NEIL HINDMAN, IMRE LEADER, AND DONA STRAUSS

Transcription

1 TRNSCTIONS OF THE MERICN MTHEMTICL SOCIETY Volume 00, Number 0, Pages S (XX) This paper was published in Trans mer Math Soc 355 (2003), To the best of my knowledge, this is the final version as it was delivered to the publisher NH INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS NEIL HINDMN, IMRE LEDER, ND DON STRUSS bstract finite or infinite matrix is image partition regular provided that whenever N is finitely colored, there must be some x with entries from N such that all entries of x are in the same color class In contrast to the finite case, infinite image partition regular matrices seem very hard to analyze: they do not enjoy the closure and consistency properties of the finite case, and it is difficult to construct new ones from old In this paper we introduce the stonger notion of central image partition regularity, meaning that must have images in every central subset of N We describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices It turns out that the centrally image partition regular matrices are closed under some natural operations, and this allows us to give new examples of image partition regular matrices In particular, we are able to solve a vexing open problem by showing that whenever N is finitely colored, there must exist injective sequences x n n=0 and zn n=0 in N with all sums of the forms x n + x m and z n + 2z m with n < m in the same color class This is the first example of an image partition regular system whose regularity is not guaranteed by the Milliken-Taylor Theorem, or variants thereof 1 Introduction In 1933, R Rado [9] characterized those (finite) matrices with rational entries which are kernel partition regular, that is, those matrices with the property that whenever N is finitely colored, there exists some x with monochrome entries such that x = 0 He showed that is kernel partition regular if and only if satisfies a computable property called the columns condition (See [4] or [7] for a presentation and proof of Rado s Theorem) Sixty years later, several characterizations of (finite) image partition regular matrices were obtained [5] These are the matrices with the property that whenever N is finitely colored, there will be some x (with entries from N) such that the entries of x are monochrome Image partition regular matrices are of special interest because many of the classical theorems of Ramsey Theory are naturally stated as statements about image partition regular matrices For example, Schur s Theorem [10] and the length 4 version of van der Waerden s Theorem [12] amount to the 2000 Mathematics Subject Classification Primary 05D10; Secondary 2215, 54H13 The first author acknowledges support received from the National Science Foundation (US) via grant DMS c 2003 merican Mathematical Society

2 2 NEIL HINDMN, IMRE LEDER, ND DON STRUSS assertions that the matrices and are image partition regular In [6] additional characterizations of finite image partition regular matrices were obtained Some of these involve the notion of central sets Central sets were introduced by Furstenberg [3] and defined in terms of notions of topological dynamics These sets enjoy very strong combinatorial properties (See [3, Proposition 821] or [7, Chapter 14]) They have a nice characterization in terms of the algebraic structure of βn, the Stone-Čech compactification of N We shall present this characterization below, after introducing the necessary background information Let (S, +) be an infinite discrete semigroup We take the points of βs to be the ultrafilters on S, the principal ultrafilters being identified with the points of S Given a set S, = {p βs : p} The set { : S} is a basis for the open sets (as well as a basis for the closed sets) of βs There is a natural extension of the operation + of S to βs making βs a compact right topological semigroup with S contained in its topological center This says that for each p βs the function ρ p : βs βs is continuous and for each x S, the function λ x : βs βs is continuous, where ρ p (q) = q + p and λ x (q) = x + q See [7] for an elementary introduction to the semigroup βs ny compact Hausdorff right topological semigroup (T, +) has a smallest two sided ideal K(T ) which is the union of all of the minimal left ideals of T, each of which is closed [7, Theorem 28], and any compact right topological semigroup contains idempotents Since the minimal left ideals are themselves compact right topological semigroups, this says in particular that there are idempotents in the smallest ideal There is a partial ordering of the idempotents of T determined by p q if and only if p = p + q = q + p n idempotent p is minimal with respect to this order if and only if p K(T ) [7, Theorem 159] Such an idempotent is called simply minimal Definition 11 Let (S, +) be an infinite discrete semigroup set S is central if and only if there is some minimal idempotent p such that p See [7, Theorem 1927] for a proof of the equivalence of the definition above with the original dynamical definition We present in Theorem 12 a few known characterizations of finite image partition regular matrices (There and elsewhere we take N = {1, 2, 3, } and ω = {0, 1, 2, } lso, ω is the first infinite cardinal) We use the x notation throughout to represent both column and row vectors, expecting the reader to rely on the context to tell which is meant Theorem 12 Let u, v N and let be a u v matrix with entries from Q The following statements are equivalent (a) is image partition regular (b) For every central subset C of N, there exists x N v such that x C u (c) For every central subset C of N, { x N v : such that x C u } is central in N v

3 INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 3 (d) For each r Q v \{ 0} there exists b Q\{0} such that ( ) b r is image partition regular (e) For every central subset C of N, there exists x N v such that y = x C u, all entries of x are distinct, and for all i, j {1, 2,, u}, if rows i and j of are unequal, then y i y j Proof [6, Theorem 210] Infinite image partition regular matrices are also of significant interest For example, the Finite Sums Theorem (see [4, Theorem 315] or [7, Corollary 59]) is the assertion that the matrix , (whose rows are all vectors with entries from {0, 1} with only finitely many 1 s and not all 0 s) is image partition regular The fact, guaranteed by statement (d) of Theorem 12, that finite image partition regular matrices can be almost arbitrarily extended is very useful (and a property we would hope to maintain with infinite image partition regular matrices) nother important property was originally established by W Deuber in [1] in terms of (m, p, c)-sets That property is an immediate consequence of Theorem 12(b), namely that if and B are finite image partition regular matrices, then the matrix ( ) O O is also image partition regular The question of which infinite matrices are image partition regular seems to be significantly more complicated than the finite case For example, it was shown in [2] that( there are ) infinite image partition regular matrices and B such that the O matrix is not image partition regular (See Theorem 22 below) Not O B many examples of infinite image partition regular matrices are known The matrices which we shall describe now have been essentially the only known examples, all based on the Milliken-Taylor Theorem ([8], [11], or see [7, Section 181]) or variants thereof Definition 13 Let a be a (finite or infinite) sequence in Q with only finitely many nonzero entries Then c( a) is the finite sequence obtained by first deleting all occurrences of 0 and then deleting any term equal to its predecessor compressed sequence is a (necessarily finite) sequence a such that c( a) = a For example, if a = (1, 0, 0, 1, 1, 2, 0, 2, 0, 3, 0, 0, 0, 0, ), then c( a) = (1, 2, 3) In [2, Theorem 25] it was shown that if is an ω ω matrix with entries from B

