2 A new structure and Stone–Čech compactification

Let $\Sigma = \{a^2 + b^2 : a, b \in \mathbb {N}_0\} \subseteq \mathbb {N}_0$ denote the set of nonnegative integers expressible as sums of two squares. Under standard multiplication, $\Sigma $ forms a commutative semigroup, as the product of two elements in $\Sigma $ remains in $\Sigma $ by the identity

$$ \begin{align*} (a^2 + b^2)(c^2 + d^2) = (ad - bc)^2 + (ac + bd)^2. \end{align*} $$

We order $\Sigma $ increasingly as

$$ \begin{align*} \Sigma = \{s_0 < s_1 < s_2 < s_3 < \cdots\}; \end{align*} $$

the first few elements are $s_0 = 0$ , $s_1 = 1$ , $s_2 = 2$ , $s_3 = 4$ , $s_4 = 5$ , $s_5 = 8$ , $s_6 = 9$ , $s_7 = 10$ , $s_8 = 13$ , $s_9 = 16$ , $s_{10} = 17$ , $s_{11} = 18$ , $s_{12} = 20$ , $s_{13} = 25$ , $s_{14} = 26$ , $s_{15} = 29$ , $s_{16} = 32$ .

For $s \in \Sigma $ , define the predecessor set $\Sigma _{<s} = \{y \in \Sigma : y < s\}$ . Let $g: \Sigma \to \mathbb {N}_0$ be the function

$$ \begin{align*} g(s) = |\Sigma_{<s}|, \end{align*} $$

which counts the number of elements in $\Sigma $ strictly less than s; g is a bijection, with inverse $f: \mathbb {N}_0 \to \Sigma $ given by $f(n) = s_n$ , the nth term in the ordered sequence of $\Sigma $ .

Definition 2.1 (Induced operation)

The operation $\ast _f$ on $\mathbb {N}_0$ is defined by

$$ \begin{align*} m \ast_f n = g(f(m) \cdot f(n)) = |\Sigma_{<s_m \cdot s_n}|. \end{align*} $$

Define the power $x^{(n)}$ in $(\mathbb {N}_0, \ast _f)$ recursively by:

• • $x^{(0)} = 1$ ;

• • $x^{(n)} = x \ast _f x^{(n-1)} = x \ast _f x \ast _f \cdots \ast _f x = |\Sigma _{<s_x^n}|$ .

We briefly recall the algebraic structure of the Stone–Čech compactification $\beta S$ for a discrete semigroup $(S, \cdot )$ . The elements of $\beta S$ are the ultrafilters on S. By identifying the principal ultrafilters with the points of S, we may regard S as a subset of $\beta S$ , that is, $S \subseteq \beta S$ .

For a subset $A \subseteq S$ , define

$$ \begin{align*} \overline{A} = \{p \in \beta S \mid A \in p\}. \end{align*} $$

The collection $\{\overline {A} \mid A \subseteq S\}$ forms a basis for a topology on $\beta S$ . The semigroup operation $\cdot $ on S can be extended to $\beta S$ , making $(\beta S, \cdot )$ a compact right topological semigroup. This means:

• • for any $p \in \beta S$ , the right translation map $\rho _p: \beta S \to \beta S$ defined by $\rho _p(q) = q \cdot p$ is continuous;

• • the semigroup S is contained in the topological centre of $\beta S$ , meaning that for any $x \in S$ , the left translation map $\lambda _x: \beta S \to \beta S$ defined by $\lambda _x(q) = x \cdot q$ is continuous.

The extended operation on $\beta S$ is characterised as follows: for $p, q \in \beta S$ and $A \subseteq S$ ,

$$ \begin{align*} A \in p \cdot q \quad\mbox{if and only if}\quad \{x \in S \mid x^{-1} A \in q\} \in p, \end{align*} $$

where $x^{-1} A = \{y \in S \mid x \cdot y \in A\}$ . A fundamental result due to Ellis states that every compact right topological semigroup contains an idempotent element. Thus, there exists an idempotent ultrafilter $p \in \beta S$ satisfying $p \cdot p = p$ .

