SOME NEW EXAMPLES OF INFINITE IMAGE PARTITION REGULAR MATRICES

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1 #A5 INTEGERS 9 (29) SOME NEW EXAMPLES OF INFINITE IMAGE PARTITION REGULAR MATRIES Neil Hindman Department of Mathematics, Howard University, Washington, D nhindman@aolcom Dona Strauss Department of Pure Mathematics, University of Leeds, Leeds, UK dstrauss@hullacuk Received: /23/8, Accepted: /22/8, Published: /5/9 Abstract A matrix A, finite or infinite, is image partition regular (over the set N of positive integers) if and only if, whenever N is finitely colored, there is a vector ~x of the appropriate size with entries in N such that all entries of A~x are the same color (or monochromatic) A large number of characterizations of finite matrices that are image partition regular are known There is no known characterization of infinite image partition regular matrices, and the classes of infinite matrices that are known to be image partition regular have been rather limited; we present a list of those classes of which we are aware Extending an idea of Patra and Ghosh, we produce several new classes of infinite image partition regular matrices Introduction We shall be concerned in this paper with finite or infinite matrices whose entries are rational numbers Definition A matrix (finite or infinite) is admissible if and only if it has entries from Q, no row equal to ~, and finitely many nonzero entries in each row We will index the rows and columns of a matrix A by ordinals (here always countable) The first infinite ordinal is! = N [ {} Recall that an ordinal is the set of its predecessors, so for an ordinal, the statements x < and x 2 are synonymous If A is an matrix, B is a matrix, and O represents a matrix with all A O entries equal to of the appropriate size, then is an ( + ) ( + ) O B matrix

2 INTEGERS: 9 (29) 2 We will follow the custom of denoting the entries of a matrix by the lower case letter corresponding to the upper case letter which is the name of the matrix Definition 2 Let and be positive ordinals An matrix A is image partition regular (IPR) if and only if A is admissible and whenever N is finitely colored, there is some ~x 2 N such that the entries of A~x are monochromatic Many of the classical results of Ramsey Theory can be naturally viewed as the assertion that certain matrices are image partition regular For example, Schur s Theorem and the length 4 version of van der Waerden s theorem are the assertions that the A and 2 3 are IPR The first class of matrices shown to be image partition regular were the matrices that produced Deuber s (m, p, c) sets [2] Deuber used these sets to prove Rado s onjecture This was the assertion that if a large set was finitely colored, then one of the color classes must be large, where a large set was defined to be one that contains solutions to every (finite) partition regular system of homogeneous linear equations The matrices producing the (m, p, c) sets are special cases of first entries matrices Definition 3 Let, 2 N An matrix A is a first entries matrix if and only if A is admissible and for i, j <, if k = min{t < : a i,t 6= } = min{t < : a j,t 6= }, then a i,k = a j,k > Using Deuber s methods, it was easy to see that all first entries matrices are IPR The first characterizations of image partition regular matrices were found in [7] One of these was that a finite matrix A is image partition regular if and only if it has among its images all of the images of some first entries matrix B There are now at least a dozen known characterizations of IPR matrices see [3, Theorem 524] Of these, one that is important from the point of view of this paper is that a finite matrix A is IPR if and only if it has images contained in every central subset of N This has the important consequence that if A and B are finite A O IPR matrices, then is also IPR O B In order to define the notion of central that we just mentioned, we need to give a brief introduction to the algebraic structure of N And we will need to use that structure for most of our proofs in Section 3 In particular, we will use the structures of both ( N, +) and ( N, ), so we will write here about an arbitrary semigroup S Given a discrete space S, we take the Stone-Čech compactification of S of S to be the set of ultrafilters on S, the principal ultrafilters being identified with the A

