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New York Journal of Mathematics

New York J. Math.17_{(2011) 149–161.}

Image partition regularity of matrices

near 0 with real entries

Dibyendu De

and

Ram Krishna Paul

Abstract. We prove that many of the known results regarding image partition regularity and image partition regularity near zero for finite and infinite matrices with rational or integer entries have natural ana-logues for matrices with real entries over the reals, extending work by N. Hindman.

Contents

1. Introduction 149

2. Finite matrices 151

3. Infinite matrices 154

References 161

1. Introduction

One of the famous classical results of Ramsey Theory is van der Waerden’s Theorem [12], which states that if r, l ∈ N and N = Sr

i=1Ai, there exist i∈ {1,2, . . . , r}and a, d∈Nsuch that{a, a+d, . . . , a+ld} ⊆Ai. In other

words it says that the entries of M~x, where

M =



     

1 0 1 1 1 2

.. . ... 1 l



     

and

~x=

a d

∈N2,

are monochromatic.

This suggests the following definition:

Received January 23, 2011.

2000Mathematics Subject Classification. 05D10, 22A15.

Key words and phrases. Ramsey Theory, Image partition regularity of matrices, Central set near zero, infinite matrices.

The second named author acknowledges support, received from the CSIR, India via Ph.D. grant.

ISSN 1076-9803/2011

Definition 1.1. Letu, v∈Nand letM be au×vmatrix with entries from

Q. The matrixM is image partition regular over Nif and only if whenever

r ∈ N and N =Sr

i=1Ci, there existi∈ {1,2, . . . , r} and ~x ∈Nv such that M~x∈C_iu.

Characterizations of those matrices with entries from Q that are image partition regular overNwere obtained in [8]. The underlying fact behind the characterization of finite image partition regular matrices is that wheneverN

is finitely colored one color should contain acentral set[5, Theorem 8.8] and hence satisfies the conclusion of the Central Sets Theorem [5, Proposition 8.21]. Algebraic definition of central set is also available in [10, Definition 4.42]. After the algebraic definition of central sets it becomes immediate that any set containing a central set is central. A natural extension of this notion tocentral set near zerowas introduced in [7, Definition 4.1]. For any dense subsemigroup of (R,+) it was observed that there are some subsets of R living near zero which also satisfy some version of the Central Sets Theorem. Motivated by this observation, the notion of Image Partition Regularity near zero was introduced in [1].

Definition 1.2. Let S be a subsemigroup of (R,+) with 0 ∈ cℓS, let

u, v ∈ N, and let M be a u×v matrix with entries from Q. Then M is

image partition regular over S near zero (abbreviated IPR/S0) if and only

if, whenever S\ {0} is finitely colored and δ > 0, there exists ~x∈Sv _such

that the entries of M~xare monochromatic and lie in the interval (−δ, δ).

The matrices which are image partition regular near zero are very in-teresting in themselves. Image partition regularity near zero over various subsemigroups of (R,+) containing 0 in the closure has been investigated extensively in [1]. One of the main objectives was that for finite matrices with rational entriesimage partition regularity andimage partition regular-ity near zero are equivalent. But when we turn our attention to admissible infinite matrices these two notions are not equivalent. Here, an admissible infinite matrix is anω×ω matrix, each row of which contains only finitely many nonzero elements, whereω is the first infinite ordinal number.

The characterization of finite matrices with real entries that are image partition regular over R+ were obtained in [6]. The definitions of image partition regularity andimage partition regularity near zero for finite matri-ces with entries from R have natural generalizations for admissible infinite matrices.

In this paper we shall investigate matrices with real entries that areimage partition regular near zero overR+. In Section 2 we shall characterize the finite matrices with real entries that are image partition regular near zero

Definition 1.3. Letu, v∈NandM be au×v matrix with entries fromR. ThenM isimage partition regular overR+near zero(abbreviated IPR/R+₀) if and only if, whenever R+ is finitely colored and δ > 0, there exists~x ∈ (R+)v such that the entries ofM~xare monochromatic and lie in the interval (0, δ).

In Section 3 we shall prove that the Milliken–Taylor matrices and in-sertion matrices (which we shall define in Definition 3.3 and Definition 3.7 respectively) with real entries areimage partition regular near zero overR+. We complete this section with some necessary facts regarding the com-pact right topological semigroup (βR+_d,+), where R+_d means R+ with the discrete topology. For details regarding the algebraic structure of βSd for

an arbitrary dense subsemigroupS of R+ one can see [7].