4 4 NEIL HINDMN, IMRE LEDER, ND DON STRUSS ω (and only finitely many nonzero entries on each row) such that the compressed forms of all rows were equal, then is image partition regular, and it is a consequence of Corollary 36 below, that the same statement holds if the entries are allowed to come from Z, provided the rightmost nonzero entries are positive It has not even been known if even the simplest diagonal sums are image partition regular For example, let = and let B = It is an immediate consequence of Ramsey s Theorem that both and B are image partition regular by way ( of vectors ) x and z without repeated terms, but it has not O been known whether is image partition regular in the same fashion O B That is, it has not been known whether whenever N is finitely colored, there must exist injective sequences x n n=0 and z n n=0 in N with all sums of the forms x n + x m and z n + 2z m with n < m in the same color class In Section 2 we shall present some contrasts between finite and infinite partition regular matrices and, motivated by considerations presented there, introduce the notions of centrally image partition regular matrices (those having images in any central subset of N) and strongly centrally image partition regular matrices (those centrally image partition regular matrices for which distinct rows produce distinct entries of the image) It turns out that these classes do have some of the closure properties that one would like In particular, if and B are both centrally image ( partition ) regular or both strongly centrally image partition regular, then O has the corresponding property It is not true that matrices whose O B rows have the same compressed form are centrally image partition regular However, we show in Section 3 that matrices whose rows have the same compressed form and the same nonzero row sum are strongly centrally image partition regular, and we show that the corresponding statement for matrices with row sums of zero is not true We obtain as a consequence, in Corollary 38, new results about (ordinary) image partition regularity, such as the result mentioned in the abstract 2 Contrasts between Finite and Infinite Matrices We take as the principal good properties of finite partition regular matrices that we would like infinite partition regular matrices to share, the characterization of Theorem 12(d) and the result of [6, Corollary 211] That is, we would like it to be true that whenever is an infinite image partition regular matrix and r Q ω \{ 0} (with( only ) finitely many nonzero entries) there should be some b Q\{0} such b r that is image partition regular We would also like it to be the case that ( ) O whenever and B are infinite image partition regular matrices, so is ( ) O B O (If and B are ω ω matrices, then is an (ω + ω) (ω + ω) matrix O B We shall blissfully ignore this and similar distinctions throughout this paper)

5 INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 5 We shall see in Theorems 21 and 22 that neither of our main wishes with respect to infinite image partition regular matrices can be granted Theorem 21 Let be a matrix whose rows are all rows a Q ω with only finitely many nonzero entries such that c( a) = (1, 2) Let r = ( ) ( Then ) b r is image partition regular, but there does not exist b Q such that is image partition regular Proof s we remarked above, that is image partition regular follows from [2, Theorem 25] ( ) b r Suppose we have b Q such that is image partition regular, and notice that trivially b > 0 Write b = m n where m and n are relatively prime positive integers Case 1 m 1 Given x N, let α(x) = max{t ω : m t x} For i {0, 1}, let B i = {x N : α(x) i(mod 2)} Choose i {0, 1} and x N ω such that ( ) b r x B ω i Then the first row says that m n x 0 B i and so α( m n x 0) i(mod 2) lso α( m n x 0) = α(mx 0 ) since m and n are relatively prime Consequently, α(x 0 ) i(mod 2) Let s = α(x 0 ) and pick F P f (N) such that m s+1 Σ t F x t, where P f (N) is the set of finite nonempty subsets of N Pick G P f (N) such that max ( F ) < min G b r and m s+1 Σ t G 2x t Then x 0 + Σ t F x t + Σ t G 2x t is an entry of x while α(x 0 + Σ t F x t + Σ t G 2x t ) = s i(mod 2), a contradiction Case 2 m = 1 and n 1 Given x N, let α(x) = max{t ω : n t x} For i {0, 1}, let B i = {x N : α(x) i(mod 2) Choose i {0, 1} and x N ω such that ( b r ) x B ω i Then the first row says that 1 n x 0 B i and so α( 1 n x 0) i(mod 2) Consequently, α(x 0 ) i(mod 2) Choosing F and G as in Case 1, we again obtain a contradiction Case 3 b = 1 For x N, let ϕ(x) count the number of odd length blocks of 0 s interior to the binary expansion of x (More formally, if x = Σ t F 2 t and k = max F, then ϕ(x) = {t F \{k} : (min{s F : s > t} t) is even}) For j {0, 1, 2}, let B j = {x N : ϕ(x) j(mod 3)} Choose j {0, 1, 2} and x N ω such that ( r ) x Bj ω The first row of this equation says that ϕ(x 0 ) j(mod 3) Choose inductively a sequence H t t=1 in P f (N) such that for each t, max H t < min H t+1 and, if 2 s Σ i Ht x i, then 2 s+1 Σ i Ht+1 x i For each t N, let z t = Σ i Ht x i and choose F t P f (N) such that z t = Σ i Ft 2 i Notice that for each t, max F t < min F t+1 Pick s N such that 2 s > x 0 Choose u < k < l in N such that ϕ(x u ) ϕ(x k ) ϕ(x l )(mod 3), min F u min F k min F l (mod 2), and max F u max F k max F l (mod ( 2) ) Notice that 2z l + z k + z u and 2z l + z k + z u + x 0 are both entries of x and so 2z r l + z k + z u B j and 2z l + z k + z u + x 0 B j