3 INTEGERS: 9 (29) 3 points of S Given A S, let A = {p 2 S : A 2 p} Then {A : A S} is a basis for the open sets and a basis for the closed sets of S If (S, ) is a discrete semigroup, the operation extends to S so that ( S, ) becomes a right topological semigroup with S contained in its topological center That is, for any p 2 S, the function q 7! q p from S to itself is continuous and for any x 2 S, the function q 7! x q from S to itself is continuous Since ( S, ) is a compact Hausdor right topological semigroup, it has a smallest two sided ideal, K( S, ), which is the union of all of the minimal right ideals of ( S, ) and is also the union of all of the minimal left ideals of ( S, ) The intersection of any minimal left ideal with any minimal right ideal is a group and any two such groups are isomorphic Given p, q 2 S and A S, A 2 p q if and only if {x 2 S : x A 2 q} 2 p where x A = {y 2 S : x y 2 A} Any compact Hausdor right topological semigroup has an idempotent Given idempotents p and q in ( S, ), p apple q if and only if p = p q = q p An idempotent p is minimal with respect to this ordering if and only if p 2 K( S, ) See Part I of [3] for much more information about the structure of S One fact about ( Z, +) and ( Z, ) that we will repeatedly use is that if p, q 2 Z and a 2 Z, then by [3, Lemma 3], a(p + q) = ap + aq Definition 4 Let (S, ) be a discrete semigroup and let A S Then A is central if and only if A is a member of some minimal idempotent in ( S, ) In Section 2 of this paper we present, in historical order of their discovery, most of the known examples of infinite image partition regular matrices of which we are aware, concluding with a recent class of IPR matrices discovered by Patra and Ghosh [6] In Section 3 we present several new classes, many of which involve extensions of the class produced by Patra and Ghosh 2 Known lasses of Infinite IPR Matrices For each l 2 N \ {} let W l = 2 A l Then van der Waerden s Theorem [8] says that each W l is IPR (In the context of this paper this is immediate from the fact that each W l is a first entries matrix) It is an easy consequence of van der Waerden s Theorem that given any finite coloring of N, there is one color which contains arbitrarily long arithmetic progressions

4 INTEGERS: 9 (29) 4 onsequently, the infinite matrix W = W 2 O 3 O 3 O 4 W 3 O 4 O 5 O 5 W 4 is IPR, where O m is the m 2 matrix with all entries equal to The Finite Sums Matrix F is a matrix whose rows are all possible rows with entries from {, }, at least one, and only finitely many s (Of course there are many such matrices, di ering only by the order of their rows To be definite, one can specify a particular well ordering of the rows) By the Finite Sums Theorem [6, Theorem 3], F is image partition regular Given ~x 2 N!, the entries of F~x are all sums of the form P t2f x t for F 2 P f (!) (For a set X, we let P f (X) be the set of finite nonempty subsets of X) Fix an enumeration hb n i n= of the finite IPR matrices For each n, assume that B n is a u(n) v(n) matrix For each i 2 N, let ~ i be the vector with i entries Let D be an!! matrix with all rows of the form ~r _ ~r _ ~r _ 2 where each ~r i is either ~ v(i) or is a row of B i, and all but finitely many are ~ v(i) Again, if one wants to be definite, one can specify a well ordering of the rows of D) It is a consequence of the theorem of [3] and the fact that every finite IPR matrix has its images contained in the images of some (m, p, c) set that D is image partition ~x regular Given ~x 2 N! B, let ~x ~x A where each ~x i 2 N v(i) Then the entries of D~x are all sums of the form P t2f y t where F 2 P f (!) and for t 2 F, y t is an entry of B t ~x t As we mentioned in the introduction, we are proceeding in historical order; [8] was published in 927, [6] was published in 974, and [3] was published in 987 Our next example is a whole class of matrices, whose image partition regularity is a consequence of the Milliken-Taylor Theorem obtained independently by Milliken [5] and Taylor [7] and published in 975 and 976 So we might seem to be departing from our historical order However, it was not until the publication of [4] in 995 that it was noticed that the Milliken-Taylor Theorem guaranteed the existence of a class of infinite partition regular matrices Definition 2 Let k 2! and let ~a = ha, a,, i be a sequence in Z \ {} (a) c(~a) is the sequence obtained from ~a by deleting any term equal to its immediate predecessor (b) ~a is said to be compressed if and only if ~a = c(~a) A

