# Large recurrence of the polynomial Van der Waerden’s theorem and its
application to monochromatic exponential patterns

Sayan Goswami
<EMAIL_ADDRESS>Institute of Mathematical Sciences, A CI of
Homi Bhabha National Institute, CIT Campus, Taramani, Chennai 600113, India.

###### Abstract

Recently Di Nasso and Ragosta proved for every finite partition of
$\mathbb{N}$, there exists a monochromatic infinite exponential pattern of a
single sequence. In this article, we prove these patterns are abundant. We
show each term of this pattern can be chosen from a finite set, consists of a
multiplicative $IP_{r}$ set and a geometric progression of finite length. To
prove this, first we improve the size of the recurrence set of the polynomial
Van der Waerden’s Theorem, and then applying the size of the recurrence set,
we prove the existence of some new monochromatic polynomial patterns. Then,
applying methods of induction, we construct such infinitary patterns.
Throughout our article, we use the methods from the Algebra of the Stone-Čech-
Compactification of the set of natural numbers.

Keywords: Exponential Hindman theorem, central set, ultrafilters, Algebra of
the Stone-Čech compactification
Mathematics subject classification 2020: 05D10, 05C55, 22A15, 54D35

## 1 Introduction

For any nonempty set $X$, let $\mathcal{P}_{f}(X)$ be the set of all nonempty
finite subsets of $X.$ Arithmetic Ramsey theory deals with the monochromatic
patterns found in any given finite coloring of the integers or of the natural
numbers $\mathbb{N}$. Here “coloring” means disjoint partition and a set is
called “monochromatic” if it is included in one piece of the partition. A
cornerstone result in this field of research is Van der Waerden’s Theorem,
which states that for any finite coloring of the natural numbers one always
finds arbitrarily long monochromatic arithmetic progressions. For any
injective sequence $\langle x_{n}\rangle_{n}$, a set of the form $FS(\langle
x_{n}\rangle_{n})=\left\\{\sum_{t\in H}x_{t}:H\text{ is a nonempty finite
subset of }\mathbb{N}\right\\}$ is called an $IP$ set. Hindman’s finite sum
theorem [12] is one of the fundamental theorems in Ramsey theory which says
that for every finite coloring of $\mathbb{N}$, there exists a monochromatic
$IP$ set.

Recent research has focused on finding exponential patterns in the Ramsey
theory. Answering Sisto’s conjecture in [18], Sahasrabudhe proved that for any
finite coloring of $\mathbb{N}$, there exists a monochromatic pattern of the
form $\\{x,y,x^{y}\\}$. For further development we refer the article [19]. In
[14, 9], Di Nasso and Ragosta proved a key theorem [9, Theorem 3.1], and as a
consequence they proved an exponential version of Hindman’s theorem: for every
finite partition of $\mathbb{N}$, there exists an injective sequence $\langle
x_{n}\rangle_{n\in\mathbb{N}}$ such that $FEP(\langle
x_{n}\rangle_{n})=\left\\{x_{i_{n}}^{x_{i_{n-1}}^{\cdot^{\cdot^{\cdot^{x_{i_{1}}}}}}}:1\leq
i_{1}<\cdots<i_{n-1}<i_{n},n\in\mathbb{N}\right\\}$ is monochromatic. We refer
such pattern as Hindman Tower.

Before we proceed let us recall the notions of some fundamental large sets in
Ramsey theory. Let $(S,+)$ be any commutative semigroup. For any set
$A\subseteq S$ and $t\in S,$ $-t+A=\\{x:x+t\in A\\}$. A set $A\subseteq S$
will be called a syndetic set if there exists a finite set $F\subset S$ such
that $S=\cup_{t\in F}(-t+A).$ A set $A$ is called a thick set if, for any
finite set $G\subset S$, there exists $x\in S$ such that $G+x\subset A.$ A set
$A$ is called piecewise syndetic if it is an intersection of a thick set and a
syndetic set. It can be shown that $A$ is piecewise syndetic if and only if
there exists a finite set $F\subset S$ such that $\cup_{t\in F}(-t+A)$ is a
thick set. This can be proved that for any finite partition of $\mathbb{N}$,
one of the partitions is a piecewise syndetic set. A set is called
$IP^{\star}$ if it intersects every $IP$ sets. For some $r\in\mathbb{N}$, a
set will be called an $IP_{r}$ set (resp. multiplicative $IP_{r}$ set,
$MIP_{r}$ set in short) if there exists an injective sequence $\langle
x_{n}\rangle_{n=1}^{r}$ such that it is of the form $FS(\langle
x_{n}\rangle_{n=1}^{r})$ (resp. $FP(\langle x_{n}\rangle_{n=1}^{r})$). A set
$A$ is called $IP_{r}^{\star}$ (resp. $MIP_{r}^{\star}$) set if $A$ intersects
every $IP_{r}$ (resp. $MIP_{r}$) set. Clearly every $IP_{r}^{\star}$ set is an
$IP^{\star}$ set, but the converse is not true. Polynomial extension of
classical Ramsey theoretic results is hard to prove. In [4], Bergelson and
Leibman proved the polynomial extension of the Van der Waerden’s theorem.
Their proof uses the methods from topological dynamics. Later in [21], Walter
found a combinatorial proof, and then in [12], Hindman found an algebraic
proof. Here we quote the version of the polynomial Van der Waerden’s theorem
from [12].

From now on, we let $\mathbb{P}$ be the set of all polynomials from
$\mathbb{Z}$ to $\mathbb{Z}$ having no constant term.

######  Theorem 1.1.

[12] Let $A\subseteq\mathbb{Z}$ be a piecewise syndetic set, and
$F\in\mathcal{P}_{f}(\mathbb{P})$. Then

$\left\\{n:\left\\{m:\left\\{m,m+p\left(n\right):p\in F\right\\}\subset
A\right\\}\text{ is piecewise syndetic }\right\\}\text{}$

is an $IP^{\star}$ set.

An improvement of this theorem play a major role to prove monochromatic
Hindman towers are abundant, where each element of these towers is coming from
“nice”222for example multiplicative $IP_{r}$ sets, Sets containing geometric
progressions of finite length etc. sets.

In section 3, we improve Theorem 1.1 by proving that the collection of such
$n$ is an $IP_{r}^{*}$ set for some $r\in\mathbb{N}.$ Then we derive some new
applications of this theorem. Introducing a new induction technique in [10],
the authors found the polynomial extension of the key theorem of Di Nasso and
Ragosta [9, Theorem 3.1 ]. In section 4, we improve [10, Theorem 4.1] in two
different direction, and as a consequence we show the existence of
monochromatic Hindman’s towers where each element of these towers comes from a
set containing an $MIP_{r}$ set (for any given $r\in\mathbb{N}$), and a set of
geometric progression of length $r.$

## 2 Ultrafilters and it’s connections with Ramsey theory

We pause now to briefly introduce the algebra of the Stone-Čech
compactification of discrete semigroups.

For any nonempty set $X,$ let $\mathcal{P}(X)$ be the power set of $X.$
$\mathcal{F}\subset\mathcal{P}(X)$ is said to a filter if,

  * •

$\emptyset\notin\mathcal{F}$,

  * •

$A,B\in\mathcal{F}$ implies $A\cap B\in\mathcal{F},$ and

  * •

$A\in\mathcal{F}$ and $A\subset B$, implies $B\in\mathcal{F}$.

Using Zorn’s lemma one can confirm the existence of the maximal filters under
inclusion. These filters are called ultrafilters. One can show that if $p$ is
an ultrafilter, then for any $r\in\mathbb{N},$ if
$\mathbb{N}=\cup_{i=1}^{r}A_{i}$ is an $r$-coloring of $\mathbb{N}$, then one
of $A_{i}\in p.$ Therefore to prove a certain structure is monochromatic, it
is sufficient to study ultrafilters each of whose members contain that
structure.

Let $(S,+)$ be a discrete commutative semigroup. Let $\beta S$ be the
collection of all ultrafilters of $S.$ For any $(\emptyset\neq)A\subseteq S,$
let $\overline{A}=\\{p\in\beta S:A\in p\\}.$ The collection
$\\{\overline{A}:(\emptyset\neq)A\subseteq S\\}$ forms a basis for a topology.
One can show with this topology $\beta S$ is the Stone-Čech compactification
of $S.$ For any $p,q\in\beta S$, let us define $p\cdot q$ by

$A\in p\cdot q\iff\\{x:x^{-1}A\in q\\}\in p.$

With this definition $(\beta S,\cdot)$ is a compact right topological
semigroup. From Ellis theorem any closed subsemigroup of $(\beta S,\cdot)$
contains idempotents. For any subset $T\subset(\beta S,\cdot)$, we let $E(T)$
be the collection of all idempotents of $T.$ One can show that $A$ contains an
$IP$ set if and only if there exists an element $p\in E(\beta S,\cdot)$ such
that $A\in p.$

One can show if $A\subseteq S$ is a piecewise syndetic set, then there exists
a minimal left ideal $L$ of $(\beta S,\cdot)$, and $p\in L$ such that $A\in
p.$ Let $K(\beta S,\cdot)$ be the union of all minimal left ideal of $(\beta
S,\cdot)$. As each minimal left ideals are closed, it contains idempotents.
Hence $E\left(K(\beta S,\cdot)\right)\neq\emptyset$. A set $A\subseteq S$ is
said to be central if there exists $p\in E\left(K(\beta S,\cdot)\right)$ such
that $A\in p.$

Throughout our article, a set is additively (resp. multiplicatively) “large”
if this set is large in $(\mathbb{N},+)$ (resp. in $(\mathbb{N},\cdot)$). Note
that if we have a family $\mathcal{F}$ of finite sets satisfying

  * •

for each $n\in\mathbb{N}$ and $F\in\mathcal{F}$, $n\cdot F\in\mathcal{F}$, and

  * •

for any finite coloring of $\mathbb{N}$ there exists a monochromatic element
of the form $F\in\mathcal{F}$,

then we call this family a linear Rado family. Using Rado’s color compactness
argument one can show that every multiplicatively piecewise syndetic set will
contain elements from $F\in\mathcal{F}$. Hence for every $r\in\mathbb{N},$
every multiplicatively piecewise syndetic set will contain set of the form
$\\{d,a+id:i\in\\{0,1,\ldots,r\\}\\}$, and an $IP_{r}$ set.

## 3 Large recurrence of the polynomial Van der Waerden’s theorem

Hales-Jewett theorem [11] is one of the fundamental pillar of Ramsey theory.
Many classical results in Ramsey theory is directly followed by this theorem.
The strength of this theorem is that the statement of this theorem is set
theoretical in nature, and different algebraic operations induce different
Ramsey theoretic structures. In [5], V. Bergelson and A. Leibman established
the Polynomial extension of the Hales-Jewett theorem. The proof uses methods
from Topological dynamics. Then in [21], M. Walter found a combinatorial proof
and in [13], N. Hindman and R. McCutcheon found an algebraic proof. In this
section we will refine Theorem 1.1, and to do so we use the Polynomial Hales-
Jewett theorem. Then we will use our new theorem to deduce some new results.
Later we will use these results to deduce several new exponential Ramsey
theoretic patterns.

###  Revisiting Polynomial Hales-Jewett Theorem

Now we pause to recall the Hales-Jewett Theorem and it’s polynomial extension.
Let $\omega=\mathbb{N}\cup\left\\{0\right\\}$, where $\mathbb{N}$ is the set
of positive integers. Given a nonempty set $\mathbb{A}$ called alphabet, a
finite word is an expression of the form $w=a_{1}a_{2}\ldots a_{n}$ with
$n\geq 1$ and $a_{i}\in\mathbb{A}$. The quantity $n$ is called the length of
$w$ and denoted $\left|w\right|$. Let $v$ (a variable) be a letter not
belonging to $\mathbb{A}$. By a variable word over $\mathbb{A}$ we mean a word
$w$ over $\mathbb{A}\cup\left\\{v\right\\}$ that has at least one occurrence
of $v$. For any variable word $w$, $w\left(a\right)$ is the result of
replacing each occurrence of $v$ by $a$.