6 6 NEIL HINDMN, IMRE LEDER, ND DON STRUSS Now ϕ(2z l +z k +z u ) = ϕ(z l )+ϕ(z k )+ϕ(z u )+1 (Exactly one odd block of 0 s is added in addition to those interior to the expansions of 2z l, z k, and z u It is between z k and z u if max F u min F u (mod 2) and it is between 2z l and z k if max F u min F u (mod 2)) Consequently, ϕ(2z l + z k + z u ) 1(mod 3) and thus j = 1 and thus ϕ(x 0 ) 1(mod 3) Now ϕ(2z l + z k + z u + x 0 ) = ϕ(2z l + z k + z u ) + ϕ(x 0 ) + 1 or ϕ(2z l + z k + z u + x 0 ) = ϕ(2z l + z k + z u ) + ϕ(x 0 ) depending on whether or not an odd block of 0 s is introduced between the expansions of z u and x 0 Consequently, ϕ(2z l + z k + z u + x 0 ) 0(mod 3) or ϕ(2z l + z k + z u + x 0 ) 2(mod 3) In either case, we have a contradiction Theorem 22 Let b be a compressed sequence with entries from N such that b (1) Let be a matrix whose rows are all rows a Q ω with only finitely many nonzero entries such that c( a) = b Let B be the finite sums matrix (a) The matrices and B are image partition regular (b) There is a subset C of N which is a member of every idempotent in βn (and is thus, in particular, central) such that for no x N ω does one have x C ω (c) The matrix ( ) O is not image partition regular O Proof Note that B is a matrix whose rows are all rows a Q ω with only finitely many nonzero entries such that c( a) = (1) That and B are image partition regular follows from [2, Theorem 25] By [2, Theorem 314], there exist C 1 and C 2 such that N = C 1 C 2 and there does not exist x N ω with x C1 ω and there does not exist y N ω with B y C2 ω and thus ( ) O O B is not image partition regular By [7, Theorem 58], C 2 is not a member of any idempotent in βn, and thus C 1 is a member of every idempotent in βn By way of contrast with Theorem 22(c), we see that infinite image partition regular matrices can be extended by finite ones (We are grateful to V Rödl for providing us with this result and its proof) Lemma 23 Let and B be finite and infinite image partition regular matrices respectively (with rational coefficients) Then ( ) O O B is image partition regular Proof ssume that is a u v matrix Let r N and let N be r-colored by ϕ Let (by compactness) n be large enough so that whenever {1, 2,, n} is r-colored, there exists x N v such that the entries of x are monochrome Now color N with r n colors via ψ, where ψ(x) = ψ(y) if and only if for all t {1, 2,, n}, ϕ(tx) = ϕ(ty) Pick y N ω such that the entries of B y are monochrome with respect to ψ Pick an entry a of B y and define γ : {1, 2,, n} {1, 2,, r} by γ(i) = ϕ(ia) Pick x N v such that the entries of x are monochrome with respect to γ Pick an entry i of x and let j = γ(i) B

7 INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 7 ( ) ( ) a x O Let z = We claim that for any row w of, ϕ( w z) = j To i y O B see this, first assume that w is a row of ( O ), so that w = s 0, where s is a row of Then w z = a( s x) and j = γ( s x) = ϕ ( a( s x) ) = ϕ( w z) Next assume that w is a row of ( O B ), so that w = 0 s where s is a row of B Then w z = i( s y) Now ψ( s y) = ψ(a) so ϕ ( i( s y) ) = ϕ(ia) = γ(i) = j Definition 24 Let be a finite or infinite matrix with entries from Q Then C() = {p βn : for every P p, there exists x with entries from N such that all the entries of x are in P } Lemma 25 Let be a matrix with entries from Q (a) The set C() is compact and C() if and only if is image partition regular (b) If is a finite image partition regular matrix, then C() is a subsemigroup of βn Proof (a) The fact that C() is compact is trivial as is the fact that is image partition regular if C() ssume that is image partition regular and let C = {B N : for every x with entries from N and the same number of entries as has columns, some entry of x is in B} We claim that C has the finite intersection property To see this, suppose instead that we have n N and B 1, B 2,, B n C with n i=1 B i = Then N = n i=1 (N\B i) and so there are some i {1, 2,, n} and some x with all entries of x in N\B i, contradicting the fact that B i C Since C has the finite intersection property, pick p βn with C p To see that p C(), let P p If there were no x with entries from N such that all entries of x are in P, we would have N\P C p, a contradiction (b) Let u, v N such that is a u v matrix We have that C() by (a) Let p, q C() and let B p + q Then C = {y N : y + B q} p so pick x N v such that y = x C u Then D = u i=1 ( y i + B) q so pick z N v such that z D u Then ( x + z) B u The set C() need not be a semigroup if is an infinite image partition regular matrix, as can be seen from Theorem 22 wherein the matrix is image partition regular, but C() contains no idempotents Corollary 26 Let F denote the set of finite image partition regular matrices over Q If B is an infinite partition regular matrix, then C(B) F C() Proof Let 1, 2,, n be a finite number of elements of F Let 1 O O O O 2 O O M = O O n O O O O B By Lemma 23, M is image partition regular and so C(M) C(B) n i=1 C( i) Our claim now follows from compactness

8 8 NEIL HINDMN, IMRE LEDER, ND DON STRUSS We saw in Theorem 22(b) that the characterization of Theorem 12(b) need not be valid for infinite image partition regular matrices This leads us to hope that perhaps matrices with this stronger property are better behaved Definition 27 Let be an ω ω matrix Then is centrally image partition regular if and only if whenever C is a central set in N, there exists x N ω such that x C ω It is trivial that whenever and B are centrally image partition regular matrices, then so is ( ) O O B Unfortunately, our other desired characteristic does not hold Proposition 28 There is an ω 2 centrally image partition ( regular matrix ) b b with entries from Q such that there does not exist b Q making image ( ) x0 partition regular In fact, there do not exist b Q and x = N x 2 with 1 ( b b Proof Let ) x N ω = ( ) a Given any central set C, pick a C and let x = Then all entries of x are a equal to a ( ) ( ) x0 b b Suppose that we have b Q and x = N x 2 with x N 1 ω Since b x 0 +b x 1 N, we have that x 0 x 1 Pick n N such that x 0 x 1 < n Then 1 n x 0 + n 1 n x 1 N while 1 n x 0 + n 1 n x 1 = x 1 + x0 x1 n But x 1 N and 0 < x0 x1 n < 1, a contradiction We now turn our attention to the characterization of Theorem 12(e) Considering the matrix of Proposition 28 which could only produce entries in N if the entries of x were equal, it seems reasonable to ask that the entries of x be required to be distinct However, we see now that this also would do no good (We also see that requiring the entries of the matrix to come from Z rather than Q is of no use either) Proposition 29 There is an ω 2 matrix with entries from Z with the property that whenever C is a central set in N, there exists x 0 x 1 such that all entries