5 INTEGERS: 9 (29) 5 Definition 22 Let ~r 2 Z! Then d(~r) is the sequence obtained from ~r by deleting all occurrences of ~ Definition 23 Let k 2! and let ~a = ha, a,, i be a compressed sequence in Z \ {} with > Then MT (~a) is a matrix consisting of all rows ~r with finitely many nonzero entries such that c d(~r) = ~a Again, if one wishes one can specify a particular well-ordering of the rows of MT (~a) In [4] the terms of ~a in the definition of MT (~a) were required to be positive, but the proof of image partition regularity from the Milliken-Taylor Theorem given in [4] can be taken almost verbatim As we noted in the introduction, any finite IPR matrix has images in any given central subset of N Also it is known that the matrices W, F, and D all have images in any central subset of N By way of contrast, we have the following theorem Theorem 24 Let k, m 2! and let ~a = ha, a,, i and ~ b = hb, b,, b m i be compressed sequences in Z \ {} with > and b m > If there does not exist a positive rational such that ~a = ~b, then there exist subsets A and B of N such that N = A [ B, there does not exist ~y 2 N! with MT ( ~ b)~y 2 A!, and there does not exist ~x 2 N! with MT (~a)~x 2 B! Proof This follows from [8, orollary 39] and the fact that the assumptions that > and b m > force the conclusion that there does not exist ~x 2 N! with all entries of MT (~a)~x less than or equal to or with all entries of MT ( ~ b)~x less than or equal to Definition 25 Let and be positive ordinals An matrix A is centrally image partition regular (centrally IPR) if and only if A is admissible and whenever is a central subset of N, there is some ~x 2 N such that A~x 2 We thus have that if k 2! and ~a = ha, a,, i is a compressed sequence in Z \ {} with >, then MT (~a) is centrally IPR if and only if k = Definition 26 Let A be an!! matrix Then A is a restricted triangular matrix if and only if all entries of A are from Z and there exist d 2 N and an increasing function j :!!! such that for all i 2!, () a i,j(i) 2 {, 2,, d}, (2) for all l > j(i), a i,l =, and (3) for all k > i and all t 2 {, 2,, d}, t,j(i) It was shown in [2] that any restricted triangular matrix is centrally IPR In particular, if each a i,j(i) =, then condition (3) is automatically satisfied The next class of matrices begins a process of extending given IPR matrices

6 INTEGERS: 9 (29) 6 Theorem 27 Let A be an!! centrally image partition regular matrix and let hb n i n= be a sequence in N Let b b B = b 2 A O B A O A A B is centrally image partition regular Proof [2, Theorem 47] Theorem 28 Let A be a finite IPR matrix and let B be an infinite IPR matrix A O Then is IPR O B Proof A combinatorial proof attributed to V Rödl is in [9, Lemma 23] A short proof using the algebra of N can be found in [6, Theorem 24] We have seen that Milliken-Taylor matrices determined by compressed sequences with more than one entry are not centrally IPR However, by choosing rows of such matrices with fixed nonzero sums, one does get centrally IPR matrices If m > the conclusion of Theorem 29 is trivial, since any central set will include a multiple of m and one can let ~x be a constant sequence (Theorem 37 in [9] asserts more than just that M is centrally IPR) Theorem 29 Let k 2 N, let ha, a,, i be a compressed sequence in Z \ {} with >, and let m 2 Z \ {} Let M be an admissible matrix such that () the compressed form of each row is ha, a 2,, i and (2) the sum of each row is m Then M is centrally IPR Proof [9, Theorem 37] In the following theorem, if P m i= a i > and P n i= b i >, then the conclusion of the theorem is trivial (although [4, Theorem 2] asserts more than that the matrix is IPR) But in fact P m i= a i is allowed to be or negative and P n i= b i is allowed to be negative Theorem 2 Let m, n 2! and let ~a = ha, a,, a m i and ~ b = hb, b,, b n i be sequences in Z \ {} Assume that a m >, b n >, and P n i= b i 6= Let A be an!! matrix with all rows whose nonzero entries are a, a,, a m in order and let B be an!! matrix with all rows whose nonzero entries are b, b,, b m in A O order Then is IPR O B

7 INTEGERS: 9 (29) 7 Proof [4, Theorem 2] The fact that the matrix in the next theorem is image partition regular established that it is possible to have an IPR matrix which does not have the property that whenever R is finitely colored, there must exist monochromatic images arbitrarily close to zero Theorem 2 The following matrix is IPR A Proof [, Theorem 6] In fact, the matrix of Theorem 2 is centrally IPR The proofs in [] used the notion of upper asymptotic density The properties of upper asymptotic density used are shared by Banach density and any central subset of N has positive Banach density (See [3, Definition 2 and Theorems 25 and 26]) Theorem 22 Let k 2! and let ~a = ha, a,, i be a sequence in Z \ {} Let I be the!! identity matrix, let A be an!! matrix with all rows whose I nonzero entries are a, a,, a m in order, and let M = If A () P k i= a i = and =, (2) P k i= a i =, or (3) a = a = = =, then M is IPR Proof [5, Lemma 27] It is conjectured in [5] that the conditions of Theorem 22 are also necessary, and this conjecture is proved in special cases We have seen that Milliken-Taylor matrices determined by compressed sequences of length greater than are not centrally IPR However, if the final terms are all equal to, a shift su ces, as seen in the following theorem