The following theorem is known as Hales-Jewett theorem, is due to A. W. Hales
and R. I. Jewett.

######  Theorem 3.1.

[11, Hales-Jewett Theorem (1963)] For all values $t,r\in\mathbb{N}$, there
exists a number $\text{HJ}\left(r,t\right)$ such that, if
$N\geq\text{HJ}\left(r,t\right)$ and $\left[t\right]^{N}$ is $r$ colored then
there exists a variable word $w$ such that
$\left\\{w\left(a\right):a\in\left[t\right]\right\\}$ is monochromatic.

The word space $\left[t\right]^{N}$ is called Hales-Jewett space or H-J space.
The number $\text{HJ}\left(r,t\right)$ is called Hales-Jewett number.

For $q,N\in\mathbb{N}$, $Q=[q]^{N}$, $\emptyset\neq\gamma\subseteq[N]$ and
$1\leq x\leq q$, $a\oplus x\gamma$ is defined to be the vector $b$ in $Q$
obtained by setting $b_{i}=x$ if $i\in\gamma$ and $b_{i}=a_{i}$ otherwise.

In the statement of Theorem [21, Polynomial Hales-Jewett Theorem], we have
$a\in Q$ so that $a=\langle\vec{a}_{1},\vec{a}_{2},\ldots,\vec{a}_{d}\rangle$
where for $j\in\\{1,2,\ldots d\\}$, $\vec{a}_{j}\in[q]^{N^{j}}$ and we have
$\gamma\subseteq[N]=\\{1,2,\ldots,N\\}$. Given $j\in\\{1,2,\ldots,d\\}$, let
$\vec{a}_{j}=\langle a_{j,\vec{i}}\rangle_{\vec{i}\in N^{j}}$. Then $a\oplus
x_{1}\gamma\oplus x_{2}(\gamma\times\gamma)\oplus\ldots\oplus
x_{d}\gamma^{d}=b$ where
$b=\langle\vec{b}_{1},\vec{b}_{2},\ldots,\vec{b}_{d}\rangle$ and for
$j\in\\{1,2,\ldots,d\\}$, $\vec{b}_{j}=\langle
b_{j,\vec{i}}\rangle_{\vec{i}\in N^{i}}$ where

$b_{j,\vec{i}}=\left\\{\begin{array}[]{rl}x_{j}&\hbox{if}\,\vec{i}\in\gamma^{i}\\\
a_{j,\vec{i}}&\hbox{otherwise.}\end{array}\right.$

######  Theorem 3.2.

[21, Polynomial Hales-Jewett Theorem] For any $q,k,d$ there exists
$N\left(q,k,d\right)\in\mathbb{N}$ such that whenever
$Q=Q\left(N\right)=\left[q\right]^{N}\times\left[q\right]^{N\times
N}\times\cdots\times\left[q\right]^{N^{d}}$ is $k$-colored there exist $a\in
Q$ and $\gamma\subseteq\left[N\right]$ such that the set of points

$\left\\{a\oplus x_{1}\gamma\oplus
x_{2}\left(\gamma\times\gamma\right)\oplus\cdots\oplus x_{d}\gamma^{d}:1\leq
x_{i}\leq q\right\\}$

is monochromatic.

###  $IP_{r}^{\star}$ Polynomial van der Waerden’s theorem

As promised above, in this section we will strengthen Theorem 1.1. We provide
both algebraic and combinatorial proofs. Both of these proofs have their own
pros and cons. Algebraic proof is short but uses Zorn’s lemma, whereas the
combinatorial proof is lengthy, but does not use Zorn’s lemma.

######  Theorem 3.3.

Let $A\subseteq\mathbb{N}$ be an additive piecewise syndetic set and
$F\in\mathcal{P}_{f}(\mathbb{P})$. Then

$\left\\{n:\left\\{m:\left\\{m,m+p\left(n\right):p\in F\right\\}\subset
A\right\\}\text{ is piecewise syndetic }\right\\}$

is an $IP_{r}^{\star}$ set for some $r\in\mathbb{N}.$

###### Proof.

(Combinatorial proof:) As $A$ is a piecewise syndetic set, there exists a
finite set $F_{1}\subset\mathbb{N}$ such that $B=\cup_{t\in
F_{1}}\left(-t+A\right)$ is a thick set. Assume $|F_{1}|=k.$ Let $\mathbb{A}$
be the set of all coefficients of the polynomials of $F,$ and $d$ be the
maximum degree of the polynomials of $F.$ Let
$N=N\left(|\mathbb{A}|,k,d\right)\in\mathbb{N}$ guaranteed by Theorem 3.2. Let
$\langle x_{n}\rangle_{n=1}^{N}$ be any sequence in $\mathbb{N}.$ Let
$Q=Q\left(N\right)$ be as in Theorem 3.2. Define the map
$\gamma:Q\rightarrow\mathbb{Z}$ by

$\gamma\left(\left(a_{1},\ldots,a_{N},a_{11},\ldots,a_{NN},\ldots,a_{11\cdots
1(d\text{ times})},\ldots,a_{NN\cdots N(d\text{ times})}\right)\right)=$

$\sum_{i=1}^{n}a_{i}x_{i}+\sum_{i,j=1,1}^{N,N}a_{ij}x_{i}x_{j}+\cdots+\sum_{i_{1},i_{2},\ldots,i_{d}=1,1,\cdots,1}^{N,N,\cdots,N}a_{i_{1}i_{2}\cdots
i_{d}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{d}}.$

If necessary, translate the set
$G=\left\\{\gamma\left(\overrightarrow{a}\right):\overrightarrow{a}\in
Q\right\\}$ by a sufficiently large number $M$ to assume that
$G\subset\mathbb{N}$. Then for any finite coloring of $G,$ there exists
$a\in\mathbb{N},$ and $d\in FS\left(\langle x_{n}\rangle_{n=1}^{N}\right)$
such that $\left\\{a+p\left(d\right):p\in F\right\\}\cup\left\\{a\right\\}$ is
monochromatic.

As $G$ is finite, the number of such patterns in $G$ is finite. Let
$\mathscr{H}$ be the collection of such patterns in $G$. Let
$|\mathscr{H}|=l$, $\mathscr{H}=\left\\{H_{i}:1\leq i\leq l\right\\}$ and
$F_{1}=\\{f_{i}:1\leq i\leq k\\}.$ As $G$ is finite, the collection
$C=\left\\{x:G+x\subset B\right\\}$ is a thick set.

Now construct a $[l]\times[k]$ coloring of $C$ by $x\in C$ has color
$\left(i,j\right)\in[l]\times[k]$ if and only if there exists $f_{j}\in F_{1}$
and $H_{i}\in\mathscr{H}$ such that $\left\\{x+f_{j}+m:m\in
H_{i}\right\\}\subset A$ (one can choose the least $\left(i,j\right)$ in
dictionary ordering). Let $x\in C_{\left(i,j\right)}$ if and only if $x\in C$
and $x$ has color $\left(i,j\right).$ So,
$C=\cup_{\left(i,j\right)\in[l]\times[k]}C_{\left(i,j\right)}.$ Hence at least
one of $C_{(i,j)}^{\prime}s$, say $C_{\left(s,t\right)}$ is a piecewise
syndetic set. Hence there exists $H=\left\\{q+p\left(r\right):p\in
F\right\\}\cup\left\\{q\right\\}\in\mathcal{H}$ such that
$\left\\{x+\left(t+q\right)+p\left(r\right):p\in
P\right\\}\cup\left\\{x+\left(t+q\right)\right\\}\subset A$ for all $x\in
C_{\left(s,t\right)}$.

As $C_{\left(s,t\right)}$ is a piecewise syndetic set, we have
$C_{\left(s,t\right)}+\left(t+q\right)$ is a piecewise syndetic set. This
implies for all $m\in C_{\left(s,t\right)}+\left(t+q\right),$ we have
$\left\\{m\right\\}\cup\left\\{m+p\left(d\right):p\in F\right\\}\subset A,$
for some $d\in FS\left(\langle x_{n}\rangle_{n=1}^{N}\right).$ Hence

$\left\\{n:\left\\{m:\left\\{m,m+p\left(n\right):p\in F\right\\}\subset
A\right\\}\text{ is piecewise syndetic }\right\\}$

is $IP_{N}^{\star}$. This completes the proof. ∎

###### Proof.

(Algebraic proof:) Let $A$ be a piecewise syndetic set. Hence there exists
$p\in K\left(\beta\mathbb{N},+\right)$ such that $A\in p.$ By [14, Theorem
4.39], the set $S=\left\\{x:-x+A\in p\right\\}$ is syndetic. Hence there
exists a finite set $F$ such that $\mathbb{N}=\cup_{t\in F}-t+S.$ Let $|F|=r.$
Proceeding along the same line of the above proof, let us choose
$N=N\left(\mathbb{A},r,d\right)\in\mathbb{N}$, a sequence $\langle
x_{n}\rangle_{n=1}^{N}$, and corresponding $G\subset\mathbb{N}$. As
$G\subset\cup_{t\in F}-t+S,$ there exists
$(\emptyset\neq)H\subseteq\left\\{1,2,\ldots,N\right\\}$, and $b\in\mathbb{N}$
such that $\left\\{b,b+f\left(x_{H}\right):f\in F\right\\}\subset S.$ In other
words

$D=\left\\{d:\text{ there exists }m\text{ such that
}\left\\{m,m+f\left(d\right):f\in F\right\\}\subset S\right\\}$

is $IP_{N}^{\star}$ set. Now for each $n\in D,$ there exists $m$ such that
$\left\\{m,m+f\left(n\right):f\in F\right\\}\subset S.$ Now
$B=A\cap\left(-m+A\right)\cap\bigcap_{f\in
F}-\left(m+f\left(n\right)\right)+A\in p.$ As $B$ is piecewise syndetic, $B+m$
is piecewise syndetic, and for each $a\in B+m,$
$\left\\{a,a+f\left(n\right):f\in F\right\\}\subset A.$ This completes the
proof. ∎

It is easy to check that $\overline{E\left(K(\beta\mathbb{N},+)\right)}$ is a
left ideal of $(\beta\mathbb{N},\cdot)$. Hence we can choose $p\in
E\left(K(\beta\mathbb{N},\cdot)\right)\cap\overline{E\left(K(\beta\mathbb{N},+)\right)}.$
By choice of $p,$ each member of $p$ is additive and multiplicative piecewise
syndetic set. The following corollary immediately follows from Theorem 3.3.

######  Corollary 3.4.

Let $F\in\mathcal{P}_{f}(\mathbb{P})$, and $p\in
E\left(K(\beta\mathbb{N},\cdot)\right)\cap\overline{E\left(K(\beta\mathbb{N},+)\right)}.$
Then for every $A\in p,$

$\left\\{d\in A:\left\\{a:\left\\{a,a+p\left(d\right):p\in F\right\\}\subset
A\right\\}\text{ is additive piecewise syndetic }\right\\}$

is additive and multiplicative piecewise syndetic.

###### Proof.

As $A\in p,$ we have $A$ is additive piecewise syndetic set. Hence

$D=\left\\{d:\left\\{a:\left\\{a,a+p\left(d\right):p\in F\right\\}\subset
A\right\\}\text{ is additive piecewise syndetic }\right\\}$

is an $IP_{r}^{\star}$ set for some $r\in\mathbb{N}.$ As for each
$r\in\mathbb{N}$, every multiplicative piecewise syndetic set contains an
$IP_{r}$ set, we have
$\overline{K\left(\beta\mathbb{N},\cdot\right)}\subset\overline{D}.$ So, $D\in
p$ and as $A\in p$, we have $A\cap D\in p.$ Hence we have our desired
conclusion. ∎

Let $\mathbb{N}_{>1}=\mathbb{N}\setminus\\{1\\},$ and $n\in\mathbb{N}_{>1}.$ A
set $A\subseteq\mathbb{N}$ will be called $n$-exponential piecewise syndetic
if there exists a multiplicative piecewise syndetic set $B$ such that
$n^{B}=\\{n^{b}:b\in B\\}\subseteq A.$ For any $p\in\beta\mathbb{N},$ denote
by $E_{1}(n,p)\in\beta\mathbb{N}$ such that $A\in E_{1}(n,p)$ if and only if
$B=\\{m:n^{m}\in A\\}\in p.$ Clearly $A$ is $n$-exponential piecewise syndetic
if and only if there exists $p\in\overline{K(\beta\mathbb{N},\cdot)}$ such
that $A\in E_{1}(n,p).$

Passing to the map $\overrightarrow{a}$ to $\gamma(\overrightarrow{a})$ in the
proof of Theorem 3.3, one can prove the following theorem and so we leave it
to the reader.

