Monochromatic products and sums in the rationals

Abstract

We show that every finite coloring of the rationals contains monochromatic sets of the form $\{x,y,xy,x+y\}$.

1 Introduction

In this paper, we prove the following:

Theorem 1.1. Any finite coloring of $\mathbb {Q}$ contains monochromatic sets of the form $\{x,y,xy,x+y\}$ with all terms distinct and nonzero.

The classical theorem of Schur [Sch16] states that any finite coloring of the set $\mathbb {N}$ of positive integers contains a monochromatic set of the form $\{x,y,x+y\}$ and, as observed by Graham, this immediately implies the analogous result for sets of the form $\{x,y,xy\}.$ In 1974, Hindman [Hin74] proved a strong generalization of this theorem and showed that finite colorings of $\mathbb {N}$ admit monochromatic sets which consist of all finite distinct sums of an infinite set. An enlightening proof of Hindman’s theorem using ultrafilters was found by Galvin and Glazer, see, for example, the book of Todorčević [Tod10, Chapter 2].

In [Hin79], Hindman famously conjectured that any finite coloring of $\mathbb {N}$ contains monochromatic sets of the form $\{x,y,xy,x+y\}$ , see also the textbook [GRS91] or the recent Quanta article on this conjecture and the recent advances in the field [Qua23]. In the two color case, Graham showed that any $2$ -coloring of $\{1,\ldots ,252\}$ contains a monochromatic configuration of the form $\{x,y,xy,x+y\}$ , and Hindman showed the same for any $2$ -coloring of $\{2,\ldots ,990\}$ [Hin79]. However, even the three color case remains open.

Hindman’s conjecture and its infinite versions have been studied also with $\mathbb {N}$ replaced by the rationals since at least the late 1990s, see, for example, the recent paper of Hindman et al. [HIL23] for more background. The question of whether Hindman’s conjecture holds over the rationals has been also asked by Moreira [Mor13]. Theorem 1.1 answers this in the affirmative.

In fact, in Theorem 5.1, we prove an extension of the above result involving arithmetic progressions and several variables. In particular, we can find monochromatic sets of the form $\{x,y,xy,x+iy:i\leq k\}$ and $\{x_i, \sum x_i, \prod x_i: i\leq k\}$ for any $k\in \mathbb {N}$ .

Hindman’s conjecture with $\mathbb {N}$ replaced by a field has been the subject of significant study recently. In [Shk10], Shkredov showed that finite colorings of large enough prime fields $\mathbb {F}_p$ contain monochromatic sets of the form $\{x,xy,x+y\}$ and Green and Sanders [GS16], building on this work, completely solved Hindman’s conjecture for colorings of finite prime fields, even obtaining good quantitative bounds on the number of solutions. For large enough finite fields, Cilleruelo [Cil12], Hanson [Han13] and Bergelson and Moreira [BM17] gave different proofs that any large subset of the field, and, in particular, a color class in any finite coloring, contains sets of the form $\{xy,x+y\}.$ For infinite fields, Bergelson and Moreira [BM17] proved that any finite coloring of an infinite field contains a monochromatic set of the form $\{x,xy,x+y\}$ . Later [BM18], they generalized this result to a wider class of rings.

In [Mor17], Moreira gave a beautiful proof that any finite coloring of $\mathbb {N}$ contains monochromatic sets of the form $\{x,xy,x+y\}$ . The first author [Bow22] was also able to refine Moreira’s result to also include the y term in the two color case. Recently, Xiao [Xia22] used the structural technique of the present paper to obtain other extensions of Moreira’s theorem.

By a standard compactness argument, Theorem 1.1 extends to other fields, giving a (nonquantitative) extension of the Green and Sanders theorem [GS16] to nonprime fields.

Corollary 1.2. For every n, there exists a prime p, such that whenever a field of characteristic at least p is colored with n colors, there exists a monochromatic set of the form $\{x,y,xy,x+y\}$ with all terms distinct and nonzero.

In fact, our proof of Theorem 1.1 can be adapted to work for infinite fields of arbitrary characteristic through routine modifications, however, throughout the paper, we will work only with $\mathbb {Q}$ for ease of exposition.

Our proof of Theorem 1.1 uses a recent result due to Bergelson and Glasscock [BG20] that is a quantitative version of Szemerédi’s theorem [Sze75] and is based on the density version of the Hales–Jewett theorem [FK91].

The proof of our result in the case of two colors is simpler than in the general case. Namely, in the case of two colors, we use an analysis of colorings where each class is thick, or each class is syndetic and apply the result of Bergelson and Glasscock [BG20]. However, for more than two colors, the complexity of the problem grows considerably, and we need a systematic analysis of how thick and syndetic sets interact with the partition classes of a given finite coloring.

The main new technique of our paper is devised to work with colorings with an arbitrary finite number of colors. It seeks to localize certain thick and syndetic sets with respect to the given finite coloring of the rationals. More precisely, in Section 3, we prove a structure result that decomposes a coloring into pieces that globally appear thick and locally appear syndetic, allowing us to extend the simpler proofs that apply to colorings where each class is thick or syndetic to colorings with arbitrary classes.