8 INTEGERS: 9 (29) 8 Theorem 23 Let m 2! and for each i 2 {,,, m}, let k(i) 2 N, let ~a i = ha i,, a i,,, a i,k(i) i be a compressed sequence in Z \ {} with a i,k(i) = Let and be the length! constant column vectors Then MT (~a ) MT (~a ) MT (~a m ) A F is centrally IPR Proof [, orollary 64] A generalization of Theorem 23 (minus the conclusion that the matrix is centrally IPR) is a consequence of Theorem 326 below In that generalization F is replaced by an arbitrary Milliken-Taylor matrix and the requirement that a i,k(i) = is replaced by the requirement that all of the final entries are the same positive number The most recent class of infinite matrices known by us to be IPR (before the results of the present paper) is given in the following theorem Theorem 24 Let u, v 2 N and let D be a u v IPR matrix Let A be an! v matrix all of whose rows are rows of D, let B be an!! centrally IPR matrix, A B O and let M be a Milliken-Taylor matrix Then is image partition O O M regular Proof [6, Theorem 29] 3 New lasses of Infinite IPR Matrices Our first result replaces the assumption in Theorem 24 that B is centrally IPR by the assumption that B is simply IPR As did the authors of [6], we utilize the following set introduced in [9] Definition 3 Let and be positive ordinals and let A be an admissible matrix Then (A) = {p 2 N : for every P 2 p, there exists ~x 2 N such that A~x 2 P } Lemma 32 Let A be an admissible matrix () The set (A) is compact and (A) 6= ; if and only if A is IPR

9 INTEGERS: 9 (29) 9 (2) If A is a finite IPR matrix, then (A) is a subsemigroup of ( N, +) (3) If A is IPR, then (A) is a left ideal of ( N, ) (4) If A is a finite IPR matrix, then (A) is a two sided ideal of ( N, ) (5) If A is a finite IPR matrix, then all minimal idempotents of ( N, +) are members of (A) Proof Statements () and (2) are [9, Lemma 25] Statements (3) and (4) are [6, Lemma 23] Statement (5) holds because finite IPR matrices are centrally IPR Lemma 33 Let u, v 2 N and let A be a u v IPR matrix Let r 2 be a minimal idempotent in ( N, +) () If r 2 (A), then r + p 2 (A) Z and let p (2) If r 2 (A), then r + p 2 (A) (3) If r 2 (A), then r + ( p) 2 (A) (4) If r 2 (A), then r + ( p) 2 (A) Proof [6, Lemma 28] In the next lemmas, the products of the form a b p are computed in ( Q d, ), where Q d is the set of rationals with the discrete topology Lemma 34 Let u, v 2 N and let A be a u v IPR matrix Let k 2! and let ha, a,, i be a sequence in Z \ {} with > Let p be a minimal idempotent in ( N, +) Then a p+a p+ + p 2 (A) and a p+ a p+ + p+p 2 (A) Proof By [3, Lemma 592], each a i p is a minimal idempotent in ( N, +) as is each ai p Since A is centrally IPR, each a i p is in (A) or (A) as is each a i p depending on whether a i is positive or negative The results then follow by repeated applications of Lemma 33 Lemma 35 Let k 2! and let ~a = ha, a,, i be a compressed sequence in Z \ {} with > Let p be an idempotent in ( N, +) Then () a p + a p + + p 2 MT (~a) and (2) a p + a p + + p + p 2 MT (~a) Proof onclusion () is an immediate consequence of [9, orollary 36] To verify conclusion (2), let q = a p + a p + + p + p and let Q 2 q By (), q 2 MT (~a) Since p is an idempotent, N 2 p Also Q 2 q Thus by [9, orollary 36], we may pick ~x 2 ( N)! such that MT (~a)~x 2 ( Q)! Then ~x 2 N! and MT (~a) ~x 2 Q!