######  Theorem 3.5.

Let $n\in\mathbb{N}_{>1}$, $F\in\mathcal{P}_{f}(\mathbb{P})$ and
$A\subseteq\mathbb{N}$ be an multiplicative piecewise syndetic set. Then there
exists $r\in\mathbb{N}$ such that

$\left\\{b:\left\\{a:\left\\{a,a\cdot n^{p\left(b\right)}:p\in
F\right\\}\subset A\right\\}\text{ is multiplicative piecewise syndetic
}\right\\}$

is an $IP_{r}^{\star}$ set.

The following corollary of the above theorem shows that exponential Schur
patterns are abundant.

######  Corollary 3.6.

Let $n\in\mathbb{N}_{>1}$, $p\in\overline{K(\beta\mathbb{N},\cdot)}$ and $A\in
E_{1}(n,p)$. Then

$\\{y\in A:\\{x:\\{x,x^{y}\\}\subset A\\}\text{ is a }n\text{-exponential
piecewise syndetic}\\}\in E_{1}(n,p).$

###### Proof.

Let $A\in E_{1}(n,p)$ and so $B=\\{m:n^{m}\in A\\}\in p.$ From Theorem 3.5,

$C=\left\\{b:\left\\{a:\left\\{a,a\cdot n^{b}\right\\}\subset A\right\\}\text{
is piecewise syndetic }\right\\}$

is an $IP_{r}^{\star}$ set for some $r\in\mathbb{N}.$ As $C$ is an
$IP_{r}^{\star}$ set, we have $D=A\cap C\in p.$ Hence

$D=\left\\{b\in A:\left\\{a:\left\\{a,a\cdot n^{b}\right\\}\subset
A\right\\}\text{ is piecewise syndetic }\right\\}\in p.$

Now for every $m\in\mathbb{N}$, passing to the map $m\rightarrow n^{m}$ we
have our desired result. ∎

####  Applications to new polynomial patterns

In this section, we deduce two new applications of Theorem 3.3. The first one
involves patterns in additively piecewise syndetic sets, where the domain of
the polynomials can be chosen from any given multiplicative central set. And
the second one involves the evaluation of a polynomial over the “sum
subsystem” of any given $IP$ set.

######  Theorem 3.7.

Let $F\in\mathcal{P}_{f}(\mathbb{P})$, $A$ be an additively piecewise syndetic
set, and $B$ be a multiplicatively central set. Then there exists a sequence
$\langle x_{n}\rangle_{n=1}^{\infty}$ such that

  1. 1.

$FP(\langle x_{n}\rangle_{n=1}^{\infty})\subset B$, and

  2. 2.

for each $N\in\mathbb{N}$, there exists $a(N)\in\mathbb{N}$ such that

$\\{a(N),a(N)+P(y):P\in F,y\in FS(\langle x_{n}\rangle_{n=1}^{N})\cup
FP(\langle x_{n}\rangle_{n=1}^{N})\\}\subset A.$

In addition for each $N\in\mathbb{N}$, such collection of $a(N)$ is piecewise
syndetic.

###### Proof.

Let $q\in E(K(\beta\mathbb{N},\cdot))$ be such that $B\in q$. Let
$B^{*}=\\{n\in B:n^{-1}B\in q\\}\in q$. As $A$ is additively piecewise
syndetic set, from Theorem 3.3 we have

$D=\\{d:\\{a:\\{a,a+P(d):P\in F\\}\subset A\\}\text{ is a piecewise
syndetic}\\}$

is an $IP_{r}^{*}$ for some $r\in\mathbb{N}$. Hence,
$\overline{K(\beta\mathbb{N},\cdot)}\subset\overline{D}$, and this implies
$B^{*}\cap D\in q$. Let $x_{1}\in B^{*}\cap D$. Then there exists
$a(1)\in\mathbb{N}$ such that $\\{a(1),a(1)+P(x_{1}):P\in F\\}\subset A$ and
such collection of $a(1)$ is piecewise syndetic. Inductively assume that for
some $N\in\mathbb{N}$, there exists a finite sequence $\langle
x_{n}\rangle_{n=1}^{N}$ such that

  1. 1.

$FP(\langle x_{n}\rangle_{n=1}^{N})\subset B^{*}$, and

  2. 2.

for $1\leq k\leq N$, $\exists$ $a(k)\in\mathbb{N}$ such that
$B_{k}=\\{a:\\{a,a+P(y):P\in F,y\in FS(\langle x_{n}\rangle_{n=1}^{k})\cup
FP(\langle x_{n}\rangle_{n=1}^{k})\\}\subset A\\}$ piecewise syndetic.

Now $B_{N}$ is a piecewise syndetic set. For each $y\in FP(\langle
x_{n}\rangle_{n=1}^{N})$, $y\in FS(\langle x_{n}\rangle_{n=1}^{N})$, and $P\in
F$, define new polynomials $P_{y},P^{z}\in\mathbb{P}$ by $P_{y}(n)=P(ny)$, and
$P^{z}(n)=P(z+n)-P(z)$ $\forall n\in\mathbb{Z}$. Define $G=F\cup\\{P_{y}:P\in
F,y\in FP(\langle x_{n}\rangle_{n=1}^{N})\\}\cup\\{P^{z}:P\in F,z\in
FS(\langle x_{n}\rangle_{n=1}^{N})\\}$, and so
$G\in\mathcal{P}_{f}(\mathbb{P})$. Again from Theorem 3.3,

$D_{1}=\\{d:\\{b:\\{b,b+f(d):f\in G\\}\subset B_{N}\\}\text{ is a piecewise
syndetic set}\\}$

is an $IP_{s}^{*}$ for some $s\in\mathbb{N}$. From [14, Lemma 4.14], we know
that for each $y\in FP(\langle x_{n}\rangle_{n=1}^{N})$, $y^{-1}B^{*}\in q.$
Hence we have $E=D_{1}\cap\bigcap_{y\in FP(\langle
x_{n}\rangle_{n=1}^{N})}y^{-1}B^{*}\cap B^{*}\in q$. Choose $x_{N+1}\in E$,
and so $FP(\langle x_{n}\rangle_{n=1}^{N+1})\subset B^{*}$. Again $x_{N+1}\in
D_{1}$ and so there exists $a(N+1)$ such that

$\left\\{a(N+1),a(N+1)+f(x_{N+1}):f\in G\right\\}\subset B_{N}.$

Again the collection of such $a(N+1)$ is piecewise syndetic. Now,

  1. 1.

$a(N+1)\in B_{N}$ implies $a(N+1)\in A$, and for all $y\in FP(\langle
x_{n}\rangle_{n=1}^{N})\cup FS(\langle x_{n}\rangle_{n=1}^{N})$, and $P\in F$
we have $a(N+1)+P(y)\in A$, (A)

  2. 2.

for all $P\in F,y\in FP(\langle x_{n}\rangle_{n=1}^{N})$,
$a(N+1)+P_{y}(x_{N+1})\in A$ implies $\forall y\in FP(\langle
x_{n}\rangle_{n=1}^{N+1})\setminus FP(\langle x_{n}\rangle_{n=1}^{N})$,
$a(N+1)+P(y)\in A$, and (B)

  3. 3.

for all $P\in F,z\in FS(\langle x_{n}\rangle_{n=1}^{N})$,
$a(N+1)+P^{z}(x_{N+1})\in A$ implies $\forall z\in FP(\langle
x_{n}\rangle_{n=1}^{N+1})\setminus FP(\langle x_{n}\rangle_{n=1}^{N})$,
$a(N+1)+P(z)\in A$. (C)

Combining (A), (B), and (C), for all $P\in F$, and $y\in FS(\langle
x_{n}\rangle_{n=1}^{N+1})\cup FP(\langle x_{n}\rangle_{n=1}^{N+1})$, we have
$a(N+1)+P(y)\in A$. As the collection of such $a(N+1)$ is piecewise syndetic,
we proved the induction hypothesis, and this proves our result.

∎

For any finite coloring of $\mathbb{N}$, the following special case of Theorem
3.7 says that we can choose the set $FP(\langle x_{n}\rangle_{n=1}^{\infty})$
from the same color as of $a(N)+P(y)$.

######  Corollary 3.8.

Let $p\in
E\left(K(\beta\mathbb{N},\cdot)\right)\cap\overline{\left(K(\beta\mathbb{N},+)\right)}$,
and $A\in p.$ Then from Theorem 3.7, for any
$F\in\mathcal{P}_{f}(\mathbb{P})$, there exists a sequence $\langle
x_{n}\rangle_{n=1}^{\infty}$ such that $FP(\langle
x_{n}\rangle_{n=1}^{\infty})\subset A$, and for each $N\in\mathbb{N}$, there
exists $a(N)\in\mathbb{N}$ such that

$\\{a(N),a(N)+P(y):P\in F,y\in FS(\langle x_{n}\rangle_{n=1}^{N})\cup
FP(\langle x_{n}\rangle_{n=1}^{N})\\}\subset A.$

In addition for each $N\in\mathbb{N}$, such collection of $a(N)$ is piecewise
syndetic.

Sum subsystems of IP sets play important role in Ramsey theory. These systems
appear in many Ramsey theoretic patterns to strengthen classical statements.
Here we recall the definition of sum subsystems.

######  Definition 3.9.

(Sum subsystem) For any $IP$ set $FS\left(\langle x_{n}\rangle_{n}\right)$, a
sum subsystem of $FS\left(\langle x_{n}\rangle_{n}\right)$ is a set of the
form $FS\left(\langle y_{n}\rangle_{n}\right)$, where for each
$n\in\mathbb{N},$ $y_{n}$ is defined as follows:

  1. 1.

there exists a sequence $\langle H_{n}\rangle_{n}$ in
$\mathcal{P}_{f}(\mathbb{N})$ such that for each $n\in\mathbb{N}$, $\max
H_{n}<\min H_{n+1}$, and

  2. 2.

$y_{n}=\sum_{t\in H_{n}}x_{t}$ for all $n\in\mathbb{N}$.

The following theorem is a variant of Theorem 3.7. In the later section we
will use this Theorem to prove the first improvement of [10, Theorem 4.1.].

######  Theorem 3.10.

Let $A$ be an additive piecewise syndetic set and
$F\in\mathcal{P}_{f}(\mathbb{P})$. Then for every injective sequence $\langle
x_{n}\rangle_{n}$ there exists a sum subsystem $FS\left(\langle
y_{n}\rangle_{n}\right)$ of $FS\left(\langle x_{n}\rangle_{n}\right)$ such
that for every $N\in\mathbb{N},$ there exists $a(N)\in\mathbb{N},$ such that

$\left\\{a(N)+p\left(y\right):y\in FS\left(\langle
y_{n}\rangle_{n=1}^{N}\right)\cup FP\left(\langle
y_{n}\rangle_{n=1}^{N}\right)\text{and }p\in F\right\\}\subset A.$

In addition for each $N\in\mathbb{N}$, such collection of $a(N)$ is piecewise
syndetic.