For a sequence $\langle x_n \rangle _{n=1}^\infty $ in $(S, \cdot )$ , the set of finite products is defined by

$$ \begin{align*} \textit{FP}(\langle x_n \rangle_{n=1}^\infty) = \{x_{n_1} \cdot x_{n_2} \cdots x_{n_k} \mid n_1 < n_2 < \cdots < n_k\}. \end{align*} $$

Idempotent ultrafilters play a crucial role in Ramsey theory. This is exemplified by Galvin’s theorem, which states that for any semigroup $(S, \cdot )$ and an idempotent ultrafilter $p = p \cdot p$ in $(\beta S, \cdot )$ , every set $A \in p$ contains the set of finite products $\textit {FP}(\langle x_n \rangle _{n=1}^\infty )$ for some sequence $\langle x_n \rangle _{n=1}^\infty $ in S. For further details on the Stone–Čech compactification and its applications in Ramsey theory, see [Reference Hindman and Strauss16, Reference Todorcevic21].

Since $(\mathbb {N}_0, \ast _f)$ is a discrete commutative semigroup, its Stone–Čech compactification $(\mathbb {N}_0, \ast _f)$ gives rise to numerous Ramsey-type results in $\mathbb {N}_0$ induced by the operation $\ast _f$ .

3 Monochromatic configurations

In this section, we introduce new monochromatic configurations within the semigroup $(\mathbb {N}, \ast _f)$ , which encodes information about sums of two squares. We begin with the following definition.

For any infinite sequence of natural numbers $\langle x_n \rangle _{n=1}^{\infty }$ , the corresponding set of finite sums is denoted and defined by

$$ \begin{align*}\textit{FS}(\langle x_n\rangle_{n=1}^{\infty}) = \{ x_{n_1} + x_{n_2} + \cdots + {x_{n_k}} : n_1 < n_2 < \cdots < n_k \}.\end{align*} $$

A fundamental result in arithmetic Ramsey theory due to Hindman guarantees the existence of infinite monochromatic patterns within such sets of finite sums.

A similar result holds when considering finite products instead of finite sums and, more generally, in any semigroup [Reference Hindman and Strauss16, Theorem 5.8]. Analogously, for the semigroup $(\mathbb {N}_0, \ast _f)$ , we define the set of finite $\ast _f$ -operations by

$$ \begin{align*} \mathfrak{FP}_f(\langle x_n \rangle_{n=1}^{\infty}) & = \{x_{n_1} \ast_f x_{n_2} \ast_f \cdots \ast_f x_{n_k} : n_1 < n_2 < \cdots < n_k\} \\ & = \{|\Sigma_{< s_{x_{n_1}} s_{x_{n_2}} \cdots s_{x_{n_k}}}| : n_1 < n_2 < \cdots < n_k\}, \end{align*} $$

where $\Sigma $ denotes the set of sums of two squares.

Theorem 3.2. For any $r \geq 1$ and any finite r-colouring of the nonnegative integers, $\mathbb {N}_0 = C_1 \cup C_2 \cup \cdots \cup C_r$ , there exist a colour $C_i$ and a sequence $\langle x_n \rangle _{n=1}^{\infty }$ in $\mathbb {N}$ such that

$$ \begin{align*}\mathfrak{FP}_f(\langle x_n \rangle_{n=1}^{\infty}) = |\Sigma_{< s_{x_{n_1}} s_{x_{n_2}} \cdots s_{x_{n_k}}}| : n_1 < n_2 < \cdots < n_k\} \subseteq C_i.\end{align*} $$

A cornerstone result in arithmetic Ramsey theory is van der Waerden’s theorem (1927), which guarantees the existence of arbitrarily long monochromatic arithmetic progressions. A year later, Brauer strengthened this result by showing that one can also ensure that the common difference belongs to the same colour class as the elements of the progression.

We establish an analogous version of Brauer’s theorem in our setting.

Theorem 3.5. For every finite colouring $\mathbb {N}_0 = C_1 \cup C_2 \cup \cdots \cup C_r$ and for every ${k \in \mathbb {N}}$ , there exist a colour $C_i$ and elements $x, z \in \mathbb {N}$ such that

$$ \begin{align*}\{x, z, |\Sigma_{<{s_x^j}s_z}|: j=1,2, \ldots, k\} \subseteq C_i.\end{align*} $$

Deuber established a result concerning the generalised partition regularity of homogeneous systems of linear Diophantine equations. In particular, he demonstrated the partition regularity of the so-called $(m,p,c)$ -sets.

We present an analogous version of Deuber’s theorem in our context.