10 INTEGERS: 9 (29) Lemma 36 Let u, v 2 N and let D be a u v IPR matrix Let and be positive ordinals, let A be an v matrix, all of whose rows are rows of D, and let B be an matrix which is IPR If r 2 (A) and p 2 (B), then r + p 2 A B Proof Let H 2 r + p Then X = {z 2 N : z + H 2 p} 2 r so pick ~w 2 N v such that ~y = A~w 2 X Then {y i : i < } is finite so P = T i< ( y i + H) 2 p Pick ~w ~z 2 N such that B~z 2 P Then A B 2 H ~z Theorem 37 Let u, v 2 N and let D be a u v IPR matrix Let and be positive ordinals, let A be an v matrix whose rows are the rows of D, let B be an matrix which is IPR, and let M be a Milliken-Taylor matrix Then A B O is IPR O O M Proof Note that (A) = (D) so (A) is an ideal of ( N, ) and a compact subsemigroup of ( N, +) Also (B) is a left ideal of ( N, ) so pick p 2 (A) \ (B) Pick a compressed sequence ha i i k i= in Z \ {} with > such that M = MT (~a) We need to show that A B \ (M) 6= ; Now p 2 (A) so (A) + p is a left ideal of (A) Pick a minimal left ideal L of (A) such that L ((A) + p) and pick an idempotent q 2 L Now all minimal idempotents of ( N, +) are in (A) so K( N, +) \ (A) 6= ; and therefore q is minimal in ( N, +) by [3, Theorem 65] Pick x 2 (A) such that q = x + p Let r = a q + a q + + q By Lemma 35, r 2 (M), so it su ces to show that r 2 A B By Lemma 34, r 2 (A) Note that q = x + p Also q + q = (q + q) = q so r = r + q = r + x + p We have that r + x 2 (A) and p 2 (B) By Lemma 36, r + x + p 2 A B Our next new result involves the notion of thick sets A subset A of N is thick if and only if it contains arbitrarily long integer intervals Equivalently, by [3, Theorem 448], A is thick if and only if there is a left ideal L of ( N, +) with L A Since any left ideal contains a minimal idempotent, it is immediate that any thick set is central, and thus contains images of any centrally IPR matrix Vitaly Bergelson has asked in a personal communication whether every thick set in N contains solutions to any partition regular system of linear homogeneous equations By [, Theorem 24] an a rmative answer to Bergelson s question would follow if one could show that any thick set contains images of any infinite IPR matrix That in turn would follow from the validity of the following conjecture onjecture 38 If and are positive ordinals, A is IPR matrix, B is a thick subset of N, r 2 N, and B = S r i= i, then there exist ~x 2 N and i 2 {, 2,, r} such that A~x 2 i

11 INTEGERS: 9 (29) Notice that the assertion in onjecture 38 that whenever B is finitely colored, there is a monochromatic image of A is stronger than the assertion that B contains an image of A onsider for example the following!! matrix: A = A The assertion that a set B is thick is precisely the assertion that there exists ~x 2 N! such that A 2 B ~x! But it is easy to color N in two colors so that neither color class contains arbitrarily long arithmetic progressions with a fixed increment Definition 39 M = {A : there exist positive ordinals and such that A is an admissible matrix and whenever T is a thick subset of N and T is finitely colored, there exists ~x 2 N such that the entries of A~x are monochromatic} onjecture 38 is the assertion that M is exactly the set of IPR matrices Lemma 3 Let and be positive ordinals, and let A be an admissible matrix The following statements are equivalent (a) A 2 M (b) For every minimal left ideal L of ( N, +), L \ (A) 6= ; (c) For every left ideal L of ( N, +), L \ (A) 6= ; Proof To see that (a) implies (b), assume that A 2 M, let L be a minimal left ideal of ( N, +), and suppose that L \ (A) = ; For each p 2 L, pick p 2 p such that there does not exist ~x 2 N such that A~x 2 p Then L S p2l p Minimal left ideals of ( N, +) are closed so pick F 2 P f (L) such that L S p2f p Let B = S p2f p Since L B, B is thick Since A 2 M, pick ~x 2 N and p 2 F such that A~x 2 p This is a contradiction Since every left ideal of ( N, +) contains a minimal left ideal, it is trivial that (b) implies (c) To see that (c) implies (a), let B be a thick subset of N and pick a left ideal L of ( N, +) such that L B Pick p 2 L \ (A) Let r 2 N and let B = S r i= i