###### Proof.

For any $v\in E\left(\beta\mathbb{N},+\right)$ and any $A\in v,$ let
$A^{\star}=\\{n\in A:-n+A\in v\\}.$ As $v$ is an idempotent, we have
$A^{\star}\in v$. Let $v$ be an idempotent such that $FS\left(\langle
x_{n}\rangle_{n}\right)\in v,$ and so $FS\left(\langle
x_{n}\rangle_{n}\right)^{\star}\in v.$ From Theorem 1.1, we have $y_{1}\in
FS\left(\langle x_{n}\rangle_{n}\right)^{\star}$ such that the set
$B_{1}=\left\\{a:\left\\{a,a+p(y_{1}):p\in F\right\\}\subset A\right\\}$ is
piecewise syndetic. So there exists an element $a\in B_{1}$ such that
$\left\\{a,a+p(y_{1}):p\in F\right\\}\subset A.$ Inductively assume that for
some $M\in\mathbb{N}$,

  1. 1.

$FS\left(\langle y_{n}\rangle_{n=1}^{M}\right)\subset FS\left(\langle
x_{n}\rangle_{n}\right)^{\star},$

  2. 2.

For each $1\leq N\leq M,$ we have

$\left\\{a(N)+p\left(y\right):y\in FS\left(\langle
y_{n}\rangle_{n=1}^{N}\right)\cup FP\left(\langle
y_{n}\rangle_{n=1}^{N}\right)\text{and }p\in F\right\\}\subset A,$

  3. 3.

For each $1\leq N\leq M,$ the set

$B_{N}=\left\\{a:\left\\{a,a+p(y):p\in F,y\in FS\left(\langle
y_{n}\rangle_{n=1}^{N}\right)\cup FP\left(\langle
y_{n}\rangle_{n=1}^{N}\right)\right\\}\subset A\right\\}\text{ is piecewise
syndetic. }$

For each $p\in F,$ $y\in FS\left(\langle y_{n}\rangle_{n=1}^{M}\right),$ and
$z\in FP\left(\langle y_{n}\rangle_{n=1}^{M}\right),$ define new polynomials
$p_{y},p^{\prime}_{z}:\mathbb{N}\rightarrow\mathbb{N},$ by
$p_{y}\left(n\right)=p\left(n+y\right)-p\left(y\right)$ and
$p^{\prime}_{z}\left(n\right)=p(zn).$ Choose a new collection of polynomials
$G=F\cup\left\\{p_{y}:p\in F,y\in FS\left(\langle
y_{n}\rangle_{n=1}^{M}\right)\right\\}\cup\left\\{p^{\prime}_{z}:p\in F,z\in
FP\left(\langle y_{n}\rangle_{n=1}^{M}\right)\right\\}.$ From Theorem 1.1,

$D=\left\\{x:\left\\{c:\left\\{c,c+p(x):p\in G\right\\}\subset
B_{M}\right\\}\text{ is piecewise syndetic }\right\\}$

is an $IP^{\star}$ set. Note that if $A$ is an $IP^{*}$ set, and
$n\in\mathbb{N},$ then $n^{-1}A$ is again an $IP^{*}$ set. Hence,

$D_{1}=D\cap\bigcap_{z\in FS\left(\langle
y_{n}\rangle_{n=1}^{M}\right)}z^{-1}D\cap FS\left(\langle
x_{n}\rangle_{n}\right)^{\star}\cap\bigcap_{y\in FS\left(\langle
y_{n}\rangle_{n=1}^{M}\right)}\left(-y+FS\left(\langle
x_{n}\rangle_{n}\right)^{\star}\right)\in v.$

Choose $y_{M+1}\in D_{1}$, and an element $c_{1}\in
E=\left\\{c:\left\\{c,c+p(y_{M+1}):p\in G\right\\}\subset B_{M}\right\\}.$
Hence $FS\left(\langle y_{n}\rangle_{n=1}^{M+1}\right)\subset FS\left(\langle
x_{n}\rangle_{n}\right)^{\star}.$ Now we will complete the induction.

  * •

Case 1: if $z\in FS\left(\langle y_{n}\rangle_{n=1}^{M}\right)\cup
FP\left(\langle y_{n}\rangle_{n=1}^{M}\right)\cup\left\\{y_{M+1}\right\\}$,
then for all $p\in F,$ $c_{1}+p\left(z\right)\in B_{M}\subset A,$ and

  * •

Case 2: if $z\in FS\left(\langle
y_{n}\rangle_{n=1}^{M+1}\right)\setminus\left(FS\left(\langle
y_{n}\rangle_{n=1}^{M}\right)\cup\left\\{y_{M+1}\right\\}\right)$, then we
have $z=s+y_{M+1}$, for some $s\in FS\left(\langle
y_{n}\rangle_{n=1}^{M}\right)$.

So we have

$c_{1}+p_{s}\left(y_{M+1}\right)=c_{1}+p\left(s+y_{M+1}\right)-p\left(s\right)\in
B_{M},$

But then
$c_{1}+p\left(s+y_{M+1}\right)-p\left(s\right)+p\left(s\right)=c_{1}+p\left(z\right)\in
A.$

  * •

Case 3: if $z\in FP\left(\langle
y_{n}\rangle_{n=1}^{M+1}\right)\setminus\left(FP\left(\langle
y_{n}\rangle_{n=1}^{M}\right)\cup\left\\{y_{M+1}\right\\}\right)$, then we
have and $z=ty_{M+1}$ for some $t\in FP\left(\langle
y_{n}\rangle_{n=1}^{M}\right).$ Then

$c_{1}+p^{\prime}_{t}\left(y_{M+1}\right)=c_{1}+p\left(ty_{M+1}\right)=c_{1}+p(z)\in
B_{M}.$

Now rename $E=B_{M+1}$. Now for each $a\in B_{M+1}\subset A,$ we have

$\left\\{a+p\left(z\right):p\in F\text{ and }z\in FS\left(\langle
y_{n}\rangle_{n=1}^{M+1}\right)\cup FP\left(\langle
y_{n}\rangle_{n=1}^{M+1}\right)\right\\}\subset A,$

and the set $B_{M+1}$ is piecewise syndetic. This completes the induction. ∎

The following corollaries of Theorem 3.7 and Theorem 3.10 are new.

######  Corollary 3.11.

Let $n\in\mathbb{N}_{>1},$ $p\in
E\left(K(\beta\mathbb{N},\cdot)\right)\cap\overline{\left(K(\beta\mathbb{N},+)\right)}$,
and $A\in E_{1}(n,p)$. Then

  1. 1.

there exists $x,y,\text{ and }z\in\mathbb{N}$ such that

$\\{n^{x},n^{y},n^{z},n^{yz},(n^{x})^{n^{y+z}},(n^{x})^{n^{yz}}\\}\subset A,$

and

  2. 2.

there exists $a,b,\text{ and }c\in\mathbb{N}$ such that

$\\{n^{a},n^{b},n^{c},n^{b+c},(n^{a})^{n^{b+c}}(n^{a})^{n^{bc}}\\}\subset A.$

###### Proof.

The proof of (1), directly follows from the Corollary 3.8. For the proof of
(2), first choose a sequence $\langle x_{t}\rangle_{t=1}^{\infty}$ such that
$FS(\langle x_{n}\rangle_{t=1}^{\infty})\subseteq\\{m:n^{m}\in A\\}\in p,$ and
then apply Theorem 3.10. ∎

## 4 Applications to infinite exponential patterns

As we promised, in this section we will improve [10, Theorem 4.1.] in two
different ways. As a result we will show that monochromatic Hindman towers are
abundant. In [9], Di Nasso and Ragosta considered a groupoid operation $\star$
over $\beta\mathbb{N}$ defined as follows: first, consider the operation
$f:\mathbb{N}^{2}\rightarrow\mathbb{N}$ defined as $f(n,m)=2^{n}m$. Let
$\overline{f}:\beta\left(\mathbb{N}^{2}\right)\rightarrow\beta\mathbb{N}$ be
its continuous extension. $\star$ is the restriction of $\overline{f}$ to
tensor pairs, namely: for all $p,q\in\beta\mathbb{N}$

$p\star q=\overline{f}(p\otimes q)=2^{p}\odot q.$

The following, which is Theorem 3.1 in [9] is, at the best of our knowledge,
the first known result for infinitary exponential configurations in
literature, namely Theorem 3.1 in [9].

######  Theorem 4.1.

[9, Theorem 3.1] (Di Nasso-Ragosta) Let $p\in
E\left(K(\beta\mathbb{N},\oplus)\right)$ and $A\in p\star p$. For every
$N\in\mathbb{N}$ and for every sequence $\varPhi=\left(f_{n}:n\geq 2\right)$
of functions $f_{n}:\mathbb{N}\rightarrow\mathbb{N},$ there exists a sequence
$\left(a_{k}\right)_{k\in\mathbb{N}}$ such that

$\mathcal{F}_{k,\varPhi}\left(a_{k}\right)_{k\in\mathbb{N}}=\left\\{a_{k}\cdot
2^{\sum_{i=1}^{k-1}\lambda_{i}a_{i}}:k\in\mathbb{N},0\leq\lambda_{1}\leq
N,\text{ and }0\leq\lambda_{i}\leq f_{i}\left(a_{i-1}\right)\text{ for }2\leq
i\leq k-1\right\\}\subset A.$

For each $i\in\mathbb{N}$, choosing $f_{i}$ suitably, we have the exponential
Hindman theorem.

Let $N=1$, and choose the function $f_{2}(x)=2^{x}.$ Inductively assume that
we have defined the function $f_{n}(x)$ for some $n\in\mathbb{N}_{>1}$. Now
define $f_{n+1}(x)=(2^{x})^{f_{n}(x)}$. Now choosing
$\varPhi=\left(f_{n}:n\geq 2\right)$, theorem 4.1 immediately implies the
existence of monochromatic Hindman tower. For details see [9].

For every $p\in\mathbb{P},$ let $deg(p)$ be the degree of the polynomial $p,$
and let $coef(p)=\max\\{|c|:c\text{ is a coefficient of }p\\}.$ For
$n\in\mathbb{N}_{>1},$ define a new operation $\star_{n}$ on $\mathbb{N}$ by
$a\star_{n}b=n^{a}b.$ The following theorem is the polynomial extension of
Theorem 4.1, established in [10].

######  Theorem 4.2.

[10, Theorem 4.1.] (Polynomial Di Nasso-Ragosta) Let $p\in
E\left(K(\beta\mathbb{N},\oplus)\right)$ and $q\in\beta\mathbb{N}$ be such
that for every $N\in\mathbb{N}$, each element of $q$ contains an $IP_{N}$
sets. Let $n\in\mathbb{N}_{>1},$ $A\in p\star_{n}q,$ and
$F_{1}\in\mathcal{P}_{f}(\mathbb{P})$. Let $\varPhi=\left(f_{n}:n\geq
2\right)$ be a sequence of functions $f_{n}:\mathbb{N}\rightarrow\mathbb{N}.$
For every $n(>1),x\in\mathbb{N}$, let $F_{n,x}=\\{p\in\mathbb{P}:deg(p)\leq
f_{n}(x)\text{ and }coef(p)\leq f_{n}(x)\\}.$

Under this hypothesis we have a sequence $\left(x_{k}\right)_{k\in\mathbb{N}}$
such that

$\mathcal{PF}^{n}_{k,\varPhi}\left(x_{k}\right)_{k\in\mathbb{N}}=\left\\{x_{k}\cdot
n^{\sum_{i=1}^{k-1}p_{i}\left(x_{i}\right)}:k\in\mathbb{N},p_{1}\in
F_{1},\text{ and }p_{i}\in F_{i,f_{i}\left(x_{i-1}\right)}\text{ for }2\leq
i\leq k-1\right\\}\subset A.$

###  Combined multiplicative and exponential structures I

The following theorem is the first improvement of Theorem 4.2. As a corollary
we have mixed patterns of the form of $MIP_{r}$ sets and infinite exponential
structures.