Theorem 3.7. Let $m, p, r \in \mathbb {N}$ . For every r-colouring $\mathbb {N}_0 = C_1 \cup C_2 \cup \cdots \cup C_r$ , there exists a colour $C_i$ and elements $x_0, x_1, \ldots , x_m \in C_i$ such that

$$ \begin{align*} \{|\Sigma_{<{s_{x_0}^{n_0}}{s_{x_1}^{n_1}} \cdots {s_{x_{j-1}}^{n_{j-1}}} {s_{x_j}}}| : n_0, n_1, \ldots, n_{j-1} \in \{0,1, \ldots, p\} \mbox{ and } j=1,2, \ldots, m \} \subseteq C_i .\end{align*} $$

Proof. We prove this theorem using a generalisation of Deuber’s theorem for commutative semirings, recently established by Bergelson et al. in [Reference Bergelson, Johnson and Moreira7, Corollary 3.7]. This result can be stated as follows.

Let $(S, \ast )$ be a commutative semigroup and, for each $j=1,2, \ldots , m$ , let $\mathfrak {F}_j$ be a finite set of endomorphisms $\psi : S^j \to S$ . Then, for every r-colouring $S=C_1 \cup C_2 \cup \cdots \cup C_r$ , there exist a colour $C_i$ and elements $x_0, x_1, \ldots , x_m$ distinct from the identity such that $x_0 \in C_i$ and

$$ \begin{align*} \psi(x_0, x_1, \ldots, x_{j-1}) \ast x_j \in C_i \quad \mbox{ for every } j=1,2, \ldots, m \mbox{ and for every } \psi \in \mathfrak{F}_j.\end{align*} $$

This statement generalises Theorem 3.6 for the case $c=1$ . We apply the Bergelson–Johnson–Moreira result with $(S, \ast ) = (\mathbb {N}_0, \ast _f)$ . For each j-tuple $\bar {n}=(n_0, n_1, \ldots , n_{j-1}) \in (\mathbb {N}_0)^j$ , define

$$ \begin{align*} \psi_{\bar{n}} : (x_0, x_1, \ldots, x_{j-1})\mapsto {x_0}^{(n_0)} \ast_f {x_1}^{(n_1)} \ast_f \cdots \ast_f {x_j}^{(n_j)}. \end{align*} $$

Since $(\mathbb {N}_0, \ast _f)$ is a commutative semigroup, each $\psi _{\bar {n}}$ is a semigroup homomorphism. Define

$$ \begin{align*} \mathfrak{F}_j= \{\psi_{\bar{n}} : \mathbb{N}^j \rightarrow \mathbb{N} : \bar{n}=(n_0, n_1, \ldots, n_{j-1}) \in \{0,1,2, \ldots, p\}^j\}, \end{align*} $$

which consists of homomorphisms for each $j{\kern-1pt}={\kern-1pt}1,{\kern-1pt}2, \ldots , m$ . By the Bergelson–Johnson–Moreira theorem, for every finite colouring $\mathbb {N}= C_1 \cup C_2 \cup \cdots \cup C_r$ , there exist a colour $C_i$ and elements $x_0, x_1, \ldots , x_m$ distinct from the identity such that:

• • $x_0 \in C_i$ ;

• • $\psi _{\bar {n}}(x_0, x_1, \ldots , x_{j-1}) \ast _f x_j \in C_i$ for every $j=1,2, \ldots , m$ and all $\bar {n} \in \{0,1,2, \ldots , p\}^j$ .

From the second condition, we deduce that

$$ \begin{align*} \psi_{\bar{n}}(x_0,x_1, \ldots, x_{j-1}) \ast_f x_j = {x_0}^{(n_0)} \ast_f {x_1}^{(n_1)} \ast_f \cdots \ast_f {x_j}^{(n_j)} \ast_f x_j \in C_i. \end{align*} $$

Therefore, for all $(n_0, n_1, \ldots , n_{j-1}) \in \{0,1, \ldots , p\}^j \mbox { and } j=1,2, \ldots , m$ , we obtain

$$ \begin{align*}|\Sigma_{<{s_{x_0}^{n_0}}{s_{x_1}^{n_1}} \cdots {s_{x_{j-1}}^{n_{j-1}}} {s_{x_j}}}| \in C_i.\end{align*} $$

Hence, the result follows.