Definition 1.4. Define 0+(R+) ={p∈βR_d+ : (∀ǫ >0)((0, ǫ)∈p)}.

It is proved in [7, Lemma 2.5] that 0+_(R+_{) is a compact right topological}

subsemigroup of (βR+_d,+) and therefore contains minimal idempotents of 0+(R+).

Definition 1.5. A setC ⊆R+ is central near zero if and only if there is a minimal idempotentp of 0+(R+) containingC.

For our purpose we state the following version of Central Sets Theorem which follows directly from [1, Corollary 4.7].

Theorem 1.6. Let T be the set of all those sequences hyni∞n=1 in R such that limn→∞yn = 0 and let C be central near zero in R+. Let F ∈ Pf(T).

There exist a sequence hani∞n=1 in R+ such that

P∞

n=1an converges and a

sequence hHni∞n=1 in Pf(N) such that for each n∈N, maxHn <minHn+1 and for each L∈ Pf(N) and eachf ∈F, Pn∈L(an+Pt∈Hn f(t))∈C.

The Central Sets Theorem was first discovered by H. Furstenberg [5, Proposition 8.21] for the semigroupNand considering sequences inZ. How-ever the most general version of Central Sets Theorem is available in [2, Theorem 2.2].

Acknowledgement. The authors would like to thank the referee for a constructive and helpful report, which made a serious improvement of the paper.

2. Finite matrices

We show in this section that for finite matrices with entries fromRimage partition regularity near zero depends only on the fact that they have im-ages in any central sets near zero. We start with the following well-known definition.

(1) No row ofM is row is~0.

(2) The first nonzero entry of each row is positive.

(3) If the first nonzero entries of any two rows occur in the same column, then those entries are equal.

IfM is a first entries matrix andcis the first nonzero entry of some rowM, thenc is called afirst entry ofM.

Now we prove a lemma which is important for determination of finite matrices with real entries that areimage partition regularity near zero over

R+.

Lemma 2.2. Let M be a u×v first entry matrix with entries from R and letC be a central set near zero in R+. Then there exist sequences hx1,ni∞n=1,

hx2,ni∞n=1, . . . ,hxv,nin∞=1 in R+ such that for any i∈ {1,2, . . . , v},

P∞ t=1xi,t

converges and for each F ∈ Pf(N),M~xF ∈Cu, where

~xF = 

 

P t∈F x1,t

.. .

P

t∈F xv,t 

 .

Proof. Let C be a central set near zero in R+. We proceed by induction on v. Assume first that v = 1. We can assume M has no repeated rows, so in this case we haveM = (c) for some c∈R+. Pick a sequence hyni∞n=1

[7, Theorem 3.1] withF S(hyni∞n=1) ⊆C, where

P∞

n=1yn converges and for

each n∈Nlet x1,n = y_cn. The sequencehx1,ni∞n=1 is as required.

Now letv∈Nand assume the theorem is true forv. LetM be au×(v+1) first entries matrix with entries fromR. By rearranging the rows of M and adding additional rows toM if need be, we may assume that we have some

r∈ {1,2, . . . , u−1}and some d∈R+ such that

ai,1=

(

0 ifi∈ {1,2, . . . , r},

d ifi∈ {r+ 1, r+ 2, . . . , u}.

LetBbe ther×vmatrix with entriesbi,j =ai,j+1. Pick sequenceshz1,ni∞n=1,

hz2,ni∞n=1, . . ., hzv,ni∞n=1 in R+ as guaranteed by the induction hypothesis

for the matrixB. For eachi∈ {r+ 1, r+ 2, . . . , u}and each n∈N, let

yi,n= v+1

X

j=2

ai,jzj−1,n.

Now for eachi∈ {r+ 1, r+ 2, . . . , u},P∞

t=1yi,t converges. We take yr,n= 0

for all n∈N.

Now C being central set near zero in R+, by Theorem 1.6 we can pick sequencehkni∞n=1inR+and a sequencehHni∞n=1of finite nonempty subsets of

Nsuch that maxHn<minHn+1for eachnand for eachi∈ {r, r+1, . . . , u},

F S(hkn+Pt∈Hnyi,ti

∞

n=1)⊆C, where

P∞

For each n ∈ N, let x1,n = k_dn and note that kn = kn+P_t∈Hnyr,t ∈ C ⊆ R+. For j ∈ {2,3, . . . , v+ 1}, let xj,n = P_t∈Hnzj−1,t. Certainly for

each i∈ {1,2, . . . , v+ 1},P∞

t=1xi,t converges. We claim that the sequences

3. Infinite matrices

We now prove that certain infinite matrices with real entries are also image partition regular near zero overR+. K. Milliken and A. Taylor independently proved a theorem from which it can be derived that certain infinite matrices, called Milliken–Taylor Matrices, are image partition regular over N. Some generalizations of this celebrated theorem are also available in [9, Corollary 3.6], [1, Theorem 5.7], [3, Theorem 2.6].