9 ( x0 of INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 9 x 1 partition regular Proof Let ) are in C, but there does not exist b Q making = n + 1 n ( 2b b ) image and let {D 1, D 2 } be any partition of N such that neither D 1 nor D 2 contains infinite arithmetic progressions To see that has the first claimed property, let C be a central set and pick x 0 C Let x 1 = 2x 0 Then all entries of x are equal ( to) x 0 x0 Suppose that we have b Q, i {1, 2} and x = N x 2 with 1 ( 2b b ) x D ω i Then 2bx 0 bx 1 > 0 so 2x 0 x 1 Let a = x1 x 0 Then the n th entry of x is x 0 + (2 a) x 0 n This tells us in fact that a < 2 and consequently D i contains an infinite arithmetic progression, a contradiction In view of Proposition 29, we turn our attention to the other half of the characterization of Theorem 12(e) Definition 210 Let be an ω ω matrix Then is strongly centrally image partition regular if and only if whenever C is a central set in N, there exists x N ω such that y = x C ω and for all i, j ω, if rows i and j of are unequal, then y i y j There is a simple necessary condition for a matrix to be strongly centrally image partition regular Theorem 211 Let be a strongly centrally image partition regular matrix without repeated rows Then for each k N, {i : for all j k, a i,j = 0} is finite Proof Suppose instead that {i : for all j k, a i,j = 0} is infinite Then by discarding the other rows we may presume that is an ω k matrix Let D = { x N k : all entries of x are distinct} Enumerate D as x (n) n=1 Inductively choose distinct y n and z n in x (n), with {y n, z n } ( { y t : t {1, 2,, n 1} } { z t : t {1, 2,, n 1} } ) = if n > 1 Let C = {y n : n N} Then there is no x D with x C ω and no x D with x (N\C) ω We shall see in Theorem 39 that there is a strongly centrally( image ) partition b r regular matrix such that there is no b Q\{0} for which is image partition regular, where r = ( ) We shall see in Corollary 214 that the strongly centrally image partition regular matrices do maintain one of the properties that we desire First we shall have need

10 10 NEIL HINDMN, IMRE LEDER, ND DON STRUSS of the following algebraic result which we think is of interest in its own right (The information about the minimal left and minimal right ideals involved is not needed here But it is interesting algebraically and costs us little additional effort) Notice that since, by [7, Theorem 27] R L is a group, p is the unique idempotent of R L Theorem 212 Let p be a minimal idempotent in (βn, +) and let L and R be respectively the minimal left and minimal right ideals of (βn, +) with p L R Then for each C p, there are 2 c minimal idempotents in L C and 2 c minimal idempotents in R C Proof Let C p, let C = {x C : x + C p}, and notice that, by [7, Lemma 414], for each x C, x + C p For each m ω, let S m = 2 m N C { k + C : k C {1, 2,, m} } Let V = m N S m For every m N, 2 m N p (by [7, Lemma 66]) and so S m p Thus p V We show that V is a subsemigroup of βn, using [7, Theorem 420] So, let m N and let n S m It suffices to show that n + S m+n S m Let r S m+n Certainly n + r 2 m N Since n C {1, 2,, m + n}, n + r C Let k C {1, 2,, m} Then n k +C so k +n C {1, 2,, m+n} and thus r (k +n)+c so that n + r k + C as required Since p V we have by [7, Theorem 632] that V contains a copy of H = n=1 N2n (This copy is guaranteed to be both an algebraic and topological copy, via the same function, but here we only care about the fact that it is an algebraic copy) By [7, Theorem 69], (βn, +) has 2 c minimal left ideals, so there is a set W βn of idempotents such that W = 2 c and whenever u and v are distinct members of W, u + v u and v + u v Since by [7, Lemma 66], W H and V contains a copy of H, we have a set E V of idempotents such that E = 2 c and whenever u and v are distinct members of E, u + v u and v + u v By [7, Corollary 620], if u and v are distinct members of E, then (βn + u) (βn + v) =, so in particular (V + u) (V + v) = For each u E, pick an idempotent α u (p + V ) (V + u) with α u minimal in V (By [7, Corollary 26 and Theorem 27], p + V contains a minimal right ideal of V and V + u contains a minimal left ideal of V, and the intersection of a minimal right ideal of V with a minimal left ideal of V is a group Let α u be the identity of this group) Then {α u : u E} is a set of 2 c idempotents in p + V R, each minimal in V Since p V K(βN) we have that K(V ) = V K(βN) by [7, Theorem 165], so that each α u is minimal in βn Now we verify the assertion about L For each x N define supp(x) P f (ω) by x = t supp(x) 2t Inductively choose a sequence r n n=1 in N such that, for each n N, r n S n and max supp(r n ) < min supp(r n+1 ) Let X = {r n : n N} and note that X N = βn\n V Note also that, since S n N2 n, V H Define f : N ω by f(n) = min supp(n) and let f : βn βω be its continuous extension Notice that if x βn and q H, then f(x + q) = f(x) (To see this it suffices to show that the continuous functions f ρ q and f agree on N If n N and m = f(n)+1 then f λ n is constantly equal to f(n) on N2 m and so f(n+q) = f(n)) For each y f[x N ] one has {q V : f(q) = y} is a right ideal of V so pick an idempotent δ y {q V : f(q) = y} L which is minimal in V Each δ y

11 INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 11 is minimal in V, hence in βn, and if y z, then {q V : f(q) = y} {q V : f(q) = z} = It thus suffices to show that f[x N ] = 2 c To see this, let v be any nonprincipal ultrafilter on {f(r n ) : n N} (of which there are 2 c ) For v, let B() = {r n : f(r n ) } Then {B() : v} has the finite intersection property, so pick q βn with {B() : v} q Since v B() =, q N nd f(q) = v Corollary 213 Let C be a central set in N Then there exists a sequence C n n=1 of pairwise disjoint central sets in N with n=1 C n C Proof By Theorem 212, the set of minimal idempotents in C is infinite, hence contains an infinite strongly discrete subset (lternatively, there are two minimal idempotents in C so that C can be split into two central sets, C 1 and D 1 Then D 1 can be split into two central sets, C 2 and D 2, and so on) Corollary 214 For each n N, let n be a strongly centrally image partition regular matrix Then the matrix 1 O O O 2 O M = O O 3 is also strongly centrally image partition regular Proof Let C be a central set and choose by Corollary 213 a sequence C n n=1 of pairwise disjoint central sets in N with n=1 C n C For each n N choose x (n) N ω such that y (n) = n x (n) Cn ω and if rows i and j of n are unequal, y (n) i y (n) j Let x (1) x (2) z = Then all entries of M z are in C and entries from distinct rows are unequal Of course Corollary 214 remains valid if strongly centrally image partition regular is replaced by centrally image partition regular The same proof applies and one does not need to introduce the pairwise disjoint central sets, which were required to guarantee that the entries of M z from distinct rows were distinct 3 Constant Row Sums Notice that trivially, if is an ω ω matrix with entries from Q and there is some positive m Q such that each row of sums to m, then is centrally image partition regular (Given a central set C, simply pick d N such that dm C, which one can do because for each n N, Nn is a member of every idempotent by [7, Lemma 66] Then let x i = d for each i ω) We also saw in Theorem 22(b) that if b is a compressed sequence with entries from ω such that b (1) and is a matrix whose rows are all rows a Q ω with only finitely many nonzero entries such that c( a) = b, then is not centrally image partition regular We shall show in this section (in Theorem 37) that if is a matrix with entries from Z with finitely many nonzero entries in each row such that the compressed