Proof of Theorem 3.5

The proof follows from the Bergelson–Johnson–Moreira theorem, as in Theorem 3.7. For $j=0,1,2, \ldots , k+1$ , let $\psi _j : (\mathbb {N}, \ast _f) \to (\mathbb {N}, \ast _f)$ be the endomorphism defined by $\psi _j(x) = x^{(j)}$ . Then, for every r-colouring $\mathbb {N}= C_1 \cup C_2 \cup \cdots \cup C_r$ , there exist a colour $C_i$ and elements $x, y \neq 1$ such that

$$ \begin{align*} \{x, \psi_0(x) \ast_f y, \psi_1(x) \ast_f y, \psi_2(x) \ast_f y , \ldots, \psi_{k+1}(x) \ast_f y\} \subseteq C_i. \end{align*} $$

Setting $z:= x \ast _f y$ , we obtain

$$ \begin{align*} \{x, z, x\ast_f z, x^{(2)} \ast_f z, x^{(3)} \ast_f z , \ldots, x^{(k)} \ast_f z \} = \{x, z, |\Sigma_{<s_x^js_z}|: j=1,2, \ldots, k\} \subseteq C_i. \end{align*} $$

Hence, the result follows.

4 Milliken–Taylor theorem

The Milliken–Taylor theorem simultaneously generalises Hindman’s finite sums theorem and Ramsey’s theorem. It has been widely applied in combinatorial number theory, including extensions of Szemerédi’s theorem on arithmetic progressions.

To state the theorem, we first establish some notation. Let $\mathcal {P}_f(\mathbb {N})$ denote the set of all finite nonempty subsets of $\mathbb {N}$ . For $m \in \mathbb {N}$ , define $[\mathbb {N}]^m = \{A \subseteq \mathbb {N} : |A| = m\}$ . For $F, G \in \mathcal {P}_f(\mathbb {N})$ , we write $F < G$ if $\max F < \min G$ .

Theorem 4.1 (Milliken–Taylor theorem [Reference Milliken17, Reference Taylor20])

Let $r, m \in \mathbb {N}$ and let $[\mathbb {N}]^m = C_1 \cup \cdots \cup C_r$ be an r-colouring. There exist an injective sequence $\langle x_n \rangle _{n=1}^\infty $ in $\mathbb {N}$ and a colour $C_i$ such that

$$ \begin{align*} \{(x_{F_1}, \ldots, x_{F_m}) : F_1 < \cdots < F_m\} \subseteq C_i, \end{align*} $$

where for $F = \{n_1 < \cdots < n_k\} \in \mathcal {P}_f(\mathbb {N})$ , we define $x_F = \sum _{i=1}^k x_{n_i}$ .

4.1 Ultrafilter formulation

The theorem admits an ultrafilter-theoretic formulation using tensor products. Let $(S_i, \cdot )$ be semigroups and $p_i \in \beta S_i$ ultrafilters. The tensor product $\bigotimes _{i=1}^k p_i \in \beta (\times _{i=1}^k S_i)$ is defined inductively:

• • for $k=1$ , $\bigotimes _{i=1}^1 p_i = p_1$ ;

• • for $k \geq 1$ , $A \in \bigotimes _{i=1}^{k+1} p_i$ if and only if

  $$ \begin{align*} \{(x_1, \ldots, x_k) \in \times_{i=1}^k S_i : \{x_{k+1} \in S_{k+1} : (x_1, \ldots, x_{k+1}) \in A\} \in p_{k+1}\} \in \bigotimes_{i=1}^k p_i. \end{align*} $$

Theorem 4.3 (Bergelson–Hindman–Williams [Reference Bergelson, Hindman and Williams6])

Let S be a semigroup, $m \in \mathbb {N}$ and $A \subseteq \times _{i=1}^m S$ . The following statements are equivalent.

• • There exists a sequence $\langle x_n \rangle _{n=1}^\infty $ in S with $\{(x_{F_1}, \ldots , x_{F_m}) : F_1 < \cdots < F_m\} \subseteq A$ .

• • There exists an idempotent ultrafilter $p \in \beta S$ such that $A \in \bigotimes _{i=1}^m p$ .

For a function $\phi : X \to Y$ , the induced map $\phi _*: \beta X \to \beta Y$ is defined by $\phi _*(p) = \{B \subseteq Y : \phi ^{-1}(B) \in p\}$ . Note that $A \in p$ implies $\phi (A) \in \phi _*(p)$ , but the converse fails.

Corollary 4.4. Let $(S, \cdot )$ be a semigroup, $m \in \mathbb {N}$ , $\phi : S^m \to S$ and $B \subseteq S$ . The following statements are equivalent.