Definition 3.1. Letm∈ω, let~a=haiimi=0 be a sequence inR\ {0}, and let

~x= hxni∞n=0 be a sequence in R. The Milliken–Taylor system determined

by~aand~x,M T(~a, ~x) ={Pm

i=0ai·P_t∈Fi xt: eachFi∈ Pf(ω) and ifi < m,

then maxFi <minFi+1}

If ~c is obtained from ~a by deleting repetitions, then for any infinite se-quence~x, one hasM T(~a, ~x)⊆M T(~c, ~x), so it suffices to consider sequences

~c without adjacent repeated entries.

Definition 3.2. Let~abe a finite or infinite sequence inRwith only finitely many nonzero entries. Thenc(~a) is the sequence obtained from~aby deleting all zeroes and then deleting all adjacent repeated entries. The sequencec(~a) is the compressed form of~a. If~a=c(~a), then~ais a compressed sequence. Definition 3.3. Let~a be a compressed sequence in R\ {0}. A Milliken– Taylor matrix determined by~ais anω×ω matrixM such that the rows of

M are all possible rows with finitely many nonzero entries and compressed form equal to~a.

It follows from [7, Lemma 2.5] that 0+(R) =

p∈βRd: (∀ǫ >0) (0, ǫ)∈p and

0−(R) =

p∈βRd: (∀ǫ >0) (−ǫ,0)∈p

are both right ideals of 0+(R) ∪0−(R). Further given c ∈ R\ {0} and

p ∈βRd\ {0}, the product c·p is defined in (βRd,·). One has A ⊆R is a

member ofc·p if and only if c−1A={x∈R:c·x∈A} is a member of p. Lemma 3.4. Let p ∈ 0+(R), and let c ∈ R+. Then c·p ∈ 0+(R) and

(−c)·p∈0−_(R).

Proof. Letǫ >0. We need to show that (0, ǫ)∈c·p. Now (0, ǫ/c)∈p. But (0, ǫ/c)⊆c−1_{(0, ǫ). Hence c}−1_{(0, ǫ)∈p so that (0, ǫ)∈c·p.}

The proof of the other is similar.

Definition 3.5. Let hwni∞n=0 be a sequence in R. A sum subsystem of

hwni∞n=0 is a sequencehxni∞n=0 such that there exists a sequence hHni∞n=0 in

Pf(N) such that for eachn∈ω, maxHn<minHn+1 and xn=Pt∈Hn wt. Recall F S(hxni∞_n=k) = {P_n∈F xn : F ∈ Pf(ω) and minF ≥ k}, for a

given a sequence hxni∞n=0 and k∈ω. The proof of the following theorem is

Theorem 3.6. Let m ∈ ω and ~a = haiim_i=0 be a compressed sequence in R\ {0} with a0 >0, and let M be a Milliken–Taylor matrix determined by

~a, Then M is IPR/R+₀. In fact, given any sequence hwni∞n=0 in R+ such that P∞

n=0wn converges, whenever r ∈N, R+ =Sri=1Ci, and δ >0, there

exist i ∈ {1,2, . . . , r} and a sum subsystem hxni∞n=0 of hwni∞n=0 such that

M T(~a, ~x)⊆Ci∩(0, δ).

Proof. By [10, Lemma 5.11] we can choose an idempotent

p∈

∞

\

k=0

cℓβRdF S(hwni

∞

n=k).

SinceP∞

n=0wnconverges,p∈0+(R). Letq =a0·p+a1·p+· · ·+am·p. Then

by Lemma3.4and the previously mentioned fact that 0+(R) and 0−(R) are both right ideals of 0+_(R)∪0−_{(R), we have that q ∈0}+_{(R). So it suffices}

to show that wheneverQ∈q, there is a sum subsystemhxnin∞=0 ofhwni∞n=0

such that M T(~a, ~x) ⊆ Q. Let Q ∈ q be given. Assume first that m = 0. Then (a0)−1Q∈pso by [10, Theorem 5.14] there is a sum subsystemhxni∞n=0

of hwni∞n=0 such that F S(hxni∞n=0) ⊆(a0)−1Q. Then M T(~a, ~x) ⊆Q. Now

assume thatm >0. Define

P(∅) ={x∈R:−(a0·x) +Q∈a1·p+a2·p+· · ·+am·p}.

