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1 CHAPTER 8 Monochroatic iages 1 The Central Sets Theore Lea 11 Let S,+) be a seigroup, e be an idepotent of βs and A e There is a set B A in e such that, for each v B, there is a set C A in e with v+c A Proof By the Idepotent Characterization Theore, there is a set B A in e such that, for each b B, there is a set C b A in e with b+c b A For each vector b 1 v C, b we have that C : C b1 C b e Then v+c A The following notion is stronger than being piecewise syndetic and FS Definition 12 A set A N is central if it is a eber of a inial idepotent of βn,+) Theore 13 Central Sets) Let S,+) be an abelian seigroup, A S be a central set and be a natural nuber For all v 1,v 2, S {0}), there are nonepty finite sets of natural nubers F 1 < F 2 < and eleents a 1,a 2, A such that FSa 1 +v F1,a 2 +v F2,) A Proof Let e be a inial idepotent of βs with A e We proceed as in the proof of Hindan s Theore A1 A2 A3 A4 w 1 w 2 w 3 We use Lea 11 repeatedly Let A 1 : A Choose an eleent B e as in the lea, for A 1 Let w 1 B By the lea, there is a set A 2 A 1 in e such that w 1 +A 2 A 1 Choose an eleent B e as in the lea, for A 2 Let w 2 B By the lea, there is a set A 3 A 2 in e such that w 2 +A 3 A 2 61

2 62 8 MONOCHROMATIC IMAGES Continue in the sae anner It follows, as in the proof of Hindan s Theore, that FSw 1,w 2,) A In each step n of the construction, the vector w n ay be chosen to be any eleent in a power of a central set Thus, by the Piecewise Syndetic Sets Theore, we ay request that the vector w 1 is of the for a 1 +v F1, where a 1 B In particular, a 1 A By the Piecewise Syndetic Sets Theore with the vectors {v n : n > F 1 }, we ay request that the vector w 2 is of the for a 2 +v F2, where F 1 < F 2 and a 2 B A Continuing in this anner, we see that we ay request that F n < F n+1 and a n A for all n w n a n +v Fn, Exercise 14 Prove that, in the Central Sets Theore, we ay request, in addition, that FSa 1,a 2,) A Hint: Consult the proof of the Piecewise Syndetic Sets Theore Corollary 15 Let S,+) be an abelian seigroup and be a natural nuber For each finite coloring of S and all v 1,v 2, S {0}), there are a color, nonepty finite sets of natural nubers F 1 < F 2 <, and eleents a 1,a 2, S, such that the coordinates of the vectors in the set FSa 1 +v F1,a 2 +v F2,) are all of that color Proof Let e be a inial idepotent of βs Take a onochroatic set A e and apply the Central Sets Theore We will use below that the finite sus in the Central Sets Theore are of the following for: For all i 1 < i 2 < < i k, a i1 +v Fi1 )+ +a ik +v Fik ) a i1 + +a ik +v Fi1 + +v Fik a+v F, where a a i1 + +a ik and F F i1 F ik For a finite set H N, write F H : n H F n Then a n +v Fn a H +v FH n H 2 Monochroatic iages Definition 21 Let A be a atrix of nonnegative integers An entry a ij of the atrix A is first if it is the first nonzero entry in its row A atrix A has the first entries property if it has no zero rows so that each row as a first entry) and, in each colun of A, the first entries are equal In the definition of the first entries property, we do not request that there are first entries in every colun of the atrix In this section, we will prove the following theore Theore 22 Monochroatic Iage) Let A be an n atrix of nonnegative integers with the first entries property For each finite coloring of N, there is a vector v N n such that all coordinates of the vector Av are of the sae color Moreover, for each central set A N there is a vector v N n such that Av A As usual, to see that the second part of the theore iplies the first, fix a inial idepotent e βn and recall that, given a finite coloring of N, there is in e a onochroatic set A The set A is central This provides a stronger assertion that, for each finite coloring of N,