12 INTEGERS: 9 (29) 2 Pick i 2 {, 2,, r} such that i 2 p Since p 2 (A), pick ~x 2 N A~x 2 i such that Lemma 3 Let A be an admissible matrix If A is either centrally IPR or a Milliken-Taylor matrix, then A 2 M Proof Assume first that A is centrally IPR Let L be a minimal left ideal of ( N, +) and pick a minimal idempotent p 2 L Then p 2 L \ (A) so Lemma 3 applies Now assume that we have k 2! and a compressed sequence ~a = ha, a,, i in Z \ {} with > such that A = MT (~a) Let L be a left ideal of N and let q be an idempotent in L By [3, Lemma 592], there is an idempotent p 2 N such that p = q It follows from Lemma 35 that a p + a p + + p 2 (A) \ L B O Theorem 32 Let B be an IPR matrix and let M 2 M Then O M is IPR Proof Assume that and are positive ordinals and that B is an IPR matrix Pick p 2 (B) and let L = N + p Pick q 2 L \ (M) and pick r 2 N such that q = r + p Then r 2 N = () so by Lemma 36, q 2 B The next theorem has a (possibly) stronger hypothesis and stronger conclusion than Theorem 32 (As we have noted, if onjecture 38 holds, then the content of Theorems 32 and 33 are identical) B O Theorem 33 Let B, M 2 M Then 2 M O M Proof Let L be a minimal left ideal of ( N, +) Pick q 2 L \ (M) We will show that q 2 B Pick p 2 L \ (B) Then q 2 L = N + p so pick r 2 N such that q = r + p By Lemma 36, q 2 B Since the hypothesis on B and the conclusion of Theorem 33 are the same, we can iterate the process orollary 34 Let m 2! and for i 2 {,,, m + }, let M i 2 M Then M O O O O O O O M O O O B 2 O O O O M m O A O O O O O O M m+ Proof Apply Theorem 33 m + times

13 INTEGERS: 9 (29) 3 Theorem 35 Let u, v 2 N, let D be a u v IPR matrix, and let A be an! v matrix, whose rows are the rows of D Let M and M be Milliken-Taylor matrices A M O Then 2 M O O M Proof Let k, m 2! and let ~a = ha, a,, i and ~ b = hb, b,, b m i be condensed sequences in Z \ {} with > and b m > such that M = MT (~a) and M = MT ( ~ b) Let L be a minimal left ideal of ( N, +) and let p be an idempotent in L Since D is finite and IPR, p 2 (D) Note that by [3, Exercise 438], L = N + p = Z + p Let q = a p + a p + + p + p and let q = b b m p + b b m p + + bm b m p + p Then q and q are in L By Lemma 34, q, q 2 (D) = (A) By Lemma 35, q 2 MT (~a) and q 2 MT ( ~ b) Let L = L \ (A) Then q, q 2 L By [3, Theorem 65] L is a minimal left ideal of (A) so L = (A) + p Pick r 2 (A) such that q = r + q By Lemma 36, q 2 A MT ( ~ b) so L \ A MT ( ~ b) \ MT (~a) 6= ; Theorem 36 Let u, v 2 N and let D be a u v IPR matrix Let and be positive ordinals and let A be an v matrix, whose rows are the rows of D Let B be an centrally IPR matrix, and let M be a Milliken-Taylor matrix Then A B O 2 M O O M Proof Let k 2! and let ~a = ha, a,, i be a condensed sequence in Z \ {} with > such that M = MT (~a) Let L be a minimal left ideal of ( N, +) and let p be an idempotent in L Since D is finite and IPR, p 2 (D) Note that by [3, Exercise 438], L = N + p = Z + p Let q = a p + a p + + p + p Then q 2 L By Lemma 34, q 2 (D) = (A) By Lemma 35, q 2 (M) Since p is an idempotent in L and L is minimal, q = q + p Since B is centrally IPR, p 2 (B) By Lemma 36, q 2 A B So L \ A B \ (M) 6= ; We now introduce two more classes of matrices As far as we know, it is possible that M, N and R are all equal to the set of all IPR matrices Definition 37 N = {M : M is an admissible matrix and for any finite IPR matrix D, (D) \ (M) \ K( N, +) 6= ;} Definition 38 R = {M : M is an admissible matrix and for any finite IPR matrix D and any left ideal L of (D), L \ (M) 6= ;} Lemma 39 R N Proof Let M 2 R and let D be a finite IPR matrix Pick a minimal left ideal L of (D) All minimal idempotents of ( N, +) are in (D) so by [3, Theorem