Such analysis of thick and syndetic sets is new in the context of [Hin79] conjecture, and it seems to not have been used before even outside of the scope of Ramsey theory. This result is stated in purely combinatorial terms in order to make the paper accessible to a wider audience. However, our original motivation was inspired by the ultrafilter methods of Galvin and Glazer (see [HS12]) and was focused on analyzing how minimal left ideals in $(\beta \mathbb {Q},\cdot )$ interact with a given finite coloring of the rationals.

2 IP sets and a quantitative version of Szemerédi’s theorem

In this section, we recall several notions of size in the structure of the additive group $({\mathbb {Q}},+)$ that will be useful throughout the remainder of this paper.

For a finite sequence $(a_1,\ldots ,a_n)$ of elements of ${\mathbb {Q}}$ , we use the notation $\sum (a_1,\ldots ,a_n)=a_1+\ldots +a_n$ . Given a sequence A of elements of ${\mathbb {Q}}$ , we write

$$ \begin{align*}\mathrm{FS}(A)=\left\{ \sum A_0: A_0\text{ is finite subsequence of } A\right\}.\end{align*} $$

The notion of IP sets, coined to abbreviate “infinite-dimensional parallelepiped” [FW78], is defined in terms of the sets FS $(A)$ and can be considered in any semigroup. In this paper, we will always consider IP sets with respect to the additive structure $({\mathbb {Q}},+)$ . Since we will be also concerned with the multiplicative structure on the rationals, we make it explicit in the definition below that IP sets are considered with respect to the additive structure.

Definition 2.1.

• • A subset of ${\mathbb {Q}}$ is additively IP if it contains a set of the form FS(A) for some infinite sequence A of elements of ${\mathbb {Q}}$ .

• • Given $r\in \mathbb {N}$ , a subset of ${\mathbb {Q}}$ is additively $\mathbf {IP}_r$ if it contains a set of the form FS(A) for some finite sequence A of length r consisting of elements of ${\mathbb {Q}}$ .

• • A subset of ${\mathbb {Q}}$ is additively $\mathbf {IP}_r^*$ if it has nonempty intersection with every IP $_r$ subset of ${\mathbb {Q}}$ .

The classical Folkman theorem (see [GRS91, Theorem 11]) ensures that any finite coloring of $\mathbb {N}$ contains a monochromatic additively IP $_r$ set, and Hindman’s theorem [Hin74], which has a particularly enlightening ultrafilter proof due to Galvin and Glazer (see, e.g., the textbook of Hindman–Strauss [HS12] or of Todorčević [Tod10]), gives the same result for additively IP sets. The proof of Theorem 1.1 will use a colorful extension of the IP $_r$ version that applies to thick colorings (see Lemma 3.2, which is related to the monochromatic products of sums theorem given in [Bow22, Theorem 3.1]).

Definition 2.2. An invariant mean on $({\mathbb {Q}},+)$ is a positive linear functional d of norm $1$ which is translation invariant on the space of all bounded, real-valued functions on ${\mathbb {Q}}$ with the $\|\cdot \|_\infty $ norm.

The existence of an invariant mean on $({\mathbb {Q}},+)$ , or on any commutative semigroup, is a classical result, see e.g. [Gre69, Theorem 1.2.1]. The following result of Bergelson and Glasscock [BG20, Theorem 7.5] is a quantitative strengthening of Szemerédi’s theorem that we will use in the inductive steps of our construction. We restate it here in the special case when the additive semigroup is $({\mathbb {Q}},+)$ and the homomorphisms are of the form $\varphi _i(q)=q_i\cdot q$ for some $q_i\in {\mathbb {Q}}$ and the first one is the constant zero function.

Theorem 2.3 (Bergelson, Glasscock, [BG20] Theorem 7.5).

Let d be an invariant mean on $({\mathbb {Q}},+)$ and let $s\in \mathbb {N}$ and $\alpha>0$ be given. There exists $r\in \mathbb {N}$ and $\alpha '>0$ for which the following holds. For any sequence of rational numbers $q_1,\ldots ,q_n$ and $A\subseteq {\mathbb {Q}}$ with $d(A)\geq \alpha $ , the set

$$ \begin{align*}\{q\in {\mathbb{Q}}: d(A\cap(A-q_1 q) \cap \ldots\cap (A-q_s q))\geq\alpha'\}\text{ is additively IP}_r^*.\end{align*} $$

The key point in our application is that the numbers r and $\alpha '$ depend only on s and $\alpha $ , and not on the particular choice of the numbers $q_1,\ldots ,q_n$ .

3 Localizing thick and syndetic sets in finite colorings

In this section, we recall some notions of size in semigroups, which will be stated and used only in terms of the multiplicative group $({\mathbb {Q}}\setminus \{0\},\cdot )$ throughout the paper. We then present two lemmas which will be crucial in the remainder of the paper. The first, Lemma 3.2, will be used together with Theorem 2.3 to control the term y in colorings where each color class is multiplicatively thick. The second, Lemma 3.3, is a structure result for colorings that will allow us to reduce the problem to the cases when either all color classes are multiplicatively syndetic or all color classes are multiplicatively thick.