3 2 MONOCHROMATIC IMAGES 63 there is a color such that all atrices with the first entries property have iage vectors with all entries of that color Before proving this theore, we illustrate it by drawing fro it several earlier theores Notice that all atrices in the following three exaples have the first entries property Using that x y) x y, 1 1 x+y we obtain Schur s Coloring Theore Using that 1 0 x 1 1 x+y 1 2 x x+2y y), 1 x+y 0 c cy we obtain the upgraded van der Waerden Theore Theore 733) We can also obtain the finite version of Hindan s Theore Exercise 414) For exaple, to have three natural nubers and all their finite) sus of the sae color, we use that x y z x y z x+y x+z y +z x+y +z Exercise 23 Prove, using the Monochroatic Iage Theore, that for all natural nubers,c 1 and c 2, for each finite coloring of N there are natural nubers a and d such that 1) 1) c 1 divides a 2) The nubers a,a+d,,a+d and c 2 d have the sae color Every atrix of the for a 1 0 a 2, 0 0 a n where: a 1,,a n are natural nubers, the nuber of entries in each vector a i a i a i is unliited, and the asterisk sybols ay be replaced by arbitrary vectors nonnegative integers, has the first entries property For the following reasons, it suffices to prove the Monochroatic Iage Theore for atrices of the for 1):

4 64 8 MONOCHROMATIC IMAGES 1) If a certain colun is the zero vector, then the corresponding entry in the vector v has no effect on the iage vector Av Thus, we ay assue that the atrix A has no zero coluns 2) If we perute the order of the rows of the atrix A, the entries of the iage vector Av are just peruted accordingly 3) By adding rows to the atrix while preserving the first entries property, the clai in the theore only becoes stronger: by the previous ite, we ay assue that the rows are added at the botto of the atrix, and then the old iage vector is an initial segent of the new one Thus, we ay assue that there are first entries in every colun of the given atrix To see ore clearly the connection of the following proof to the Central Sets Theore, it is recoended to read it first under the assuption that a i 1 for all i in the atrix presentation 1) Proof of the Monochroatic Iage Theore We ay assue that the atrix A is of the for 1) Let C be a central set We will find a vector v N n such that all entries of the iage vector Av are in C The proof is by induction on n In order to carry out the induction step ore easily, we will prove a stronger assertion: there are vectors v 1,v 2, N n such that, for each finite nonepty set F N, all entries of the vector Av F are in C n 1: In this case, the atrix is a vector with all entries identical As rows identical to previous rows do not contribute a new entry to the iage vector, we ay assue that each row appears exactly once In our case, this eans that the atrix is a scalar, a a 1, and we need to find scalars v 1,v 2, N such that, for each nonepty finite set F N, av F C Since the sets C and an belong to the sae idepotent ultrafilter, the set C an is an FS set Thus, there are eleents av 1,av 2, C an such that afsv 1,v 2,) FSav 1,av 2,) C an C n+1: Represent the atrix 1) in the block for ) a B 0 A By duplicating rows, if needed, we ay assue that the nuber of rows in the atrices A and B is equal, and denote it The atrix A is of the for 1), with n coluns By the inductive hypothesis, there are vectors v 1,v 2, N n such that, for each nonepty finite set F N, all entries of the vector Av F are in C For each b N and all nonepty finite sets F N, we have that ) ) ) ) a B b ab+bvf ab+bvf 0 A v F Av F Av F Consider the vectors u 1 : Bv 1,u 2 Bv 2, For each nonepty finite set F N, we have that u F n F u n n F Bv n B n F v n Bv F By the Central Sets Theore, there are nonepty finite sets of natural nubers F 1 < F 2 < and eleents ab 1,ab 2, C an such that {ab H +u FH : H [N] < } FSab 1 +u F1,ab 2 +u F2,) C an) C

5 Let 2 MONOCHROMATIC IMAGES 65 w 1 : For each nonepty finite set H N, Thus, ) a B w 0 A H b1 v F1 ),w 2 : w H bh b2 v FH ) v F2 ), ) ) ) a B bh abh +Bv FH 0 A v FH Av FH ) abh +u FH C Av 2 FH Thus, the vectors w 1,w 2, N n+1 are as required in the inductive clai There is an obstacle for generalizing the Monochroatic Iage Theore to atrices A with arbitrary integer entries: If all entries of the atrix A are negative and v N n, then all entries of the iage vector Av are negative Since we are given a coloring of N, we ust request that all entries of Av are natural nubers It turns out that this is the only obstacle Theore 24 Let A be a rational n atrix with the first entries property, such that all first entries of A are positive For each finite coloring of N, there is a vector v N n such that all entries of the iage vector Av have the sae color Proof Multiply the atrix A by a natural nuber a so that all entries of the atrix à : aa are integer, and all first entries are greater than 1 Let N be a natural nuber greater than all absolute values of eleents of the atrix à Let 1 N N 2 N n 1 1 N B : N 2 O 1 N 1 Consider the atrix ÃB All eleents of this atrix are in N {0}: For all i,j, let a i be the first entry in row i of the atrix à For appropriate d, we have that ÃB) ij a i N d + N d or 0) Since a i > 1 and the absolute value of each entry of à is saller than N, we have that N d N 1)N d 1 +N d ) N d 1 < a i N d, and thus the entry ÃB) ij is a positive integer The atrix ÃB has the first entries property: The product of each row of the atrix à with the atrix B is of the for 1 N N 2 N n 1 1 N 0,,0,a }{{} i,,, ) N 2 0,,0,a }{{} i,,, ), k O 1 N k 1