14 INTEGERS: 9 (29) 4 65] we may pick a minimal left ideal T of ( N, +) such that T \ (D) = L and in particular L K( N, +) Thus ; 6= L \ (M) = (D) \ (M) \ T (D) \ (M) \ K( N, +) Lemma 32 All centrally IPR matrices and all Milliken-Taylor matrices are in R (and thus in N ) Proof Let D be a finite IPR matrix and let L be a left ideal of (D) We may presume that L is minimal in (D) As above, pick a minimal left ideal T of ( N, +) such that T \ (D) = L Pick an idempotent p 2 T If M is centrally IPR, then we have that p 2 L \ (M) Assume we have k 2! and a condensed sequence ~a = ha, a,, i in Z \ {} with > such that M = MT (~a) Let q = a p + a p + + p + p Since T = Z + p, q 2 T and by Lemma 34, q 2 (D) By Lemma 35, q 2 (M) Therefore q 2 L \ (M) Theorem 32 Let u, v 2 N, and let D be a u v IPR matrix Let and be positive ordinals and let A be an v matrix, whose rows are the rows of D Let A B O B be an member of R and let M 2 N Then 2 N O O M A B O Proof We show first that (A) \ (M) \ K( N, +) So let O O M q 2 (A) \ (M) \ K( N, +) Let L = (A) + q Since B 2 R, pick r 2 L \ (B) Since q 2 K( N, +) \ (A) = K (A) (by [3, Theorem 65]), L is a minimal left ideal of (A), so L = L + r Pick s 2 L such that q = s + r By Lemma 36, A B O q 2 A B so q 2 O O M A B O Now let P = To see that P 2 N, let E be a finite IPR matrix O O M We need to show that (E) \ (P ) \ K( N, +) 6= ; A O The rows of are the rows of a finite IPR matrix so, since M 2 N, O E A O O E \ (M) \ K( N, +) 6= ; A O Also \ (M) \ K( N, +) (A) \ (M) \ K( N, +) (P ) O E A O Therefore ; 6= \(M)\K( N, +) (E)\(P )\K( N, +) O E As with orollary 34, we immediately get the following corollary

15 INTEGERS: 9 (29) 5 orollary 322 Let m 2! and for i 2 {,,, m}, let M i 2 R, let A i be a matrix with rows indexed by the same positive ordinal as the rows of M i, and assume that the rows of A i are all rows of a finite IPR matrix Let M m+ 2 N Then A M O O O O O O O A M O O O B 2 O O O O A m M m O A O O O O O O M m+ In orollary 322 we could let each M i be a Milliken-Taylor matrix If each one is determined by a compressed sequence with the same last term, we will get a stronger result in Theorem 326 (except that we only conclude that the resulting matrix is IPR rather than a member of N ) If F, H 2 P f (N) we write F < H to mean that max F < min H Definition 323 Let k 2! and let ~a = ha, a,, i be a compressed sequence in Z \ {} Let i 2! and let hx n i n=i be a sequence in N Then MT (~a, hx ni n=i ) = { P k t= a P t n2f t x n : for each i 2 {,,, k}, F i 2 P f (N), F < F < < F k, and min F i} Note that if ~x = hx n i n=i, then MT (~a, hx ni n=i ) is the set of entries of MT (~a)~x Lemma 324 Let k 2 N and let ha, a, i be a compressed sequence in Z\{} with > Let p be an idempotent in N, let q = a p + a p + + a n p, and let Q 2 q There exist D 2 p and a function P : S m= Nm! P(N) such that, if hx t i t= is a sequence in N and () x 2 D and (2) for each n 2!, x n+ 2 P (x, x,, x n ), then MT (~a, hx t i t=) D Proof For m 2 {,,, k} and n 2!, let F n,m = {hf, F,, F m i : for i 2 {,,, m}, F i 2 P f ({,,, n}) and if i < m, then F i < F i+ }, noting that if m > n, then F n,m = ; Let and for m 2 {,,, k } and x, x,, x m in N, let B(x, x,, x m ) be as in the proof of [3, Theorem 533] Let D =? Given n <! and hx i i n i= in N, let P (x, x,, x n ) =? \ T { P t2f x t +? : F 2 P f ({,,, n})} \ T k T P m= {B( t2f x t,, P t2f m x t ) : hf,, F m i 2 F n,m } \ T k T P m= { t2f m x t + B( P t2f x t,, P t2f m x t ) : hf,, F m i 2 F n,m }