• • There exists a sequence $\langle x_n \rangle _{n=1}^\infty $ in S with $\{\phi (x_{F_1}, \ldots , x_{F_m}) : F_1 < \cdots < F_m\} \subseteq B$ .

• • $B \in \phi _*(\bigotimes _{i=1}^m p)$ for some idempotent $p \in \beta S$ .

Proof. Apply Theorem 4.3 with $A = \phi ^{-1}(B)$ . For details, see [Reference Di Nasso12, Corollary 5.4].

Theorem 4.5. Let $r, m \in \mathbb {N}$ , $\phi : (\mathbb {N}_0)^m \to \mathbb {N}_0$ and $\mathbb {N}_0 = C_1 \cup \cdots \cup C_r$ be an r-colouring. There exists a sequence $\langle x_n \rangle _{n=1}^\infty $ in $\mathbb {N}$ and a colour $C_i$ such that

$$ \begin{align*} \{\phi(|\Sigma_{<F_1}|, \ldots, |\Sigma_{<F_m}|) : F_1 < \cdots < F_m \} \subseteq C_i,\end{align*} $$

where for $F_j = \{n_{j_1} < \cdots < n_{j_{k_j}}\} \in \mathcal {P}_f(\mathbb {N})$ , we define $\Sigma _{<F_j} = \Sigma _{<s_{x_{n_{j_1}}} \cdots s_{x_{n_{j_{k_j}}}}}$ .

Proof. Let $(S, \cdot ) = (\mathbb {N}_0, \ast _f)$ , where $\ast _f$ is defined by $x \ast _f y = |\Sigma _{<s_x s_y}|$ . Let $p \in \beta \mathbb {N}_0$ be an idempotent ultrafilter. By Corollary 4.4, with $q = \phi _*(\bigotimes _{i=1}^m p)$ , there exist $C_i \in q$ and a sequence $\langle x_n \rangle _{n=1}^{\infty }$ such that

$$ \begin{align*} \{\phi(x_{F_1}, \ldots, x_{F_m}) : F_1 < \cdots < F_m\} \subseteq C_i. \end{align*} $$

For $F_j = \{n_{j_1} < \cdots < n_{j_{k_j}}\}$ , compute

$$ \begin{align*} x_{F_j} = x_{n_{j_1}} \ast_f \cdots \ast_f x_{n_{j_{k_j}}} = |\Sigma_{<s_{x_{n_{j_1}}} \cdots s_{x_{n_{j_{k_j}}}}}| = |\Sigma_{<F_j}|. \end{align*} $$

Substituting into $\phi $ yields the result.

5 Application of the Hales–Jewett theorem

We start this section with the definition of a partial semigroup.

Definition 5.1. A partial semigroup is a triple $(S, X \subseteq S \times S, \ast )$ of a set S, a subset $X \subseteq S \times S$ and an operation $\ast $ defined on X, satisfying

$$ \begin{align*} (x \ast y) \ast z = x \ast (y \ast z) \quad \mbox{for all } x,y,z \in G, \end{align*} $$

in the sense that if either side is defined, so is the other and they are equal.

Let $\mathcal {A}$ be a nonempty, finite set called the alphabet and $v \notin \mathcal {A}$ be a symbol called a variable. A located word in the alphabet $\mathcal {A}$ is a finitely supported function $b : \mathrm {Dom}(b) \rightarrow \mathcal {A}$ , where $\mathrm {Dom}(b)$ is a (possibly empty) finite subset of $\mathbb {N}_0.$ Similarly, a located variable word in the alphabet $\mathcal {A}$ for a variable v is a finitely supported function $b : \mathrm {Dom}(b) \rightarrow \mathcal {A} \cup \{v\}$ whose range contains v. Let $L(\mathcal {A})$ be the set of located words in $\mathcal {A}$ and let $L(\mathcal {A} v)$ be the set of located variable words in $\mathcal {A}$ for the variable $v.$ Then, $S:= L(\mathcal {A}) \cup L(\mathcal {A} v)$ has a natural partial semigroup operation, obtained by letting $b_0 \ast b_1$ be defined whenever the domains of $b_0$ and $b_1$ are disjoint. In such a case, $b_0 \ast b_1$ is just $b_0 \cup b_1.$

In 2008, M. Beiglböck extended the Hales–Jewett theorem, obtaining a result stronger than the one above. His extension involves partition regular families of $ \mathbb {N}$ . A partition regular family $ \mathcal {F} $ of $ \mathbb {N} $ is a subset of $ \mathcal {P}_f(\mathbb {N}) $ such that for any partition $ \mathbb {N} = C_1 \cup \cdots \cup C_r $ , there exists an $ i \in \{1, \ldots , r\} $ such that $ C_i \in \mathcal {F} $ .