6 66 8 MONOCHROMATIC IMAGES and thus the first entries of the atrix ÃB are equal to the first entries of the atrix Ã, which has the first entries property By the Monochroatic Iage Theore, for each finite coloring of N there is a vector v of natural nubers such that the entries of the vector ÃBv aabv AaBv) have the sae color Each entry of the vector Bv, a su of products of natural nubers, is a natural nuber Since a is a natural nuber, all entries of the vector u : abv are natural We have seen that the entries of the vector Av are of the sae color Theore 25 Let A be a rational n atrix with the first entries property, such that all first entries of A are positive Assue, further, that the rows of A are distinct For each finite coloring of N, there is a vector v N n such that the entries of the iage vector Av are distinct, and have the sae color Proof We ay assue that every row i of A has a first entry a i For distinct rows r i and r j of A, we have that r i r j 0 Assue that the first entry of the latter vector is in position k Multiply this vector by a rational nuber q ij such that its first entry becoes a k, and add this new vector to the atrix A as a new row We obtain a new rational atrix à with the first entries property, with all first entries positive By Theore 24, there is a vector v N n such that the entries of the vector Ãv are natural and onochroatic In particular, the entries of Av are onochroatic, and for distinct rows r i and r j of A, we have that q ij r i r j )v N Thus, r i v r j v for all i,j, that is, the entries of Av are distinct The following result follows iediately fro the Monochroatic Iage Theore Corollary 26 For all natural nubers n, c and k, for each finite coloring of N, there are natural nubers x 1,,x n such that all eleents of all of the following sets are of the sae color where [ k,k] : { k, k +1,,k 1,k}): cx 1 +[ k,k]x 2 +x 3 +[ k,k]x 4 + +[ k,k]x n cx 2 +[ k,k]x 3 +[ k,k]x 4 + +[ k,k]x n cx 3 +[ k,k]x 4 + +[ k,k]x n cx n For exaple, in the first set there are 2k +1) n 1 eleents) A straightforward odification of the proof of the Monochroatic Iage Theore gives the following Theore 27 Let V be an infinite vector space over a field F Let A be an n over F with the first entries property For eachfinite coloringof V\{ 0}, there are vectors v 1,,v n V\{ 0} such that all vectors for i 1,, have the sae color Exercise 28 Prove Theore 27 a i1 v 1 + +a in v n,

7 3 COMMENTS FOR CHAPTER?? 67 3 Coents for Chapter 8 The forulation and proof of the Piecewise Syndetic Sets Theore Theore 723) should be considered a part of the proof of the Central Sets Theore Theore 13) The Central Sets Theore was first proved, using a different but equivalent notion of central set Theore 1927 in Hindan Strauss), in Hillel Furstenberg, Recurrence in Ergodic Theory and Cobinatorial Nuber Theory, Princeton University Press, 1981 The ethod used in the present proof of this theore is fro Hillel Furstenberg and Yitzhak Katznelson, Idepotents in copact seigroups and Rasey Theory, Israel Journal of Matheatics, 1989 Their proof was converted to the one included here by Vitaly Bergelson and Neil Hindan Nonetrizable topological dynaics and Rasey Theory, Transactions of the Aerican Matheatical Society, 1990) Corollary 26 is due to Walter Deuber, Partitionen and lineare Gleichungssystee, Matheatische Zeitschrift, 1973 Theore 27 is due to Vitaly Bergelson, Walter Deuber and Neil Hindan, Rado s Theore for finite fields, Colloquia Matheatica Societatis János Bolyai, 1992