12 12 NEIL HINDMN, IMRE LEDER, ND DON STRUSS form of all rows of are the same and all rows of have the same nonzero sum, then is strongly centrally image partition regular We shall also show (in Corollary 315) that a matrix with the same compressed form for all rows and zero sum for each row need not be centrally image partition regular We show now that any matrix with constant positive row sums and a limited number of patterns in any finite set of columns can be extended at will That the restriction on the number of patterns in a finite set of columns is needed can be seen by considering the matrix of Proposition 28 We shall see in Theorem 39 that we cannot extend the following theorem to the case in which m Z In Theorem 31 we talk of adding finitely many rows, rather than adding rows one at a time as in Theorem 12(d), because we cannot simply iterate the procedure (If the row sums of are all m and the sum of row r is not 0, one can multiply r by b so that its sum is m and iterate However, in the interesting cases, some or all of the added rows will sum to 0) Theorem 31 Let k N, let m Q with m > 0, and let be an ω ω matrix with entries from Q such that (i) the sum of each row of is m and (ii) for each l ω, { a i,0, a i,1,, a i,l : i ω} is finite Let r (1), r (2),, r (k) Q ω \{ 0} such that each r (i) has only finitely many nonzero entries Then there exist b 1, b 2,, b k Q\{0} such that is centrally image partition regular b 1 r (1) b 2 r (2) b k r (k) Proof Pick l N such that for every j {1, 2,, k} and every i l, r (j) i = 0 For each j {1, 2,, k}, let s (j) = r (j) 0, r(j) 1,, r(j) Enumerate { a i,0, a i,1,, a i,l 1 : i ω} as w (0), w (1),, w (u) For each i {0, 1,, u}, let d i = m l 1 the (u + 1) (l + 1) matrix with entries { (i) w e i,j = j if j {0, 1,, l 1} d i if j = l l j=0 w(i) j Let E be Then E has constant row sums, so is image partition regular By applying Theorem 12(d) u + 1 times, pick b 1, b 2,, b k Q\{0} such that the matrix b 1 s (1) b 2 s (2) H = b k s (k) E is image partition regular, hence, by Theorem 12(b), centrally image partition regular

13 INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 13 Let C be a central set and pick z 0, z 1,, z l N l+1 such that H z C u+1 For n {0, 1,, l 1}, let x n = z n For n {l, l + 1, l + 2, }, let x n = z l Then b 1 r (1) b 2 r (2) b k r (k) x C ω We see now that matrices with constant positive row sums need not be strongly centrally image partition regular Proposition 32 Let be an ω ω matrix whose rows are all those rows with entries from {0, 1, 2} with exactly one 1 and exactly one 2 (and no repeated rows) While is centrally image partition regular, it is not strongly centrally image partition regular In fact, there is a two cell partition {D 0, D 1 } of N such that there do not exist t {0, 1} and x N ω with y = x Dt ω and all entries of y distinct Proof For x N, let f(x) = max supp(x), where x = t supp(x) 2t (Then 2 f(x) x < 2 f(x)+1 ) For t {0, 1}, let D t = {x N : f(x) t (mod 2)} Suppose that we have t {0, 1} and x N ω with y = x Dt ω and all entries of y distinct Then also all entries of x are distinct (If i j, then 2x i + x j and x i + 2x j are distinct entries of y) Pick k < l < m in ω such that f(x k ) + 3 < f(x l ) < f(x m ) If we had i < j in {k, l, m} such that x j < 2 f(xj)+1 2x i, then we would have f(x j + 2x i ) = f(x j ) while f(2x j + x i ) = f(x j ) + 1, a contradiction Thus for i < j in {k, l, m} we have x j 2 f(xj)+1 2x i and f(x j + 2x i ) = f(x j ) + 1 In particular ( ) x l 2 f(xl)+1 2x k and ( ) x m 2 f(xm)+1 2x k If we had 2x m 2 f(xm)+2 x l, then we would have f(2x m + x l ) = f(x m ) + 2 f(x m + 2x l ) (mod 2), again a contradiction Thus, using ( ) and ( ), we have 2 f(xm)+2 4x k 2x m < 2 f(xm)+2 x l 2 f(xm)+2 2 f(x l)+1 + 2x k so that x l < 2 f(x l)+1 < 6x k < 8x k < 2 f(x k)+4 2 f(x l) x l, a contradiction Notice that any finite set of rows of the matrix in Proposition 32 form a finite image partition regular matrix (after throwing away columns with all zeroes) Thus by Theorem 12(e), given any central set C and any n N, there must exist x N ω such that the first n entries of x are distinct and lie in C It is a consequence of Theorem 37 below that if the matrix defined in Proposition 32 is modified by requiring that the occurrence of 1 come before the occurrence of 2 on each row (or vice versa) then the resulting matrix is strongly centrally image partition regular The proof of Theorem 37 uses a quite general construction which we present now Recall that if D is a discrete space, p βd, X is a topological space, y X, and f : D X, then p-lim s D f(s) = y if and only if for every neighborhood U of y in X, {s D : f(s) U} p See [7, Section 35] for a presentation of the basic properties of these limits