######  Theorem 4.3.

Let $n\in\mathbb{N}_{>1},$ $\langle N_{i}\rangle_{i\in\mathbb{N}}$ be a
sequence in $\mathbb{N}$, $F_{1}\in\mathcal{P}_{f}(\mathbb{P})$, $p\in
E(\overline{\left(K(\beta\mathbb{N},+)\right)})$ ,
$q\in\overline{E((\beta\mathbb{N},+))}$ and $A\in p\star_{n}q$. Let
$\varPhi=\left(f_{n}:n\geq 2\right)$ be a sequence of functions
$f_{n}:\mathbb{N}\rightarrow\mathbb{N}.$ For every
$n\in\mathbb{N}_{>1},G\in\mathcal{P}_{f}(\mathbb{N})$, let
$F_{n,G}=\\{p\in\mathbb{P}:deg(p)\leq f_{n}(x)\text{ and }coef(p)\leq
f_{n}(x)\text{ for }x=\max G\\}.$ Under this hypothesis we have a sequence of
finite sets $\langle G_{i}\rangle_{i\in\mathbb{N}}$ such that

  1. 1.

for each $i\in\mathbb{N},$ $|G_{i}|=N_{i}$, and

  2. 2.

$\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}=\left\\{z_{k}\cdot
n^{\sum_{i=1}^{k-1}p_{i}\left(y_{i}\right)}:k\in\mathbb{N},p_{1}\in
F_{1},y_{j}\in FS(G_{j})\cup FP(G_{j})\text{ for }1\leq j\leq k-1,\right.$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.z_{k}\in
FS(G_{k})\text{ and }\,p_{i}\in F_{i,f_{i}\left(G_{i-1}\right)}\text{ for
}2\leq i\leq k-1\right\\}\subset A.$

###### Proof.

To avoid the complexity in the calculation we will prove this theorem upto
$k=3.$ The rest of the part can be proved inductively. The technique is
verbatim. Let $p\in E\left(K(\beta\mathbb{N},+)\right)$, $q\in
E((\beta\mathbb{N},+))$ and $A\in p\star_{n}q=E_{1}(n,p)\cdot q.$ Then
$B=\left\\{m_{0}:\left\\{n_{0}:n^{m_{0}}n_{0}\in A\right\\}\in q\right\\}\in
p.$ Let $m_{1}\in B^{\star}$ and
$C_{1}=\left\\{n^{\prime}:n^{m_{1}}n^{\prime}\in A\right\\}\in q.$ Let
$F_{1}\in\mathcal{P}_{f}(\mathbb{P})$. Hence there exists an $IP$ set
$FS(\langle n_{1,i}\rangle_{i\in\mathbb{N}})\subseteq C_{1}\in q.$ For each
$i\in\mathbb{N},$ let $x_{1,i}=n^{m_{1}}n_{1,i}\in A$. Now $FS(\langle
x_{1,i}\rangle_{i\in\mathbb{N}})\subset A.$ By Theorem 3.10, there exists
$a_{1}\in B^{\star}$, and $G_{1}\subset FS(\langle
x_{1,i}\rangle_{i\in\mathbb{N}})$ such that $|G_{1}|=N_{1},$ and

$\left\\{a_{1},a_{1}+p_{1}\left(y_{1}\right):p_{1}\in F_{1},y_{1}\in
FS(G_{1})\cup FP(G_{1})\right\\}\subset B^{\star}.$

So,

$B_{2}=B^{\star}\cap\left(-a_{1}+B^{\star}\right)\cap\bigcap_{p_{1}\in
F_{1}}\bigcap_{y_{1}\in FS(G_{1})\cup
FP(G_{1})}-\left(a_{1}+p_{1}\left(y_{1}\right)\right)+B^{\star}\in p.$

Let

$C_{2}=C_{1}\cap\bigcap_{p_{1}\in F_{1}}\bigcap_{y_{1}\in FS(G_{1})\cup
FP(G_{1})}\left\\{n^{\prime}:n^{a_{1}+p_{1}\left(y_{1}\right)}n^{\prime}\in
A\right\\}\cap\left\\{n^{\prime}:n^{a_{1}}n^{\prime}\in A\right\\}\in q.$

Hence there exists an $IP$ set $FS(\langle
n_{2,i}\rangle_{i\in\mathbb{N}})\subseteq C_{2}\in q.$ For each
$i\in\mathbb{N},$ let $n^{a_{1}}n_{2,i}=x_{2,i}\in A$. Hence $FS(\langle
x_{2,i}\rangle_{i\in\mathbb{N}})\subseteq A.$ And for each $y_{1}\in
FS(G_{1})\cup FP(G_{1}),$ and $y\in FS(\langle
x_{2,i}\rangle_{i\in\mathbb{N}}),$ and $p_{1}\in F_{1}$, we have
$yn^{p_{1}\left(y_{1}\right)}\in A.$ As $FS(\langle
x_{2,i}\rangle_{i\in\mathbb{N}})\subseteq A$ is an $IP$ set, from Theorem
3.10, we can choose a $G_{2}\subset FS(\langle
x_{2,i}\rangle_{i\in\mathbb{N}})$ such that $|G_{2}|=N_{2}$ and

$\left\\{a_{2},a_{2}+p_{2}\left(y_{2}\right):p_{2}\in
F_{2,f_{2}\left(y_{1}\right)},y_{2}\in FS(G_{2})\cup FP(G_{2})\right\\}\subset
B_{2}^{\star}.$

Letting $c=a_{1}+a_{2}$, we have

$\left\\{c+p_{1}\left(y_{1}\right)+p_{2}\left(y_{2}\right):p_{1}\in
F_{1},p_{2}\in F_{2,f_{2}\left(y_{1}\right)},y_{i}\in FS(G_{i})\cup
FP(G_{i})\text{ for }1\leq i\leq 2\right\\}\subset B^{\star}.$

Let

$C_{3}=C_{1}\cap\bigcap_{p_{1}\in F_{1}}\bigcap_{p_{2}\in
F_{2,f_{2}\left(x_{1}\right)}}\bigcap_{1\leq i\leq 2}\bigcap_{y_{i}\in
FS(G_{i})\cup
FP(G_{i})}\left\\{n^{\prime}:n^{c+p_{1}\left(y_{1}\right)+p_{2}\left(y_{2}\right)}n^{\prime}\in
A\right\\}\in q$

and

$B_{3}^{\star}=B^{\star}\cap\bigcap_{p_{1}\in F_{1}}\bigcap_{p_{2}\in
F_{2,f_{2}\left(x_{1}\right)}}\bigcap_{1\leq i\leq 2}\bigcap_{y_{i}\in
FS(G_{i})\cup
FP(G_{i})}-(c+p_{1}\left(x_{1}\right)+p_{2}\left(x_{2}\right))+B^{\star}\in
p.$

Choose an $IP$ set $FS(\langle n_{3,i}\rangle_{i\in\mathbb{N}})\subseteq
C_{3}\in q.$ Again for each $i\in\mathbb{N},$ let $n^{c}n_{3,i}=x_{3,i}\in A$.
So $FS(\langle x_{3,i}\rangle_{i\in\mathbb{N}})\subseteq A.$ Hence for every
$1\leq i\leq 2,$ and $y_{i}\in FS(G_{i})\cup FP(G_{i})$, $z\in FS(\langle
x_{3,i}\rangle_{i\in\mathbb{N}})$, $p_{1}\in F_{1},\text{ and }p_{2}\in
F_{2,f_{2}\left(x_{1}\right)}.$ we have
$zn^{p_{1}\left(y_{1}\right)+p_{2}\left(y_{2}\right)}\in A.$ As $FS(\langle
x_{3,i}\rangle_{i\in\mathbb{N}})$ is an $IP$ set, and $B_{3}^{*}$ is piecewise
syndetic set, we can use Theorem 3.10 as before to find a set $G_{3}\subset
FS(\langle x_{3,i}\rangle_{i\in\mathbb{N}})$ with $|G_{3}|=N_{3}.$

Now iterating this argument we have an infinite sequence of finite subsets
$\langle G_{n}\rangle_{n\in\mathbb{N}}$ such that the desired result is true.
∎

Now we have the following corollary which shows that monochromatic Hindman’s
towers are abundant.

######  Corollary 4.4.

Let $\langle N_{i}\rangle_{i}$ be a sequence in $\mathbb{N}.$ Then for every
finite coloring of $\mathbb{N},$ there exists a sequence of finite sets
$\langle G_{i}\rangle_{i}$ such that $|G_{i}|=N_{i}$ for every
$i\in\mathbb{N},$ such that the pattern

$\left\\{x_{i_{n}}^{x_{i_{n-1}}^{\cdot^{\cdot^{\cdot^{x_{i_{1}}}}}}}:x_{j}\in
FP(G_{j})\,\forall j\in\mathbb{N},\text{ and }1\leq
i_{1}<\cdots<i_{n-1}<i_{n},n\in\mathbb{N}\right\\}$

is monochromatic.

###### Proof.

Let $I:\mathbb{N}\rightarrow\mathbb{N}$ be the polynomial defined by for all
$x\in\mathbb{N}$, $I(x)=x.$ Let $f_{2}(x)=2^{x},$ and define inductively
$f_{n+1}(x)=(2^{x})^{f_{n}(x)}$ for all $n\in\mathbb{N}$. In the above theorem
if we choose these sequence of functions $(f_{n})_{n\in\mathbb{N}}$,
$F_{n,G}=\\{I\\}$ for all $G\in\mathcal{P}_{f}(\mathbb{N}),$ then we have a
sequence of finite sets $\langle G_{i}\rangle_{i\in\mathbb{N}}$ such that for
each $i\in\mathbb{N},$ $G_{i}=\langle n^{x_{i,k}}\rangle_{k=1}^{N_{i}}$ such
that the given pattern is monochromatic. ∎

Another application of Theorem 4.3 gives us a new exponential pattern. To
state this application, let us first introduce the notions of $FP_{n}$ and
$EFP_{n}$ sets.

######  Definition 4.5.

For every $n\in\mathbb{N}_{>1},$ and $F\in\mathcal{P}_{f}(\mathbb{N}),$ define

  * •

$FP_{n}(F)=\\{n^{y}:y\in FS(F)\\}$,

  * •

$EFP_{n}(F)=\\{n^{y}:y\in FP(F)\\}$, and

  * •

$FEFP_{n}(F)=FP_{n}(F)\cup EFP_{n}(F).$

Clearly $FP_{n}(F)=FP(n^{FS(F)}).$ The following corollary is a
straightforward application of Theorem 4.3.

######  Corollary 4.6.

Let $n,l,r\in\mathbb{N}$, and $\mathbb{N}=\cup_{i=1}^{r}A_{i}$ be a $r$\-
coloring of $\mathbb{N}$. Then there exist $i\in\\{1,2,\ldots,r\\}$, and two
finite set sets $F,G$ such that

$FP_{n}(F)\cup FP_{n}(G)\cup\\{y^{z}:y\in FP_{n}(G),z\in FP_{n}(F)\cup
EFP_{n}(F)\\}\subset A_{i}.$

###### Proof.

Choose $G_{1}=F$, and $G_{2}=G$ in Theorem 4.3. ∎

Let $\mathbb{N}=A\cup A^{c}$ be a two coloring such that
$n\in\mathbb{N}_{>1}\cap A$ (resp. in $A^{c}$) implies $n^{n}\in A^{c}$ (resp.
in $A$). Then this greedy construction gives us a two coloring of $\mathbb{N}$
such that none of the color will contain a pattern of the form
$\\{a,a^{a}\\},$ where $a>1.$ But the following corollary of Theorem 4.3 shows
that exponential translations of $a^{a}$ by an infinite set lie in the same
color with $a.$

######  Corollary 4.7.

Let $r\in\mathbb{N},$ and $\mathbb{N}=\cup_{i=1}^{r}A_{i}$ be a finite
coloring of $\mathbb{N}.$ Then there exist $i,j\in\\{1,2,\ldots,r\\}$
(possibly $i\neq j$), $a\in\mathbb{N}$, and a sequence $\langle
x_{n}\rangle_{n=1}^{\infty}$ such that $FS(\langle
x_{n}\rangle_{n=1}^{\infty})\subseteq A_{j}$, and $\\{a,a^{ya}:y\in FS(\langle
x_{n}\rangle_{n=1}^{\infty})\\}\subseteq A_{i}$.

