = 0, a i = 1 b i = 1, a i = 2. b i = 2 The heorem now follows by applying Corollary erm geomeric progressions. The simples non-rivial case of van der Waerden s heorem [6] saes ha for any naural number r, here is a number W (r) such ha for any r-coloring of he firs W (r) naural numbers here is a monochromaic hree-erm arihmeic progression. Finding he exac value of W (r) for large r-s is a hopelessly difficul ask. The bes upper bound follows from a recen resul of Bourgain [1]; W (r) ce r3/2. One can ask he similar problem for geomeric progressions; Wha is he maximum number of colors, denoed by r(n), ha for any r(n)-coloring of he firs N naural numbers here is a monochromaic geomeric progression. Applying Bourgain s bound o he exponens of he geomeric progression {2 i } i=0, shows ha r(n) c log log N. Using our mehod we can obain he same bound, wihou applying Bourgain s deep resul. 8

9 Observe ha if we associae he poin (a 1, a 2,..., a k,..., a n ) wih he ineger k pa k k, where p i denoes he i h prime, hen he poins (..., a,..., 0,..., b,..., 3,..., 0,..., c,..., 3,..., d,...) (..., a,..., 1,..., b,..., 2,..., 1,..., c,..., 2,..., d,...) (..., a,..., 2,..., b,..., 1,..., 2,..., c,..., 1,..., d,...) correspond o a 3-erm geomeric progression. Our bound from Corollary 2 wih an esimae for he produc of he firs n primes imply ha r(n) c log log N. 4 Concluding remarks I would be ineresing o know if we can complee he square for some of hese resuls. For example, one can use hese mehods o show ha if he poins of [N] [N] are colored wih a mos c log log N colors, hen here is always a monochromaic corner formed, i.e., 3 poins (a, b), (a, b), (a, b ) wih a + b = a + b. By projecion, his gives a 3-erm arihmeic progression (see [3]). Is i he case ha wih hese bounds (or even beer ones), we can guaranee he 4 h poin (a, b ) o be monochromaic as well? Similarly, if he diagonals of an n-cube are r-colored wih r < c log log n, is i rue ha a monochromaic mus be formed, i.e., a self-crossing 4-cycle (which is a self-crossing pah wih one more edge added)? Le denoe he srucure consising of he se of 6 edges spanned by 4 coplanar verices of an n-cube. In his case, he occurrence of a monochromaic is guaraneed once n N 0, where N 0 is a very large (bu well defined) ineger, someimes referred o as Graham s number (see [5]). The bes lower bound currenly available for N 0 is 11 (due o G. Exoo [2]). One can also ask for esimaes for he densiy analogs for he preceding resuls. For example, Shkredov has shown he following: Theorem ([4]). Le δ > 0 and N exp exp(δ c ), where c > 0 is an absolue consan. Le A by a subse of {1, 2,..., N} 2 of cardinaliy a leas δn 2. Then A conains a corner. I would be ineresing o know if he same hypohesis implies ha A conains he 4 h poin of he corner, for example. References [1] J. Bourgain, Roh s heorem on progressions revisied, Journal d Analyse Mahémaique 104(1) (2008),

10 [2] G. Exoo (personal communicaion). [3] R. Graham and J. Solymosi, Monochromaic righ riangles on he ineger grid, Topics in Discree Mahemaics, Algorihms and Combinaorics 26 (2006), [4] I. D. Shkredov, On a Generalizaion of Szemerédi s Theorem, Proceedings of he London Mahemaical Sociey (3): [5] hp://en.wikipedia.org/wiki/grahams Number. [6] B.L. van der Waerden, Beweis einer Baudeschen Vermuung, Nieuw. Arch. Wisk. 15 (1927),