14 14 NEIL HINDMN, IMRE LEDER, ND DON STRUSS Given a sequence x k k=0 in a semigroup (S, ), we write F P ( x k k=0 ) = { k F x k : F P f (ω)} for the set of finite products of terms of the sequence If the operation is denoted by +, then we write F S( x k k=0 ) = { k F x k : F P f (ω)} Theorem 33 Let S and T be discrete spaces, let n N, and let f : S n T Define g : βs βt by g(p) = p- lim p- lim p- lim f(s s1 S 1, s 2,, s n ) s 2 S s n S Let p S = βs\s, let h : k=1 Sk p, let Q g(p), and let p (1) There exists a sequence x k k=0 in such that, whenever r 1 < r 2 < < r n, f(x r1, x r2,, x rn ) Q, and for each k N, x k h(x 0, x 1,, x k 1 ) (2) If (S, ) is a semigroup and p is an idempotent, then the sequence x k k=0 can be chosen so that, in addition to the above conclusions, F P ( x k k=0 ) and whenever F 1, F 2,, F n are finite nonempty subsets of ω with max F k < min F k+1 for every k {1, 2,, n 1}, one has f(π t F1 x t, Π t F2 x t,, Π t Fn x t ) Q Proof Let P = {z : p- lim p- lim p- lim f(z, s s2 S 2, s 3,, s n ) Q} and notice s 3 S s n S that P p For k {1, 2,, n 2} and y 1, y 2,, y k S, let P y1,y 2,,y k = {z S : p- lim p- lim p- lim f(y 1, y 2,, y k, z, s k+2,, s n ) Q} and for s k+2 S s k+3 S s n S y 1, y 2,, y n 1 S, let P y1,y 2,,y n 1 = {z S : f(y 1, y 2,, y n 1, z) Q} Notice that, if y P, then P y p and, if k {1, 2,, n 2} and y k+1 P y1,y 2,,y k, then P y1,y 2,,y k+1 p To establish the conclusions in (1), pick x 0 P, let m ω, and assume that we have chosen x 0, x 1,, x m P such that (i) for each k {1, 2,, m}, x k h(x 1, x 2,, x k 1 ), and (ii) for each k {1, 2,, m}, if k n and 0 r 1 < r 2 < < r k m, then x rk P xr1,x r2,,x rk 1 For k {1, 2,, n 1}, let Choose V k = {(x r1, x r2,, x rk ) : 0 r 1 < r 2 < < r k m} x m+1 P h(x 0, x 1,, x m ) min{n 1,m+1} k=1 (y 1,y 2,,y k ) V k P y1,y 2,,y k The induction hypotheses guarantee that the set on the right is a member of p, and is therefore nonempty Now, to verify the conclusions in (2), assume that (S, ) is a semigroup and p = p p For any B p, let B = {x B : x 1 B p}, where x 1 B = {y S : x y B} Then B p and by [7, Lemma 414], whenever x B, one has x 1 B p Choose x 0 P Let m N, and assume that we have chosen x 0, x 1,, x m such that (i) for each k {1, 2,, m}, x k h(x 1, x 2,, x k 1 ), (ii) if F {0, 1,, m}, then Π t F x t P, and (iii) if k {2, 3,, n} and F 1, F 2,, F k P f ({1, 2,, m}) with max F j < min F j+1 for each j {1, 2,, k 1}, then Π t Fk x t (P Πt F1 x t,π t F2 x t,,π t Fk 1 x t )

15 INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 15 For r {0, 1,, m}, let E r = { Π t F x t : = F {r, r + 1,, m} } and for k {1, 2,, n 1} and r {0, 1,, m}, let W k,r = { (Π t F1 x t, Π t F2 x t,, Π t Fk x t ) : F 1, F 2,, F k P f ({0, 1,, r}) and max F j < min F j+1 for every j {1, 2,, k 1} } Note that W k,r if and only if k m + 1 Hypothesis (ii) tells us that if y E 0 (equivalently if (y) W 1,m ), then y P so that y 1 P p and P y p Hypothesis (iii) tells us that whenever k {2, 3,, n 1} and (y 1, y 2,, y k ) W k,m, one has y k P y1,y 2,,y k 1 so that P y1,y 2,,y k p Hypothesis (iii) also tells us that if r {0, 1,, m 1}, k {1, 2,, n 1}, (y 1, y 2,, y k ) W k,r, and z E r+1, then z (P y1,y 2,,y k ) and thus Thus we may choose x m+1 z 1 (P y1,y 2,,y k ) p h(x 0, x 1,, x m ) P y E 0 y 1 P min{m+1,n 1} k=1 (y 1,y 2,,y k ) W k,m (P y1,y 2,,y k ) m 1 r=0 (y 1,y 2,,y k ) W k,r z E r+1 z 1 (P y1,y 2,,y k ) min{r+1,n 1} k=1 because this set is a member of p and is therefore nonempty Hypothesis (i) holds directly To verify hypothesis (ii), let = F {1, 2,, m + 1} with m + 1 F If F = {m + 1}, then Π t F x t = x m+1 P Otherwise, let G = F \{m + 1} and let y = Π t G x t Then y E 0 and so x m+1 y 1 P and thus Π t F x t P To verify hypothesis (iii), let k {2, 3,, n} and let F 1, F 2,, F k P f ({1, 2,, m+1}) with max F j < min F j+1 for each j {1, 2,, k 1} and with m+1 F k If F k = {m + 1}, then we have t F k x t = x m+1 (P Πt F1 x t,π t F2 x t,,π t Fk 1 x t ), so assume that F k {m + 1}, let G = F \{m + 1}, and let r = min G 1 Let z = Π t G x t and for j {1, 2,, k 1}, let y j = Π t Fj x t Then (y 1, y 2,, y k 1 ) W k 1,r and z E r+1 so that Π t Fk x t = z x m+1 (P y1,y 2,,y k 1 ) as required s we have previously remarked, in [2, Theorem 25] it was shown that if is an ω ω matrix with entries from ω (and only finitely many nonzero entries on each row) such that the compressed forms of all rows were equal, then is image partition regular We extend this result now to allow negative entries Notice that in the following lemma and beyond, if p βn and a Z, the product a p refers to multiplication in the semigroup (βz, ) In particular, if a N, a p is not the sum of p with itself a times (If, as here, p = p + p, that sum is just p) In our remaining results we shall be assuming that the entries of our matrices come from Z rather than Q This is a convenient, but not essential, restriction because we shall also be assuming that the compressed forms of all rows are equal, so that any such matrix with rational entries can be turned into one with integer entries by multiplying by a constant Since multiplying a central set by a constant produces another central set [6, Lemma 21], the corresponding results hold Lemma 34 Let a 1, a 2,, a k be a sequence in Z\{0} Let m = max { a j : j {1, 2,, k} }, let p be an idempotent in βn, and let q = a 1 p + a 2 p + + a k p βz If q and P p, then there is a sequence x n n=0 in N such that