###### Proof.

Let $\mathbb{N}=\cup_{i=1}^{r}B_{i}$ be a $r$ coloring induced by the formulae
$m\in B_{i}\iff n^{m}\in A_{i}.$ Let $I:\mathbb{N}\rightarrow\mathbb{N}$ be
the polynomial defined by $I(x)=x$ for all $x\in\mathbb{N}.$ Let $q\in
E((\beta\mathbb{N},+))$ and $p\in E\left(K(\beta\mathbb{N},+)\right)$ and let
$B_{i}\in p*_{n}q$ for some $i\in\\{1,2,\ldots,r\\}.$ Let $F_{1}=\\{I\\}.$ Now
from the first step of the proof of Theorem 4.3, there exist an element $x$,
and a sequence $\langle z_{n}\rangle_{n=1}^{\infty}$ such that $\\{z\cdot
n^{x}:z\in FS(\langle z_{n}\rangle_{n=1}^{\infty})\\}\subseteq B_{i}$. Using
pigeonhole principle we have a sub $IP$ set $\langle
y_{n}\rangle_{n=1}^{\infty}$ such that $x\cdot FS(\langle
y_{n}\rangle_{n=1}^{\infty})\subset FS(\langle z_{n}\rangle_{n=1}^{\infty}).$
Let $n^{x}=a.$ Then $\\{a,a^{ya}:y\in FS(\langle
y_{n}\rangle_{n=1}^{\infty})\\}\subset A_{i}.$

Again from the Hindman theorem [12], $FS(\langle
y_{n}\rangle_{n=1}^{\infty})\subseteq\cup_{i=1}^{r}A_{i}$ implies there exists
a sub $IP$ set $FS(\langle x_{n}\rangle_{n=1}^{\infty})$ of $FS(\langle
y_{n}\rangle_{n=1}^{\infty})$ such that $FS(\langle
y_{n}\rangle_{n=1}^{\infty})\subseteq A_{j}.$ Whereas $\\{a,a^{ya}:y\in
FS(\langle x_{n}\rangle_{n=1}^{\infty})\\}\subset A_{i}.$ This completes the
proof. ∎

Let $n\in\mathbb{N},$ $\langle N_{i}\rangle_{i\in\mathbb{N}}$ be any sequence
in $\mathbb{N}$, $F_{1}\in\mathcal{P}_{f}(\mathbb{P})$, and
$\varPhi=\left(f_{n}:n\geq 2\right)$ be any sequence of functions
$f_{n}:\mathbb{N}\rightarrow\mathbb{N}.$ For every $n(>1)$ and
$G\in\mathcal{P}_{f}(\mathbb{N})$, let $F_{n,G}$ be defined as in the
hypothesis of Theorem 4.3. Let $\mathcal{L}_{\mathcal{PF}}$ be the collection
of $p\in\beta\mathbb{N}$ such that each $A\in p$ contains an element of the
form $\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}$ for
some sequence $\langle G_{k}\rangle_{k\in\mathbb{N}}$ of finite sets such that
$|G_{k}|=N_{k}$ for each $k\in\mathbb{N}$. Then the following lemma shows that
$\mathcal{L}_{\mathcal{PF}}$ is a left ideal of $(\beta\mathbb{N},\cdot).$

######  Lemma 4.8.

$\mathcal{L}_{\mathcal{PF}}$ is a left ideal of $(\beta\mathbb{N},\cdot).$

###### Proof.

Let $p\in\mathcal{L}_{\mathcal{PF}}$, and $q\in(\beta\mathbb{N},\cdot).$ Let
$A\in q\cdot p.$ Then $B=\\{x:x^{-1}A\in p\\}\in q.$ Choose $b\in B$, and so
$b^{-1}A\in p.$ Let $\langle N_{i}\rangle_{i\in\mathbb{N}}$ be any sequence in
$\mathbb{N}$, $F_{1}\in\mathcal{P}_{f}(\mathbb{P})$, and
$\varPhi=\left(f_{n}:n\geq 2\right)$ be any sequence of functions
$f_{n}:\mathbb{N}\rightarrow\mathbb{N}.$ Then define

  1. 1.

for each $i\geq 2,$ define $\varphi^{\prime}=(f^{\prime}_{i}:i\geq 2)$ by
$f^{\prime}_{i}(x)=f_{i}(bx)$ for all $x\in\mathbb{N}$,

  2. 2.

for each $i\in\mathbb{N}$, $F^{\prime}_{i}\in\mathcal{P}_{f}(\mathbb{P})$ is
defined by $p\in F^{\prime}_{i}$ iff for all $x\in\mathbb{N}$, $p(x)=q(bx)$
for some $q\in F_{i}$ or, $p(x)=r(b^{N_{1}}x)$ for some $r\in F_{i}$.

Now for these new $\varphi^{\prime}$ and $(F^{\prime}_{i}:i\in\mathbb{N})$
there exists a sequence of finite sets $\langle H_{i}\rangle_{i\in\mathbb{N}}$
such that
$\mathcal{PF}^{n}_{k,\varPhi}\left(H_{k}\right)_{k\in\mathbb{N}}\subset
b^{-1}A.$ Let $\langle G_{i}\rangle_{i\in\mathbb{N}}$ be a sequence of finite
sets such that for each $i\in\mathbb{N}$, $G_{i}=\\{by:y\in H_{i}\\}.$ Then we
have $\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\subset
A.$ This completes the proof.

∎

Using Zorn’s lemma we have every left ideal contains a minimal left ideal and
as every minimal left ideal is closed, from Ellis’s theorem we have
$\mathcal{L}_{\mathcal{PF}}$ contains a minimal idempotent. The following
theorem is an application of Lemma 4.8 which shows that we can mix certain
partition regular structures with exponential patterns.

######  Theorem 4.9.

Let $p\in\mathcal{L}_{\mathcal{PF}}$ be a minimal idempotent and $A\in p$.
Then for every linear Rado family $\mathcal{F}$, there exists
$F\in\mathcal{F}$ such that for any sequence $\langle
N_{i}\rangle_{i\in\mathbb{N}}$ in $\mathbb{N}$, sequence of functions
$\varPhi=\left(f_{n}:n\geq 2\right)$ and $F_{1}\in\mathcal{P}_{f}(\mathbb{P})$
we have a sequence of finite sets $(G_{i})_{i\in\mathbb{N}}$ such that

  1. 1.

for each $i\in\mathbb{N}$, $|G_{i}|=N_{i}$, and

  2. 2.

$F\cup\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\cup
F\cdot\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\subseteq
A.$

###### Proof.

As $p\in\mathcal{L}_{\mathcal{PF}}$ is a idempotent, $A\in p$ implies
$A^{*}=\\{x\in A:x^{-1}A\in p\\}.$ As $p$ is minimal, there exists
$G\in\mathcal{F}$ such that $G\subset A^{*}.$ Hence $B=A\cap\bigcap_{x\in
G}x^{-1}A\in p.$ Then $B^{*}=\\{y\in B:y^{-1}B\in p\\}\in p.$ Let $b\in
B^{*}$. Hence $C=b^{-1}B\cap B^{*}\in p.$ Let $(G_{k})_{k}$ be a sequence of
finite sets such that for each $k\in\mathbb{N}$, $|G_{k}|=N_{k}$, and
$\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\subset
C\subseteq A.$ Hence
$b\cdot\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\subset
B.$ Let $F=b\cdot G.$ Then $F\in\mathcal{F}$ and $F\subset A.$ Again
$b\cdot\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\subset\bigcap_{x\in
G}x^{-1}A$, and this implies
$F\cdot\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\subset
A.$ ∎

The proof of the following corollary is similar to the proof of Corollary
4.4333one has to redefine the sequence of function $\varPhi$ accordingly, so
we leave it to the reader.

######  Corollary 4.10.

Let $n(>1)\in\mathbb{N},$ $\langle N_{i}\rangle_{i}$ be a sequence in
$\mathbb{N},$ $\mathcal{F}$ be a linear Rado family,
$p\in\mathcal{L}_{\mathcal{PF}}$ be a minimal idempotent and $A\in
E_{1}(n,p)$. Then there exists $F\in\mathcal{F}$, a sequence of finite sets
$(G_{i})_{i\in\mathbb{N}}$ and a sequence $(x_{n})_{n}$ such that

  1. 1.

for each $i\in\mathbb{N}$, $|G_{i}|=N_{i}$, and

  2. 2.

$n^{F}\cup\left\\{x_{i_{m}}^{x_{i_{m-1}}^{\cdot^{\cdot^{\cdot^{x_{i_{1}}}}}}}:x_{i_{m}}\in
FP_{n}(G_{i_{m}}),x_{i_{j}}\in FEFP_{n}(G_{i_{j}})\text{ for }1\leq j\leq
m-1\right.$

$\,\,\,\,\,\,\,\,\Biggl{.}\text{ and }1\leq
i_{1}<\cdots<i_{m-1}<i_{m},m\in\mathbb{N}\Biggl{\\}}\cup$
$\left\\{x_{i_{m}}^{(a_{k_{m-1}}x_{i_{m-1}})^{\cdot^{\cdot^{\cdot^{(a_{k_{1}}x_{i_{1}})}}}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:x_{i_{m}}\in
FP_{n}(G_{i_{m}}),x_{i_{j}}\in FEFP_{n}(G_{i_{j}})\text{ for }1\leq j\leq
m-1\right.$

$\Biggl{.}\text{ and }1\leq i_{1}<\cdots<i_{m},(a_{k_{j}})_{j=1}^{m-1}\subset
F,m\in\mathbb{N}\Biggl{\\}}\subset A.$

The following example is new and immediately follows from Corollary 4.10. This
example shows the existence of monochromatic Hindman tower mixed with
arithmetic progressions.

######  Example 4.11.

Let $\mathcal{F}$ be the family of finite arithmetic progressions of arbitrary
length. Let $p\in\mathcal{L}_{\mathcal{PF}}$ be a minimal idempotent and $A\in
E_{1}(n,p)$ where $n\in\mathbb{N}_{>1}$. Then for any $l\in\mathbb{N}$, from
Corollary 4.10 we have $F=\\{a,a+d,\ldots,a+ld\\}\in\mathcal{F}$ such that

$\\{n^{a+id}:0\leq i\leq
l\\}\cup\left\\{x_{i_{m}}^{x_{i_{m-1}}^{\cdot^{\cdot^{\cdot^{x_{i_{1}}}}}}}:1\leq
i_{1}<\cdots<i_{m-1}<i_{m},m\in\mathbb{N}\right\\}\cup$
$\left\\{x_{i_{m}}^{((a+j_{m-1}d)x_{i_{m-1}})^{\cdot^{\cdot^{\cdot^{((a+j_{1}d)x_{i_{1}})}}}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:1\leq
i_{1}<\cdots<i_{m-1}<i_{m},\,and\,j_{1},\ldots,j_{m-1}\in\\{0,1,\ldots,l\\},m\in\mathbb{N}\right\\}\subset
A.$

###  Combined multiplicative and exponential structures II

In this section we find the second improvement of Theorem 4.2. As a corollary,
we have a mixed pattern of arithmetic progressions, geometric progressions,
and infinite exponential structures.

######  Theorem 4.12.

Let $F_{1}\in\mathcal{P}_{f}(\mathbb{P})$, $n\in\mathbb{N}_{>1}$, and $\langle
r_{i}\rangle_{i=1}^{\infty}$, $\langle l_{i}\rangle_{i=1}^{\infty}$ be two
sequences in $\mathbb{N}$. Choose $p\in E(\overline{K(\beta\mathbb{N},+)})$,
$q\in K(\beta\mathbb{N},\cdot)$, and $A\in p\star_{n}q$. Let
$\phi=(f_{n}:n\geq 2)$ be a sequence of functions
$f_{n}:\mathbb{N}\longrightarrow\mathbb{N}$. For every
$n(>1),G\in\mathcal{P}_{f}(\mathbb{N})$, let
$F_{n,G}=\\{p\in\mathbb{P}:deg(p)\leq f_{n}(x)\text{ and }coef(p)\leq
f_{n}(x)\text{ for }x=\max G\\}.$

Then for each $m\in\mathbb{N}$, there exists elements $x_{m},y_{m},\langle
x_{i}^{n}\rangle_{i=1}^{r_{m}}$ in $\mathbb{N}$ such that

  1. 1.