Theorem 5.3 (Beiglböck, 2008)

Let $ \mathcal {F} $ be a partition regular family of finite subsets of $ \mathbb {N} $ that contains no singletons and let $ \mathcal {A} $ be a finite alphabet set. For any finite colouring of $ L(\mathcal {A}) $ , there exist $ \alpha \in L(\mathcal {A}) $ , $ \gamma \in \mathcal {P}_f(\mathbb {N}) $ and $ F \in \mathcal {F} $ such that $ \mathrm {Dom}(\alpha ) $ , $ \gamma $ and $ F $ are pairwise disjoint sets, and

$$ \begin{align*} \{\alpha \cup (\gamma \cup \{t\} \times \{s\}) : s \in \mathcal{A}, t \in F\} \end{align*} $$

is monochromatic.

5.1 Geo-arithmetic progressions

Bergelson proved that for any finite colouring of $ \mathbb {Z} $ , there exists a monochromatic geo-arithmetic progression of arbitrary length. This result can be viewed as a combined extension of the additive and multiplicative versions of van der Waerden’s theorem. Bergelson initially proved this property using ergodic theory [Reference Bergelson4]. Later, Beiglböck et al. [Reference Beiglböck, Bergelson, Hindman and Strauss3] provided an alternative proof using the algebra of the Stone–Čech compactification.

Theorem 5.4 (Geo-arithmetic progression)

If $ n, r \in \mathbb {N} $ and $ \mathbb {Z} $ is $ r $ -coloured, then there exist $ a, b, d \in \mathbb {N} $ such that the set $ \{a \cdot (b + i \cdot d)^j : 0 \leq i, j \leq n\} $ is monochromatic.

In [Reference Beiglböck2], Beiglböck demonstrated that the extension of the Hales–Jewett theorem is sufficiently strong to imply Theorem 5.4. We will use this variant of the Hales–Jewett theorem (Theorem 5.3) to prove a geo-arithmetic result in our structure.

Theorem 5.5. If $ k, r \in \mathbb {N} $ and $ \mathbb {N} $ is $ r $ -coloured, then there exist $ a, b, d \in \mathbb {N} $ and $ {\gamma \in \mathcal {P}_f(\mathbb {N}) }$ such that the set

$$ \begin{align*} \{|\Sigma_{<s_b (\prod_{t \in \gamma} s_t \cdot s_{a + i d})^j}| : i, j = 0, 1, 2, \ldots, k\} \end{align*} $$

is monochromatic, where $ \Sigma $ is the set of sums of two squares.

Proof. Assume that $ \mathbb {N} $ is finitely coloured. Fix $ k \in \mathbb {N} $ and consider the set $ {\mathcal {F} = \{\{a, a + d, \ldots , a + k d\} : a, d \in \mathbb {N}\} }$ of all $ (k + 1) $ -term arithmetic progressions. Let $ \mathcal {A} = \{0, 1, \ldots , k\} $ and define the function

$$ \begin{align*} h : L(\mathcal{A}) \rightarrow \mathbb{N} \quad \text{by} \quad h(\alpha) = \underset{t \in \mathrm{Dom}(\alpha)}{\ast_f} t^{(\alpha(t))}. \end{align*} $$

We colour each $ \alpha \in L(\mathcal {A}) $ with the colour of $ h(\alpha ) $ , and choose $ \alpha $ , $ \gamma $ and ${F = \{a, a + d, \ldots , a + k d\}}$ .