Since Q∈q we have

a−₀1· {x∈R:−x+Q∈a1·p+a2·p+· · ·+am·p} ∈p

which shows that

P(∅) ={x∈R:−(a0·x) +Q∈a1·p+a2·p+· · ·+am·p} ∈p.

Givenx0 define

P(x0) ={y∈R:−(a0·x0+a1·y) +Q∈a2·p+a3·p+· · ·+am·p}.

Ifx0∈P(∅), then −(a0·x0) +Q∈a1·p+a2·p+· · ·+am·p and so

{y∈R:−(a1·y) + (−(a0·x0) +Q)∈a2·p+a3·p+· · ·+am·p} ∈p

and thus P(x0)∈p.

Givenn∈ {1,2, . . . , m−1}andx0, x1, . . . , xn−1, letP(x0, x1, . . . , xn−1) =

{y∈R:−(a0·x0+· · ·+an−1·xn−1+an·y) +Q∈an+1·p+· · ·+am·p}. If x0 ∈P(∅) and for eachi∈ {1,2, . . . , n−1}, xi ∈ P(x0, x1, . . . , xi−1), then

P(x0, x1, . . . , xn−1)∈p.

Now givenx0, x1, . . . , xm−1, let

P(x0, x1, . . . , xm−1)

={y∈R:a0·x0+a1·x1+· · ·+am−1·xm−1+am·y∈Q}.

Ifx0∈P(∅) and for eachi∈ {1,2, . . . , m−1},xi ∈P(x0, x1, . . . , xi−1), then

P(x0, x1, . . . , xm−1)∈p. Given any B∈p, let B⋆ ={x∈B :−x+B∈p}.

Choose x0 ∈ P(∅)⋆ ∩F S(hwni∞n=0) and choose H0 ∈ Pf(N) such that x0 =Pt∈H0 wt. Let n∈ω and assume that we have chosen x0, x1, . . . , xn

and H0, H1, . . . , Hn such that:

(1) If k∈ {0,1, . . . , n}, thenHk∈ Pf(ω) and xk =P_t∈Hk wt.

(2) If k∈ {0,1, . . . , n−1}, then maxHk <minHk+1.

(3) If ∅ 6=F ⊆ {0,1, . . . , n}, thenP

t∈F xt∈P(∅)⋆.

(4) If k ∈ {1,2, . . . ,min{m, n}}, F0, F1, . . . , Fk ∈ Pf({0,1, . . . , n}), and

for eachj ∈ {0,1, . . . , k−1}, maxFj <minFj+1, then

X

t∈Fk

xt∈P X

t∈F0

xt, X

t∈F1

xt. . . , X

t∈Fk−1

xt !⋆

.

All hypotheses hold atn= 0, (2) and (4) vacuously. Letv= maxHn. Forr∈ {0,1, . . . , n}, let

Er={Pt∈F xt:∅ 6=F ⊆ {r, r+ 1, . . . , n}}.

Fork∈ {0,1, . . . , m−1} and r∈ {0,1, . . . , n}, let

Wk,r = (

X

t∈F0

xt, . . . , X

t∈Fk

xt !

:F0, F1, . . . , Fk∈ Pf({0,1, . . . , r})

and for eachi∈ {0,1, . . . , k−1},maxFi <minFi+1

)

Note thatWk,r 6=∅if and only if k≤r.

Ify ∈E0, then y∈P(∅)⋆, so−y+P(∅)⋆ ∈p and P(y)∈p. If k∈ {1,2,

. . . , m−1} and (y0, y1, . . . , yk) ∈ Wk,m, then yk ∈ P(y0, y1, . . . , yk−1) so

P(y0, y1, . . . , yk)∈pand thusP(y0, y1, . . . , yk)⋆ ∈p. Ifr ∈ {0,1, . . . , n−1}, k ∈ {0,1, . . . ,min{m−1, r}}, (y0, y1, . . . , yk) ∈ Wk,r, and z ∈ Er+1, then

z∈P(y0, y1, . . . , yk)⋆ so−z+P(y0, y1, . . . , yk)⋆ ∈p.