16 16 NEIL HINDMN, IMRE LEDER, ND DON STRUSS x n+1 > 2m n t=0 x t for each n ω, F S( x n n=0) P and { k t=1 a t n F t x n : F 1, F 2,, F k P f (N) and max F t < min F t+1 for t {1, 2,, k 1} } Proof Define h : n=1 Nn p and f : N k Z by h(x 0, x 1,, x n 1 ) = {z N : z > 2m n 1 t=1 x t} and f(y 1, y 2,, y k ) = a 1 y 1 + a 2 y a k y k Then p- lim p- lim p- lim f(y y1 N 1, y 2,, y k ) = a 1 p + a 2 p + + a k p y 2 N y k N so Theorem 33 applied to the semigroup (N, +) yields the desired conclusion We observe now that the sequence produced in Lemma 34 satisfies a strong uniqueness of sums property Lemma 35 Let a 1, a 2,, a k be a compressed sequence in Z\{0}, let m = max { a j : j {1, 2,, k} }, and let x n n=0 be a sequence in N such that x n+1 > 2m n t=0 x t for each n ω Then whenever F 1, F 2,, F k, G 1, G 2,, G k P f (N), max F t < min F t+1 and max G t < min G t+1 for t {1, 2,, k 1}, and k t=1 a t n F t x n = k t=1 a t n G t x n, one must have F t = G t for each t {1, 2,, k} Proof Suppose instead that we have some F 1, F 2,, F k, G 1, G 2,, G k P f (N) such that max F t < min F t+1 and max G t < min G t+1 for t {1, 2,, k 1}, k t=1 a t n F t x n = k t=1 a t n G t x n, but F t G t for some t {1, 2,, k} Pick the largest l {1, 2,, k} such that F l G l Then l t=1 a t n F t x n = l t=1 a t n G t x n Let r = max(f l G l ) By subtracting any larger terms from both sides of the last equation, we may presume that r = max(f l G l ) ssume without loss of generality that r F l We may also assume that a l > 0 (Otherwise, multiply both sides by 1) Then l t=1 a t n F t x n a r x r r 1 n=1 mx n > r 1 n=1 mx n l t=1 a t n G t x n, a contradiction Corollary 36 Let a 1, a 2,, a k be a compressed sequence in Z\{0} with a k > 0 Let M be a matrix, with finitely many nonzero entries in each row, such that the compressed form of each row is a 1, a 2,, a k Then M is image partition regular Indeed, given any idempotent p βn and any function h : t N Nt p, if q = a 1 p + a 2 p + + a k p, then q n=1 nn and for any q and any P p, there is an increasing sequence x N ω such that F S( x n n=0) P, x n+1 h(x i1, x i2,, x im ) for every n N and every choice of 0 i 1 < i 2 < i m n, M x ω, and entries which correspond to distinct rows are distinct Proof Let p be an idempotent in βn and let q = a 1 p + a 2 p + + a k p Let T = n=1 nz We show first that q T βn = n=1 nn By [7, Lemma 66] p T Since, by [7, Theorems 215 and 217] T is an ideal of (βz, ), we have that each a i p T By [7, Exercise 232] T is a subsemigroup of (βz, +), and so q T Since a k > 0, a k p N and so q βn by [7, Exercise 435] Let q and P p and choose a sequence x n n=0 as guaranteed by Lemma 34 for and P If x = x n n=0, then all the entries of M x are in By Lemma 35, entries which correspond to distinct rows are distinct

17 INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 17 Notice that if x is as guaranteed by Corollary 36 and z is a subsequence of x, then also M z ω, and entries which correspond to distinct rows are distinct, because any entry of M z is also an entry of M x We remark that the possibility of choosing x n+1 h(x i1, x i2,, x im ), guaranteed by Corollary 36, yields ( non-trivial ) information about simple finite matrices 1 2 For example, let = Then is image partition regular Color N 0 1 according to the parity of max ( supp(n) ) (Recall that supp(n) P f (ω) is defined for n N by n = i supp(n) 2i ) We cannot choose x N 2 such that min ( supp(x 2 ) ) > max ( supp(x 1 ) ) ( ) and the entries of x are monochrome However, if B =, then, defining h(x) = {y N : min ( supp(y) ) > max ( supp(x) )} and h(x, y) = {z N : min ( supp(z) ) > max ( supp(y) )}, Corollary 36 guarantees that, in any finite coloring of N, there exists x N 3 such that min ( supp(x i+1 ) ) > max ( supp(x i ) ) if i {1, 2} and the entries of B x are monochrome Theorem 37 Let k N, let a 1, a 2,, a k be a compressed sequence in Z\{0} with a k > 0, and let m Z\{0} Let M be a matrix, with finitely many nonzero entries in each row, such that (i) the compressed form of each row is a 1, a 2,, a k and (ii) the sum of each row is m Then M is strongly centrally image partition regular Proof Let L = {q βn : for every q and every k N, there exists x {k + 1, k + 2, k + 3, } ω such that M x ω and entries corresponding to distinct rows of M are distinct} It is obvious that L is closed By Corollary 36, L m βn We claim that L m βn is a left ideal of m βn To this end, let q L m βn It suffices to show that m N + q L m βn We have immediately that m N+q m βn To see that m N+q L, let n N, let m n+q, and let k N Pick x {k+n+1, k+n+2, k+n+3, } ω such that y = M x ( m n+) ω and the entries of y corresponding to distinct rows are distinct Let s = m m and for each i ω let z i = sn + x i Then each z i > k Let w = M z Then for each j ω, w j = m n + y j and entries of w corresponding to distinct rows of M are distinct Let C be a central set in N and pick a minimal idempotent r in βn such that C r Let p = 1 a k r (multiplication in βq d, where Q d is the set Q with the discrete topology) It is routine to check that p βn, p + p = p, and a k p = r Let q = a 1 p + a 2 p + + a k p By Corollary 36 q L m βn Since a k p = r, q βz + r = βz + r + r βn + r (the last inclusion by [7, Exercise 435]) Since q m βn and r m βn, we have that q m βn + r Now r m βn K(βN), so by [7, Theorem 165], r K( m βn) Thus, by [7, Theorem 29], m βn + r is a minimal left ideal of m βn Since q (L m βn) ( m βn + r) we have m βn + r L and so r = r + r m βn + r L