By Theorem 5.3, the set

$$ \begin{align*} \{h(\alpha \cup (\gamma \cup \{a + i d\}) \times \{j\}) : i, j \in \{0, 1, \ldots, k\}\} \end{align*} $$

is monochromatic. Expanding $ h $ , we obtain

$$ \begin{align*} h(\alpha \cup (\gamma \cup \{a + i d\}) \times \{j\}) & = \Big(\underset{t \in \mathrm{Dom}(\alpha)}{\ast_f} t^{(\alpha(t))}\Big) \ast_f \Big(\underset{t \in \gamma}{\ast_f} t^{(j)}\Big) \ast_f (a + i d)^{(j)} \\ & = |\Sigma_{<(\prod_{t \in \mathrm{Dom}(\alpha)} s_t^{\alpha(t)}) (\prod_{t \in \gamma} s_t \cdot s_{a + i d})^j}|. \end{align*} $$

Let $ s_b = \prod _{t \in \mathrm {Dom}(\alpha )} s_t^{\alpha (t)} $ . Then, the set

$$ \begin{align*} \{|\Sigma_{<s_b (\prod_{t \in \gamma} s_t \cdot s_{a + i d})^j}| : i, j \in \{0, 1, \ldots, k\}\} \end{align*} $$

is monochromatic, as required.

5.2 Polynomial van der Waerden’s theorem

The polynomial extension of van der Waerden’s theorem relies on the polynomial version of the Hales–Jewett theorem. In 1988, Bergelson and Liebman proved the polynomial extension of the Hales–Jewett theorem by introducing and developing the apparatus of set-polynomials (polynomials whose coefficients are finite sets), and applying the methods of topological dynamics [Reference Bergelson and Leibman8]. Later, Walters provided short and purely combinatorial proofs of those results in [Reference Walters23]. Let us begin with the statement of the polynomial Hales–Jewett theorem, along with some relevant notation.

For fixed positive numbers $ q, N, d $ , consider the set $ X(q, N, d) = \prod _{i=1}^d [q]^{N^i} $ , where $ [q] = \{1, \ldots , q\} $ . An element $ x \in X(q, N, d) $ is of the form $ (\mathbf {b}_1, \ldots , \mathbf {b}_d) $ , where $ {\mathbf {b}_i : [N]^i \to [q] }$ . For $ a = (\mathbf {a}_1, \ldots , \mathbf {a}_d) $ , $ \gamma \subseteq [N] $ and $ (x_1, \ldots , x_d) \in [q]^d $ , define an element $ x = a \oplus x_1 \gamma \oplus \cdots \oplus x_d \gamma ^d $ as follows: if $ x = (\mathbf {b}_1, \ldots , \mathbf {b}_d) $ , then

$$ \begin{align*} \mathbf{b}_j((i_1, \ldots, i_j)) = \begin{cases} x_j & \text{if } (i_1, \ldots, i_j) \in \gamma^j, \\ \mathbf{a}_j((i_1, \ldots, i_j)) & \text{otherwise.} \end{cases} \end{align*} $$

Theorem 5.6 (Polynomial Hales–Jewett (PHJ) theorem)

For any $ q, k, d \in \mathbb {N} $ , there exists $ N \in \mathbb {N} $ such that whenever $ X(q, N, d) $ is $ k $ -coloured, there exist $ a \in X(q, N, d) $ and $ \gamma \subseteq [N] $ such that the set

$$ \begin{align*} \{ a \oplus x_1 \gamma \oplus \cdots \oplus x_d \gamma^d : (x_1, \ldots, x_d) \in [q]^d \} \end{align*} $$

is monochromatic.

We will use Theorem 5.6 to derive our version of the polynomial van der Waerden theorem, which is stated next.

Theorem 5.7. Let $ d, k, \ell \in \mathbb {N} $ and $ \{F_1, \ldots , F_\ell \} \subset \mathcal {P}_f(\mathbb {N}) $ with $ F_i = \{a_{i1}, \ldots , a_{id}\} $ for all $ i \in \{1, \ldots , \ell \} $ . Then, for any $ k $ -colouring of $ \mathbb {N} $ , there exist $ b, c \in \mathbb {N} $ such that the set

$$ \begin{align*} \{|\Sigma_{<s_b s_{a_{i1}}^{c} \cdots s_{a_{id}}^{c^d}}| : i = 1, \ldots, \ell\} \end{align*} $$

is monochromatic, where $ \Sigma $ is the set of sums of two squares.