$R_{m}=FS(\langle
x_{i}^{m}\rangle_{i=1}^{r_{m}})\cup\\{y_{m},x_{m}+iy_{m}:i=0,\cdots,l_{m}\\}\subset
A$, and

  2. 2.

$\mathcal{FG}_{k,\varPhi}^{n}(R_{k})_{k\in\mathbb{N}}=\left\\{z_{k}\cdot
n^{\sum_{i=1}^{k-1}p_{i}(y_{i})}:k\in\mathbb{N},p_{1}\in F_{1},y_{j}\in
R_{j},\text{ for }1\leq j\leq k-1,z_{k}\in R_{k},\right.\\\
\left.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p_{i}\in
F_{i,f_{i}(R_{i-1})}\text{ for }2\leq i\leq k-1\right\\}\subset A$.

###### Proof.

Similar to the proof of Theorem 4.3, we will prove this theorem upto the case
$k=3.$ The rest of the part can be proved inductively. The technique will be
verbatim.

Let $p\in E(\overline{K(\beta\mathbb{N},+)})$, $q\in K(\beta\mathbb{N},\cdot)$
and $A\in p\star_{n}q$. Let $B=\\{m_{0}:\\{n_{0}:n^{m_{0}}n_{0}\in A\\}\in
q\\}\in p$. Let $m_{1}\in B^{*}$ and
$C_{1}=\\{n^{\prime}:n^{m_{1}}n^{\prime}\in A\\}\in q$. As $C_{1}\in q$, the
set $S_{1}=\\{x:x^{-1}C_{1}\in q\\}$ is syndetic. Given
$l_{1},r_{1}\in\mathbb{N}$. Hence there exist
$a_{1},d_{1},x_{1}^{1},\cdots,x_{r_{1}}^{1}\in\mathbb{N}$ such that
$R_{1}=FS(\langle
x_{i}^{1}\rangle_{i=1}^{r_{1}})\cup\\{a_{1},d_{1},a_{1}+d_{1},\ldots,a_{1}+l_{1}d_{1}\\}\subset
S_{1}$. Hence $D_{1}=C_{1}\cap\bigcap_{r\in R_{1}}r^{-1}C_{1}\in q$. For each
$p\in F_{1}$ and $r\in R_{1}$, define a new polynomial
$p_{r}:\mathbb{Z}\longrightarrow\mathbb{Z}$ by $p_{r}(z)=p(n^{m_{1}}rz)$. Let
$G_{1}=\\{p_{r}:p\in F_{1},r\in R_{1}\\}$ be a new finite set of polynomials
from $\mathbb{P}$. As $B^{*}$ is piecewise syndetic, from Theorem 3.3 we have

$E_{1}=\\{c:\text{ there exists }b\text{ such that }\\{b,b+p(c):p\in
G_{1}\\}\subset B^{*}\\}$

is an $IP_{r_{1}}^{*}$ set for some $r_{1}\in\mathbb{N}$. Hence we have
$E_{1}\cap D_{1}\in q$, and let $n_{1}\in E_{1}\cap D_{1}$. As $d_{1}\in
R_{1},n_{1}d_{1}\in C_{1}$, and so $n^{m_{1}}n_{1}d_{1}=y_{1}\in A.$ Again,

  1. 1.

for $i_{1}\in\\{0,\ldots,l_{1}\\}$,
$n^{m_{1}}n_{1}(a_{1}+i_{1}d_{1})=x_{1}+i_{1}y_{1}\in A$ where
$x_{1}=n^{m_{1}}n_{1}a_{1}$, and

  2. 2.

for $i\in\\{1,\ldots,r_{1}\\}$, $n^{m_{1}}n_{1}FS(\langle
x_{i}^{1}\rangle_{i=1}^{r_{1}})=FS(\langle
z_{i}^{1}\rangle_{i=1}^{r_{1}})\subset A$ where
$z_{i}^{1}=n^{m_{1}}n_{1}x_{i}^{1}$.

Denote $H_{1}=FS(\langle
z_{i}^{1}\rangle_{i=1}^{r_{1}})\cup\\{y_{1},x_{1}+i_{1}y_{1}:i_{1}=0,\cdots,l_{1}\\}$.
As $n_{1}\in E_{1},$ choose $b_{1}\in\mathbb{N}$ such that
$\\{b_{1},b_{1}+p(n_{1}):p\in G_{1}\\}\subset B^{*}$. Now from the choice of
$G_{1}$, we have $\\{b_{1},b_{1}+p(c):p\in F_{1},c\in H_{1}\\}\subset B^{*}$.
Let

$C_{2}=C_{1}\cap\bigcap_{p\in F_{1},c\in
H_{1}}\\{n^{\prime}:n^{b_{1}+p(c)}n^{\prime}\in
A\\}\cap\\{n^{\prime}:n^{b_{1}}n^{\prime}\in A\\}\in q.$

As $C_{2}$ is piecewise syndetic, the set $S_{2}=\\{x:x^{-1}C_{2}\in q\\}$ is
syndetic. For given $l_{2},r_{2}\in\mathbb{N}$, there exist
$a_{2},d_{2},x_{1}^{2},\cdots,x_{r_{2}}^{2}\in\mathbb{N}$ such that
$R_{2}=FS(\langle
x_{i}^{2}\rangle_{i=1}^{r_{2}})\cup\\{a_{2},d_{2},a_{2}+i_{2}d_{2}:i_{2}=0,\cdots,l_{2}\\}\subset
S_{2}$. Hence $D_{2}=C_{2}\cap\bigcap_{s\in R_{2}}s^{-1}C_{2}\in q$. For each
$p\in F_{2,f_{2}(H_{1})}$ and $s\in R_{2}$, define a new polynomial
$p_{s}:\mathbb{Z}\longrightarrow\mathbb{Z}$ by $p_{s}(z)=p(n^{b_{1}}sz)$. Let
$G_{2}=\\{p_{s}:p\in F_{2,f_{2}(H_{1})},s\in R_{2}\\}$ be a new finite set of
polynomials from $\mathbb{P}$. Let

$B_{2}=B^{*}\cap\bigcap_{p\in F_{1},c\in
H_{1}}-(b_{1}+p(c))+B^{*}\cap(-b_{1}+B^{*})\in p.$

Now

$E_{2}=\\{d:\text{there exists }a\text{ such that }\\{a,a+p(d):p\in
G_{2}\\}\subset B_{2}^{*}\\}$

is an $IP_{t}^{*}$ set for some $t\in\mathbb{N}$. Choose $n_{2}\in E_{2}\cap
D_{2}$. For $i\in\\{1,\cdots,r_{2}\\},$ let
$n^{b_{1}}n_{2}x_{i}^{2}=z_{i}^{2}$, $n^{b_{1}}n_{2}a_{2}=x_{2}$, and
$n^{b_{1}}n_{2}d_{2}=y_{2}$. Clearly $H_{2}=FS(\langle
z_{i}^{2}\rangle_{i=1}^{r_{2}})\cup\\{y_{2},x_{2}+i_{2}y_{2}:0\leq i_{2}\leq
l_{2}\\}\subset A$, and $\\{h_{2}n^{p(h_{1})}:h_{2}\in H_{2},h_{1}\in
H_{1},p\in F_{1}\\}\subset A$. Again choose $b_{2}\in\mathbb{N}$ such that
$\\{b_{2},b_{2}+p(n_{2}):p\in G_{2}\\}\subset B_{2}^{*}$. Hence, for all
$p_{1}\in F_{1},p_{2}\in F_{2,f_{2}(H_{1})},h_{1}\in H_{1},$ and $h_{2}\in
H_{2}$, we have $b_{1}+b_{2}+p_{1}(h_{1})+p_{2}(h_{2})\in B^{*}$. Now
proceeding iteratively, we have our desired result. ∎

Before we state some corollaries of Theorem 4.12 let us introduce the
following notion of large set.

######  Definition 4.13.

For given $m,n\in\mathbb{N}$, a $FGP_{m,n}$ set is a set of the form
$FP(\langle
x_{i}\rangle_{i=1}^{m})\cup\\{d\\}\cup\\{ad^{i}:i\in\\{0,1,\ldots,n\\}\\}$ for
some $a,d,\langle x_{i}\rangle_{i=1}^{m}$ in $\mathbb{N}$. That is these sets
consist of an multiplicative $IP_{m}$ set and a set of geometric progressions
of length $n.$

Let $n\in\mathbb{N}_{>1},$ $\langle r_{i}\rangle_{i=1}^{\infty}$, $\langle
l_{i}\rangle_{i=1}^{\infty}$ be two sequences in $\mathbb{N}$. Let
$F_{1}\in\mathcal{P}_{f}(\mathbb{P})$, and $\varPhi=\left(f_{n}:n\geq
2\right)$ be any sequence of functions
$f_{n}:\mathbb{N}\rightarrow\mathbb{N}.$ For every $n\in\mathbb{N}_{>1},$ and
$G\in\mathcal{P}_{f}(\mathbb{N})$, let $F_{n,G}$ be defined as in the
hypothesis of Theorem 4.12. Let $\mathcal{R}_{\mathcal{F}\mathcal{G}}$ be the
collection of $p\in\beta\mathbb{N}$ such that each $A\in p$ contains a set of
the form $\mathcal{FG}_{k,\varPhi}^{n}(R_{k})_{k\in\mathbb{N}}$ for some
sequence of finite sets $(R_{k})_{k\in\mathbb{N}}$, where each $R_{k}$ is a
$FGP_{r_{k},l_{k}}$ set. Then the following lemma shows that
$\mathcal{R}_{\mathcal{FG}}$ is a left ideal of $(\beta\mathbb{N},\cdot).$

######  Lemma 4.14.

$\mathcal{R}_{\mathcal{FG}}$ is a left ideal of $(\beta\mathbb{N},\cdot).$

###### Proof.

The proof is similar to the proof of Lemma 4.8. ∎

The following theorem is an analogue of Theorem 4.9.

######  Theorem 4.15.

Let $F_{1}\in\mathcal{P}_{f}(\mathbb{P})$, $p\in\mathcal{R}_{\mathcal{FG}}$ be
a minimal idempotent and $A\in p$. Then for every linear Rado family
$\mathcal{F}$, there exists $F\in\mathcal{F}$ such that for any two sequences
$\langle r_{i}\rangle_{i=1}^{\infty}$, $\langle l_{i}\rangle_{i=1}^{\infty}$
in $\mathbb{N}$, and sequence of functions $\varPhi=\left(f_{n}:n\geq
2\right)$ we have a sequence of finite sets $(R_{k})_{k\in N}$ such that each
$R_{k}$ is a $FGP_{r_{k},l_{k}}$ set and

$F\cup\mathcal{FG}_{k,\varPhi}^{n}(R_{k})_{k\in\mathbb{N}}\cup
F\cdot\mathcal{FG}_{k,\varPhi}^{n}(R_{k})_{k\in\mathbb{N}}\subset A.$

###### Proof.

The proof is similar to the proof of Theorem 4.9. ∎

The following Corollary of the above theorem immediately extends the Example
4.11.

######  Corollary 4.16.

Let $l\in\mathbb{N}$, $\langle r_{i}\rangle_{i=1}^{\infty}$ and $\langle
l_{i}\rangle_{i=1}^{\infty}$ be two sequences in $\mathbb{N}$. Then for any
finite partition of $\mathbb{N}$, there exists a sequence of finite sets
$\langle R_{i}\rangle_{i=1}^{\infty}$ such that each $R_{i}$ is a
$FGP_{r_{i},l_{i}}$ set, and a set $F=\\{a,a+d,\ldots,a+ld\\}$ such that

$\\{n^{a+id}:0\leq i\leq
l\\}\cup\left\\{x_{i_{n}}^{x_{i_{n-1}}^{\cdot^{\cdot^{\cdot^{x_{i_{1}}}}}}}:x_{j}\in
R_{j}\,\forall j\in\mathbb{N},\text{ and
}1<i_{1}<\cdots<i_{n-1}<i_{n},n\in\mathbb{N}\right\\}\cup$
$\left\\{x_{i_{m}}^{((a+j_{m-1}d)x_{i_{m-1}})^{\cdot^{\cdot^{\cdot^{((a+j_{1}d)x_{i_{1}})}}}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:x_{j}\in
R_{j}\,\forall
j\in\mathbb{N},1<i_{1}<\cdots<i_{m},\,and\,j_{1},\ldots,j_{m-1}\in\\{0,1,\ldots,l\\},m\in\mathbb{N}\right\\}$

is monochromatic.