18 18 NEIL HINDMN, IMRE LEDER, ND DON STRUSS Observe that, if all terms of a come from N, then the number of entries of any row of a matrix M as in Theorem 37 is limited However, this need not be the case if one or more entries are negative We shall see in Theorem 314 that the requirement that m 0 cannot be eliminated (We saw in Theorem 22(b) that the restriction on the row sums cannot simply be omitted) Consider the following consequence of Theorem 37 Let M = , a matrix whose rows have somewhere a single 2 followed somewhere by a single 1 Even though matrices with constant positive row sums are trivially centrally image partition regular via a constant vector x, it does not seem to be trivial that the matrix M is even centrally image partition regular, while Theorem 37 tells us that it is in fact strongly centrally image partition regular s we have earlier promised, we obtain as a consequence of our consideration of strongly centrally image partition regular matrices, new results about ordinary image partition regular matrices (The last result mentioned in the abstract is the instance of Corollary 38 for which a = 1, b = 1, 2, m = 2, and n = 3) Corollary 38 Let k, s N, let a = a 1, a 2,, a k and b = b 1, b 2,, b s be compressed sequences in Z\{0} with a k > 0 and b s > 0, let m, n Z\{0}, and assume that there exist r, t Z ω such that c( r) = a, c( t) = b, j=0 r j = m, and j=0 t j = n Let q N and let N = q i=0 C i Then there exist i {1, 2,, q} and injective sequences x n n=0 and z n n=0 in N such that, for every r and g in Z ω with only finitely many nonzero entries, if c( r) = a, c( g) = b, j=0 r j = m, and j=0 g j = n, then j=0 r j x j C i, j=0 g j z j C i, and j=0 r j x j j=0 g j z j Proof Let M and N be matrices with finitely many nonzero entries in each row such that (a) the compressed form of each row of M is a; (b) the sum of each row of M is m; (c) all rows with compressed form a and sum m occur in M; (d) the compressed form of each row of N is b; (e) the sum of each row of N is n; and (f) all rows with compressed form b and sum n occur ( in N ) M O Then by Theorem 37 and Corollary 214, the matrix is strongly centrally image partition regular, so pick i ({1, 2,, q} ) ( such ) that C i is central and O N M O x pick x and z in N ω such that all entries of are in C O N z i and entries ( ) M O corresponding to distinct rows of are distinct One has then immediately that for any row r of M and any row g of N, O N u=0 r u x u u=0 g u z u To see, for example, that x is injective let j < l in ω and pick t 0, t 1,, t v Z\{0} such that c( t 0, t 1,, t v ) = a and v u=0 t u = m Define r and g in Z ω by r j = g l = t 0,

19 INFINITE PRTITION REGULR MTRICES SOLUTIONS IN CENTRL SETS 19 r l+u = g l+u = t u for u {1, 2,, v} (if any), and all other entries of r and g are equal to 0 Then r and g are distinct rows of m, so u=0 r u x u u=0 g u x u and thus x j x l We see now that strongly centrally image partition regular matrices need not have one of our desired properties, namely the analogue of Theorem 12(d) Proposition 39 Let r = ( ) There is an ω ω matrix M with entries from Z such that the compressed form of each row is ( 2, 1) and the sum of each row is 1 (so M is strongly( centrally ) image partition regular), but there is b r no b Q\{0} such that the matrix is image partition regular M Proof Let M = ( ) b r as described above Suppose that is image partition regular for some M b Q\{0} This implies that b > 0 Let b = k l, where k, l N Let r be a prime number satisfying r > k + 2l Each n N can be expressed uniquely as n = i ω a ir i, where each i {0, 1, 2,, r 1} Let supp r (n) = {i ω : a i 0} and m(n) = min ( supp r (n) ) We define f : N {1, 2,, r 1} by f(n) = a m(n) ( ) b r Choose x N ω and c {1, 2,, r 1} such that x C M ω, where C = f 1 [{c}] Then bx 0 C and 2x m + x n C whenever n, m ω with m < n Let d = f(x 0 ) We claim that m(x n ) m(x 0 ) for every n N To see this, observe that m(x n ) > m(x 0 ) implies that f( 2x 0 + x n ) = 2d in Z r Since f(bx 0 ) = k l d in Z r, 2d = k l d in Z r and k + 2l = 0 in Z r a contradiction Thus, by the pigeon hole principle, there exists s, t N such that t > s, m(x t ) = m(x s ) and f(x t ) = f(x s ) = e, say Then f( 2x s + x t ) = e = c in Z r We consider two cases: (i) If m(x s ) = m(2x 0 ), then f( 2x 0 + x s ) = 2d + e = c in Z r, and so d = c in Z r We also have k l d = c in Z r and thus k + l = 0 in Z r a contradiction (ii) If m(x s ) < m(2x 0 ), then f( 2x 0 + x s ) = e So e = e in Z r again a contradiction The fact that the row sums in Proposition ( ) 39 were negative is needed to prevent b r the image partition regularity of, as we shall see now We do not know M whether matrices with fixed compressed form and fix positive row sum can necessarily be arbitrarily extended to strongly centrally image partition regular matrices Corollary 310 Let k, l, m N, let a 1, a 2,, a k be a compressed sequence in Z\{0}, and let be an ω ω matrix such that (i) the compressed form of each row is a and

20 20 NEIL HINDMN, IMRE LEDER, ND DON STRUSS (ii) the sum of each row of is m Let r (1), r (2),, r (l) Q ω \{ 0} such that each r (i) has only finitely many nonzero entries Then there exist b 1, b 2,, b k Q\{0} such that is centrally image partition regular b 1 r (1) b 2 r (2) b k r (l) Proof The matrix satisfies the hypotheses of Theorem 31 We note now that the requirement in Theorem 37 that a k > 0 is essential Theorem 311 Let k N, let a 1, a 2,, a k be a compressed sequence in Z\{0} with a k < 0, and let m Z be any number expressible in the form k t=1 α ta t where each α t N Let M be a matrix (with finitely many nonzero entries in each row) such that (i) the compressed form of each row is a 1, a 2,, a k, (ii) the sum of each row is m, and (iii) any row with compressed form a 1, a 2,, a k and row sum m occurs in M Then M is not strongly centrally image partition regular, and if m 0, M is not centrally image partition regular Proof If M x N ω and either the entries of M x corresponding to distinct rows are distinct or m 0, there must be infinitely many distinct entries in x This in turn forces some entries of M x to be negative Recall that we have defined the (binary) support of a positive integer n by n = t supp(n) 2t We introduce now some special notation needed for the proof of Theorem 314 Definition 312 Let n N The set H is a block of n if and only if H is a maximal set of consecutive integers contained in supp(n) lso b(n) is the number of blocks of n Thus if, written in binary, n = , then the blocks of n are {0, 1}, {3}, {5, 6, 7}, and {10}, and so b(n) = 4 Lemma 313 Let B = {n N : b(n) 0 (mod 2)} (a) For n N, let h(n) {0, 1} be such that h(n) b(n) (mod 2) Let h : βn Z 2 be the continuous extension of h Then the restriction of h to n=1 cl (N2n ) is a homomorphism (b) If q + q = q βn, then B q (c) If p βn and for each n N, N2 n p, then B / p + p Proof (a) This is an immediate consequence of [7, Theorem 421] (b) By [7, Lemma 66] q n=1 cl (N2n ), so this follows from (a) (c) Let D 0 = {x N : min ( supp(x) ) + 1 / supp(x)} and let D 1 = {x N : min ( supp(x) ) + 1 supp(x)}