Proof. Let $ q, k, d \in \mathbb {N} $ be as in the statement of the polynomial Hales–Jewett theorem. Let $ N = N(q, k, d) $ be the natural number given by that theorem. Suppose that $ {\chi : (\mathbb {N}, \ast _f) \to [k] }$ is a $ k $ -colouring of $ \mathbb {N} $ , and let $ m : X(q, N, d) \to (\mathbb {N}, \ast _f) $ be the canonical map defined by

$$ \begin{align*} m((\mathbf{b}_1, \ldots, \mathbf{b}_d)) = \overset{d}{\underset{j=1}{\ast_f}} \Big(\underset{(i_1, \ldots, i_j) \in [N]^j}{\ast_f} \mathbf{b}_j((i_1, \ldots, i_j)) \Big). \end{align*} $$

The composite $ \chi \circ m $ is a $ k $ -colouring of $ X(q, N, d) $ . By the polynomial Hales–Jewett theorem, there exist $ a \in X(q, N, d) $ and $ \gamma \subseteq [N] $ such that the set

$$ \begin{align*} \{ a \oplus x_1 \gamma \oplus \cdots \oplus x_d \gamma^d : (x_1, \ldots, x_d) \in [q]^d \} \end{align*} $$

is monochromatic. Therefore, the image

$$ \begin{align*} m(\{ a \oplus x_1 \gamma \oplus \cdots \oplus x_d \gamma^d : (x_1, \ldots, x_d) \in [q]^d \}) \end{align*} $$

is monochromatic for the colouring $ \chi $ of $ \mathbb {N} $ . Note that

$$ \begin{align*} m&(a \oplus x_1 \gamma \oplus \cdots \oplus x_d \gamma^d) \\ & = \overset{d}{\underset{j=1}{\ast_f}} \Big( \underset{(i_1, \ldots, i_j) \in \gamma^j}{\ast_f} \mathbf{b}_j((i_1, \ldots, i_j)) \Big) \ast_f \Big( \overset{d}{\underset{j=1}{\ast_f}} \Big( \underset{(i_1, \ldots, i_j) \in [N]^j \setminus \gamma^j}{\ast_f} \mathbf{b}_j((i_1, \ldots, i_j)) \Big) \Big). \end{align*} $$

This can be rewritten as

$$ \begin{align*} x_1^{(c)} \ast_f \cdots \ast_f x_d^{(c^d)} \ast_f b = |\Sigma_{< s_b s_{x_1}^{c} \cdots s_{x_d}^{c^d}}|, \end{align*} $$

where $ c = |\gamma | $ and

$$ \begin{align*} b = \overset{d}{\underset{j=1}{\ast_f}} \Big( \underset{(i_1, \ldots, i_j) \in [N]^j \setminus \gamma^j}{\ast_f} \mathbf{a}_j((i_1, \ldots, i_j)) \Big).\\[-47pt] \end{align*} $$

Remark 5.8. Let $ (P, <) $ be a countably infinite poset. Then, there is a canonical bijection

$$ \begin{align*} \phi : P \to \mathbb{N} \quad \text{given by}\quad \phi(x) = |\{ y \in P : y < x \}|, \end{align*} $$

with an inverse $ \psi $ given by the $ \psi (n) = (n+1){\text {st}} $ smallest element of $ P $ . Since $ \psi $ does not preserve the multiplication of $ \mathbb {N} $ , if $ P $ is a countable multiplicative subgroup of $ \mathbb {N} $ , then the pullback $ \psi $ is not a semigroup homomorphism with respect to the usual multiplication. Thus, we obtain a new operation on $ \mathbb {N} $ via

$$ \begin{align*} m \ast_\psi n = \phi(\psi(m) \cdot \psi(n)) \end{align*} $$

(as we did before for the case of $ \Sigma $ ). Note that $ (\mathcal {P}_f(\mathbb {N}), <) $ is a countable ordered poset and, for the map $ \sigma : \mathcal {P}_f(\mathbb {N}) \to \mathbb {N} $ given by $ \sigma (F) = \sum _{n \in F} 2^n $ , this map is not of the above form, although it induces operations on $ \mathcal {P}_f(\mathbb {N}) $ .

Remark 5.9. The set $ \Sigma = \{ a^2 + b^2 : a, b \in \mathbb {N}_0 \} $ is symmetric with respect to $ a, b $ , so there is a canonical bijection between the set $ T := \{(a, b) \in \mathbb {N}_0^2 : a \leq b \} $ and $ \Sigma $ . Then, the map $ \psi : T \to \mathbb {N} $ given by

$$ \begin{align*} \psi(m, 2n) = (n+1)^2 - m \quad \text{and} \quad \psi(m, 2n+1) = (n+1)(n+2) - m \end{align*} $$

is a bijection that induces a new operation on $ T $ . Hence, one may find many different monochromatic configurations in $ \mathbb {N} $ .