###### Proof.

The proof is similar to the proof of Corollary 4.4. ∎

The following lemma is a strengthening of [10, Lemma 5.22.].

######  Lemma 4.17.

Let $n\in\mathbb{N}_{>1}$, and $L^{a}$ and $L^{m}$ be both left ideals of
$(\beta\mathbb{N},+)$ and $(\beta\mathbb{N},\cdot).$ Then there exist $p\in
L^{a}$ and $q\in L^{m}$ such that $p\star_{n}q=q.$

###### Proof.

In the proof of [10, Lemma 5.22.], we can choose an idempotent $p\in L^{a}$
using [15, Lemma 2.1]. The rest of the proof is same. ∎

The following corollary shows that every multiplicative piecewise syndetic set
witness the conclusion of Theorem 4.12.

######  Corollary 4.18.

$\overline{K\left(\beta\mathbb{N},\cdot\right)}\subseteq\mathcal{R}_{\mathcal{FG}}.$

###### Proof.

It will be sufficient to prove
$K\left(\beta\mathbb{N},\cdot\right)\subseteq\mathcal{R}_{\mathcal{FG}}.$ Let
$L$ be any minimal left ideal of $(\beta\mathbb{N},\cdot)$. Then from Lemma
4.17, there exists $q\in L$, and $p\in E(\overline{K(\beta\mathbb{N},+)})$
such that $p\star_{n}q=q.$ Then from Theorem 4.12,
$q\in\mathcal{R}_{\mathcal{FG}}$. Now from Lemma 4.14, $L=\beta\mathbb{N}\cdot
q\subset\mathcal{R}_{\mathcal{FG}}$. As $L$ is arbitrary, we have
$K\left(\beta\mathbb{N},\cdot\right)=\bigcup\\{L:L\text{ is a minimal left
ideal of }(\beta\mathbb{N},\cdot)\\}\subseteq\mathcal{R}_{\mathcal{FG}}.$ This
completes the proof. ∎

As a consequence we have for every $n(>1)\in\mathbb{N}$, and
$p\in\overline{K\left(\beta\mathbb{N},\cdot\right)}$, $E_{1}(n,p)$ witness the
pattern of Corollary 4.16. In [3, Theorem 2.4.], Bergelson and Hindman proved
the following combined extension of the additive and multiplicative Hindman
theorem.

######  Theorem 4.19.

If $\mathbb{N}$ is finitely colored, then there exist two sequences $\langle
x_{n}\rangle_{n}$ and $\langle y_{n}\rangle_{n}$ such that $FS(\langle
x_{n}\rangle_{n})\cup FP(\langle y_{n}\rangle_{n})$ is in the same color.

In [10, Theorem 4.5.], the authors extended the above theorem for exponential
patterns. In the following theorem we show that for any finite coloring of
$\mathbb{N}$, there exists a color witnessing the conclusion of Theorem 4.3,
4.12, and 4.19. This simultaneously extends Theorem 4.3, 4.12 and [3, Theorem
2.4.].

######  Theorem 4.20.

Let $r,n(>1)\in\mathbb{N}$, $\langle N_{i}\rangle_{i\in\mathbb{N}}$, $\langle
r_{i}\rangle_{i=1}^{\infty}$ and $\langle l_{i}\rangle_{i=1}^{\infty}$ be
three sequences in $\mathbb{N}$, $F_{1}\in\mathcal{P}_{f}(\mathbb{P})$. Let
$\varPhi=\left(f_{n}:n\geq 2\right)$ be a sequence of functions
$f_{n}:\mathbb{N}\rightarrow\mathbb{N}.$ For every
$n\in\mathbb{N}_{>1},G\in\mathcal{P}_{f}(\mathbb{N})$, let
$F_{n,G}=\\{p\in\mathbb{P}:deg(p)\leq f_{n}(x)\text{ and }coef(p)\leq
f_{n}(x)\text{ for }x=\max G\\}.$ Now for every finite coloring
$\mathbb{N}=\cup_{i=1}^{r}C_{i}$, there exists $j\in\\{1,2,\ldots,r\\},$ two
sequences of finite sets $\langle G_{i}\rangle_{i\in\mathbb{N}}$, and $\langle
R_{i}\rangle_{i\in\mathbb{N}}$, and two sequences $\langle
x_{n}\rangle_{n\in\mathbb{N}}$, and $\langle y_{n}\rangle_{n\in\mathbb{N}}$
such that

  1. 1.

for each $i\in\mathbb{N},$ $|G_{i}|=N_{i}$,

  2. 2.

for each $i\in\mathbb{N},$ $R_{i}$ is a $FGP_{r_{i},l_{i}}$ set, and

  3. 3.

$FS(\langle x_{n}\rangle_{n\in\mathbb{N}})\cup FP(\langle
y_{n}\rangle_{n\in\mathbb{N}})\cup\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\cup\mathcal{FG}^{n}_{k,\varPhi}\left(R_{k}\right)_{k\in\mathbb{N}}\subseteq
C_{j}.$

###### Proof.

Let $p\in E(\beta\mathbb{N},+)$ and $q\in\overline{E(\beta\mathbb{N},+)}$. Let
$r=p\star_{n}q$ and $L_{r}=\beta\mathbb{N}\cdot r$. Then by Lemma 4.8, $L$ is
the left ideal of $(\beta\mathbb{N},\cdot)$, and so contains a minimal left
ideal say $L.$ From Lemma 4.8, each member of $L$ witness the conclusion of
Theorem 4.3. Let $t\in L$ be a minimal idempotent and let $t=s\cdot r=(s\cdot
E_{1}(n,p))\cdot q.$ Let $C_{j}\in t.$ Now from Lemma 4.8, there exists a
sequence of finite sets $\left(G_{k}\right)_{k\in\mathbb{N}}$ satisfying for
each $i\in\mathbb{N},$ $|G_{i}|=N_{i}$, and
$\mathcal{PF}^{n}_{k,\varPhi}\left(G_{k}\right)_{k\in\mathbb{N}}\subset
C_{j}.$ Again $t\in K(\beta\mathbb{N},\cdot),$ hence from Lemma 4.18, there
exists a sequence of finite sets $\left(R_{k}\right)_{k\in\mathbb{N}}$ such
that for each $i\in\mathbb{N},$ $R_{i}$ is a $FGP_{r_{i},l_{i}}$ set, and
$\mathcal{FG}^{n}_{k,\varPhi}\left(R_{k}\right)_{k\in\mathbb{N}}\subseteq
C_{j}.$ As $t$ is an idempotent of $(\beta\mathbb{N},\cdot)$, there exists a
sequence $\langle y_{n}\rangle_{n\in\mathbb{N}}$ such that $FP(\langle
y_{n}\rangle_{n\in\mathbb{N}})\subseteq C_{j}.$ As
$q\in\overline{E(\beta\mathbb{N},+)}$, and $\overline{E(\beta\mathbb{N},+)}$
is a left ideal of $(\beta\mathbb{N},\cdot)$, we have
$t\in\overline{E(\beta\mathbb{N},+)}.$ Hence there exists a sequence $\langle
x_{n}\rangle_{n\in\mathbb{N}}$ such that $FS(\langle
x_{n}\rangle_{n\in\mathbb{N}})\subseteq C_{j}.$ This completes the proof. ∎

## 5 Open questions

In this section, we summarize some questions that we could not solve, as well
as some possible directions we believe could be interesting to pursue in the
future. The first question is regarding the refinement of Corollary 3.11. In
that corollary we are unable to insert both additive and multiplicative
structure in the exponent of $n$. So one may expect the following joint
extension of (1), and (2) of Corollary 3.11 is possible.

######  Question 5.1.

Let $n\in\mathbb{N}_{>1}$, and $\mathbb{N}$ be finitely colored. Does there
exists $x,y,\text{ and }z\in\mathbb{N}$ such that
$\\{n^{x},n^{y},n^{z},n^{yz},n^{y+z},(n^{x})^{n^{y+z}},(n^{x})^{n^{yz}}\\}$ is
monochromatic?

But if the above question has an affirmative answer, then this would imply
that the pattern $\\{x,y,x+y,xy\\}$ is partition regular. But this is a long
standing open problem. For recent development in this area, one can look into
the articles [1, 2, 6, 7, 16, 17]. But we expect a relatively weak version of
the Question 5.2 is true. So we make the following conjecture.

######  Conjecture 5.2.

Let $n\in\mathbb{N}_{>1}$, and $\mathbb{N}$ be finitely colored. Then there
exists $x,y,\text{ and }z\in\mathbb{N}$ such that
$\\{n^{x},n^{yz},n^{y+z},(n^{x})^{n^{y+z}},(n^{x})^{n^{yz}}\\}$ is
monochromatic. In other words the pattern $\\{x,y+z,yz,xn^{y+z},xn^{yz}\\}$ is
partition regular.

The second question is regarding Theorem 4.3. In Theorem 4.3, we proved that
for each $k\in\mathbb{N}$, $z_{k}\in FS(G_{k})$. But there is a drawback when
we compare it with Theorem 4.12. Because in the later case, we were able to
show $z_{k}\in R_{k}$. This immediately requires a refinement of Theorem 4.3,
asking whether $z_{k}\in FS(G_{k})\cup FP(G_{k})?$ So we leave the following
question open.

######  Question 5.3.

Let $n\in\mathbb{N}_{>1},$ $\langle N_{i}\rangle_{i\in\mathbb{N}}$ be a
sequence in $\mathbb{N}$, $F_{1}\in\mathcal{P}_{f}(\mathbb{P})$. Let
$\varPhi=\left(f_{n}:n\geq 2\right)$ be a sequence of functions
$f_{n}:\mathbb{N}\rightarrow\mathbb{N}.$ For every
$n(>1),G\in\mathcal{P}_{f}(\mathbb{N})$, let
$F_{n,G}=\\{p\in\mathbb{P}:deg(p)\leq f_{n}(x)\text{ and }coef(p)\leq
f_{n}(x)\text{ for }x=\max G\\}.$ Now for every finite coloring of
$\mathbb{N}$, does there exists a sequence of finite sets $\langle
G_{i}\rangle_{i\in\mathbb{N}}$ such that

  1. 1.

for each $i\in\mathbb{N},$ $|G_{i}|=N_{i}$, and

  2. 2.

$\left\\{y_{k}\cdot
n^{\sum_{i=1}^{k-1}p_{i}\left(y_{i}\right)}:k\in\mathbb{N},p_{1}\in
F_{1},y_{j}\in FS(G_{j})\cup FP(G_{j})\text{ for }1\leq j\leq k,\right.$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.\text{
and }\,p_{i}\in F_{i,f_{i}\left(G_{i-1}\right)}\text{ for }2\leq i\leq
k-1\right\\}$

is monochromatic?

While proving Corollary 4.18, we used the property: for any $q\in
K(\beta\mathbb{N},\cdot)$, and some suitably chosen $p\in\beta\mathbb{N}$,
$p\star_{n}q$ witness the conclusion of Theorem 4.12. But we could not prove
Theorem 4.3 for any $q\in K(\beta\mathbb{N},\cdot)$. That’s why the technique
of the proof of Corollary 4.18 can’t be adapted for the case of
$\mathcal{L}_{\mathcal{PF}}$. So we leave the following question open.

######  Question 5.4.

Does
$\overline{K\left(\beta\mathbb{N},\cdot\right)}\subseteq\mathcal{L}_{\mathcal{PF}}?$

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