If n= 0, let x1 ∈F S(hwti∞t=v+1)∩P(∅)⋆∩ {−x0+P(∅)⋆} ∩P(x0)⋆ and

pickH1 ∈ Pf(N) such that minH1> vandx1 =Pt∈H1 wt. The hypotheses

are satisfied. Now assume that n≥1 and pick

xn+1 ∈F S(hwtit∞=v+1)∩P(∅)⋆∩

\

y∈E0

{−y+P(∅)⋆}

∩

min{m−1,n}

\

k=0

\

(y0,y1,...,yk)∈Wk,m

P(y0, y1, . . . , yk)⋆

∩

n−1

\

r=0

min{m−1,r}

\

k=0

\

(y0,y1,...,yk)∈Wk,r

\

z∈Er+1

(−z+P(y0, y1, . . . , yk)⋆).

Pick Hn+1∈ Pf(N) such that minHn+1 > v and xn+1=Pt∈Hn+1 wt.

thatxn+1∈P(∅)⋆, so assume that{n+ 1}(F and letG=F\ {n+ 1}. Let

y=P

t∈G xt. Theny∈E0 soxn+1 ∈ −y+P(∅)⋆ and so Pt∈F xt∈P(∅)⋆.

To verify hypothesis (4), let k ∈ {1,2, . . . ,min{m, n+ 1}} and assume thatF0, F1, . . . , Fk ∈ Pf({0,1, . . . , n+ 1}) and for eachj∈ {0,1, . . . , k−1},

maxFj <minFj+1. We can assume thatn+1∈Fk. Forl∈ {0,1, . . . , k−1}

let yl = Pt∈Fl xt. Then k−1 ≤ min{m−1, n} and (y0, y1, . . . , yk−1) ∈

Wk−1,m. If Fk = {n+ 1}, then P_t∈Fk xt = xn+1 ∈ P(y0, y1, . . . , yk−1)⋆.

So assume that {n+ 1} ( Fk and let Fk′ = Fk \ {n + 1}. Let r =

maxFk−1. Then r < minFk′ so r ≤ n −1, k − 1 ≤ min{m − 1, r},

and (y0, y1, . . . , yk−1) ∈ Wk−1,r. Let z = P_t∈F′

k xt. Then z ∈ Er+1 so

xn+1 ∈ −z+P(y0, y1, . . . , yk−1)⋆. Hence we have

X

t∈Fk

xt∈P X

t∈F0

xt, X

t∈F1

xt. . . , X

t∈Fk−1

xt !⋆

.

Next we define another class of infinite matrices with real entries, called

insertionmatrices, which are also image partition regular near zero overR+. The notion of insertion matrix was first introduced in [11, Definition 4.8 ].

Definition 3.7. Letγ,δ∈ωS

{ω}and letC be aγ×δ matrix with finitely many nonzero entries in each row. For each t < δ, let Bt be a finite matrix

of dimensionut×vt. LetR={(i, j) :i < γ and j∈ ×t<δ{0,1, . . . , ut−1}}.

Given t < δ and k ∈ {0,1, . . . , ut−1}, denote by~b(kt) the kth row of Bt.

ThenDis an insertion matrix ofhBtit<δ intoC if and only if the rows ofD

are all rows of the form

ci,0·~b(0)_j(0) ⌢ ci,1·~b(1)_j(1) ⌢· · ·

where (i, j)∈R.

For example, if

C=

1 0 2 1

,

B0=

1 1 5 7

and

B1 =

0 1 3 3

then the insertion matrix is

following theorem is similar to that of [11, Theorem 4.12].

Theorem 3.8. Let ~a = ha0, a1, . . . , ali be a compressed sequence in R\0

withl >0anda0 >0, letC be a Milliken–Taylor matrix for~a, and for each

t < ω, let Bt be a ut×vt finite matrx with entries from R which is image

partition regular matrix overR+. Then any insertion matrix ofhBtit<ω into C is image partition regular near zero, i.e.,IP R/R+₀.

Dn∈p, where Dn=Dn−1∩Tnk=0−1

Then hypotheses (I), (III),(V), and (VI) hold directly.

for each t∈Fi, cγ,t =ai. (One can do this because C is a Milliken–Taylor

matrix for~a.) Then Pm

t=0cγ,t·yt=Pli=0ai·Pt∈Fiyt∈A.

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Department of Mathematics, University of Kalyani, Kalyani-741235, West Bengal, India

dibyendude@klyuniv.ac.in

Department of Mathematics, Jadavpur University, Kolkata-32, West Bengal, India

rmkpaul@gmail.com