16 INTEGERS: 9 (29) 6 (Here for notational convenience we are taking T ; = N) The proof of [3, Theorem 533] establishes that D and P are as required Lemma 325 For each i <!, let k(i) 2 N and let ~a i = ha i,j i k(i) j= be a compressed sequence in Z \ {} with a i,k(i) > Let p be an idempotent in ( N, +), for each i <!, let q i = a i, p+a i, p+ +a i,k(i) p, and let Q i 2 q i There exists an increasing sequence hx n i n= in N such that for each i <!, MT (~a i, hx n i n=i ) Q i Proof For i <! pick D i 2 p and P i : S m= Nm! P(N) as guaranteed by Lemma 324 Pick x 2 D, pick x 2 D \ P (x ), pick x 2 2 D 2 \ P (x, x ) \ P (x ), and in general, pick x n+ 2 D n+ \ T n i= P i(x i, x i+,, x n ) Theorem 326 Let d 2 N and let m 2! For i 2 {,,, m + }, let k(i) 2!, let ~a i = ha i,j i k(i) j= be a compressed sequence in Z \ {} with a i,k(i) = d, and let M i = MT (~a i ) For i 2 {,,, m}, let v i 2 N and let A i be an! v i matrix with all rows in some finite IPR matrix Then A O O O M O A O O M O O A 2 O M 2 N = B is O O O A m M m A O O O O M m+ Proof Pick a minimal idempotent p in ( N, +) By [3, Lemma 592] dp is a minimal idempotent so by [3, Exercise 438] Z + dp = N + dp For i 2 {,,, m + }, let q i = a i, p + a i, p + + a i,k(i) p By Lemma 34, q i 2 (A j ) for each j 2 {,,, m + } Given i 2 {,,, m}, q i and q m+ are in ( N + dp) \ (A i ) which is a minimal left ideal of (A i ) by [3, Theorem 65], so pick r i 2 (A i ) such that q m+ = r i + q i We will show that if B 2 q m+ there exists ~w with entries in N such that all entries of N ~w are in B So let B 2 q m+ be given Let i 2 {,,, m} and let D i = {x 2 N : x + B 2 q i } Then D i 2 r i and r i 2 (A i ) so pick ~z i 2 N vi such that ~y i = A i ~z i 2 D i! Now {y i,j : j <!} is finite so Q i = T j<! ( y i,j + B) 2 q i Let Q m+ = B By Lemma 325, pick a sequence hx n i n= in N such that for each i 2 {,,, m+ }, MT (~a i, hx n i n=i ) Q i For n 2!, let c n = x m+n+ ~z ~z ~w = ~z m A ~c

17 INTEGERS: 9 (29) 7 Then, given i 2 {,,, m}, O A i O M i ~w = A i ~z i + M i ~c 2 B! and O O O M m+ ~w = M m+ ~c 2 B! so ~w is as required References [] B Barber, N Hindman and I Leader, Partition regularity in the rationals, J omb Theory Ser A 2 (23), [2] W Deuber, Partitionen und lineare Gleichungssysteme, Math Z 33 (973), 9-23 [3] W Deuber and N Hindman, Partitions and sums of (m,p,c)-sets, J ombin Theory Ser A 45 (987), 3-32 [4] W Deuber, N Hindman, I Leader, and H Lefmann, Infinite partition regular matrices, ombinatorica 5 (995), [5] D Gunderson, N Hindman, and H Lefmann, Some partition theorems for infinite and finite matrices, Integers 4 (24), Article A2 [6] N Hindman, Finite sums from sequences within cells of a partition of N, J ombin Theory Ser A 7 (974), - [7] N Hindman and I Leader, Image partition regularity of matrices, ombin Probab omput 2 (993), [8] N Hindman, I Leader, and D Strauss, Separating Milliken-Taylor systems with negative entries, Proc Edinb Math Soc 46 (23), 45-6 [9] N Hindman, I Leader, and D Strauss, Infinite partition regular matrices solutions in central sets, Trans Amer Math Soc 355 (23), [] N Hindman, I Leader, and D Strauss, Extensions of infinite partition regular systems, Electron J ombin 22(2) (25), #P229 [] N Hindman, I Leader, and D Strauss, Duality for image and kernel partition regularity of infinite matrices, J omb 8 (27), [2] N Hindman and D Strauss, Infinite partition regular matrices, II extending the finite results, Topology Proc 25 (2), [3] N Hindman and D Strauss, Algebra in the Stone-Čech ompactification: Theory and Applications, 2nd edition, Walter de Gruyter & o, Berlin, 22 [4] I Leader and P Russell, onsistency for partition regular equations, Discrete Math 36 (26), [5] K Milliken, Ramsey s Theorem with sums or unions, J ombin Theory Ser A 8 (975), [6] S Patra and S Ghosh, oncerning partition regular matrices, Integers 7 (27), Article A63 [7] A Taylor, A canonical partition relation for finite subsets of!, J ombin Theory Ser A) 2 (976), [8] B van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch Wiskd 9 (927), 22-26