We will state the results of this section in the group $({\mathbb {Q}}\setminus \{0\},\cdot )$ . The only reason why we choose to work with the group $({\mathbb {Q}}\setminus \{0\},\cdot )$ rather than the semigroup $({\mathbb {Q}},\cdot )$ is because we want to obtain nonzero elements in our main result. The results hold true for more general semigroups and semirings such as $(\mathbb {N},+,\cdot )$ and may be useful in that context.

For $A\subseteq {\mathbb {Q}}\setminus \{0\}$ and $t\in {\mathbb {Q}}\setminus \{0\}$ , write $A\ t=\{a\ t: a\in A\}$ .

Definition 3.1.

• • A set $T\subseteq {\mathbb {Q}}\setminus \{0\}$ is multiplicatively thick if for any finite $F\subseteq {\mathbb {Q}}\setminus \{0\}$ , there is an $a\in {\mathbb {Q}}\setminus \{0\}$ with $F\cdot a\subseteq T$ ,

• • A set $S\subseteq {\mathbb {Q}}\setminus \{0\}$ is multiplicatively syndetic if there is a finite $F\subseteq {\mathbb {Q}}\setminus \{0\}$ , such that ${\mathbb {Q}}\setminus \{0\}=\bigcup _{f\in F} S\ f$ .

Note that a set is multiplicatively thick if and only if its complement is not multiplicatively syndetic. In the proof of our main result, we will use the following fact to control the color of the step sizes in the arithmetic progressions obtained from Theorem 2.3.

Lemma 3.2. Let $k, r, N\in \mathbb {N}$ and $T_1,\ldots ,T_k\subseteq {\mathbb {Q}}\setminus \{0\}$ be multiplicatively thick. For every $j<N$ , there are additively IP $_r$ sets $S_{1,j}\subseteq T_1,\ldots ,S_{k,j}\subseteq T_k$ , such that for any $i\leq j< N$ and for any $l_i,\ldots ,l_j\leq k$ , we have

$$ \begin{align*}S_{l_i,i}\cdot S_{l_{i+1},i+1}\cdot\ldots\cdot S_{l_j,j}\subseteq T_{l_i}.\end{align*} $$

Proof. First, observe that if $S\subseteq {\mathbb {Q}}\setminus \{0\}$ is additively IP $_r$ and $t\in {\mathbb {Q}}\setminus \{0\}$ , then $St$ is additively IP $_r$ . This implies that any multiplicatively thick set is additively IP $_r$ . Next, note that if $T\subseteq {\mathbb {Q}}\setminus \{0\}$ is multiplicatively thick and $F\subseteq {\mathbb {Q}}\setminus \{0\}$ is finite, then $\{t\in T: tF\subseteq T\}$ is also multiplicatively thick. Using these two facts, we construct the sets $S_{1,N-i},\ldots ,S_{k,N-i}$ by induction on $i\in \{0,\ldots ,N-1\}$ . Start with any additively IP $_r$ sets $S_{1,N}\subseteq T_1,\ldots S_{k,N}\subseteq T_k$ . Having constructed $S_{1,N-i},\ldots ,S_{k,N-i}$ , find additively IP $_r$ sets $S_{1,N-i-1},\ldots ,S_{k,N-i-1}$ , so that for each $j\geq N-i$ and each F of the form $S_{l_{N-i},N-i}\cdot \ldots \cdot S_{l_j,j}$ , we have

$$ \begin{align*}S_{1,N-i-1}F\subseteq T_1,\ \ldots,\ S_{k,N-i-1}F\subseteq T_k.\\[-33pt] \end{align*} $$

The following lemma is the main technical tool that allows us to analyze the structure of thick and syndetic sets within the colors in a finite coloring of ${\mathbb {Q}}$ . It will be the key ingredient in dealing with an arbitrary number of colors.

Lemma 3.3. Let ${\mathbb {Q}}\setminus \{0\}=\bigcup _{i=1}^n C_i$ be a finite coloring. There is $k\in \mathbb {N}$ , index sets $Y_1,\ldots ,Y_k\subseteq [n],$ and a finite set $F\subseteq {\mathbb {Q}}\setminus \{0\}$ , such that

1. (i) for each $l \leq k$ , the set $\bigcup _{m\in Y_l} C_m$ is multiplicatively thick,

2. (ii) for each $x\in {\mathbb {Q}}\setminus \{0\}$ , there exists $l\leq k$ , such that for each $m\in Y_l$ , we have $x\in \bigcup _{f\in F}C_m/ f$ .

Proof. For $Y\subseteq [n]$ , we denote $C_Y=\bigcup _{m\in Y}C_m$ . Write

$$ \begin{align*}\mathcal{T}=\{Y\subseteq [n]: C_Y \text{ is multiplicatively thick}\}\end{align*} $$

and

$$ \begin{align*}\mathcal{S}=\{Y\subseteq [n]: {\mathbb{Q}}\setminus\{0\}=\bigcup_{f\in F_Y} C_Y/ f\text{ for some finite }F_Y\subseteq {\mathbb{Q}}\setminus\{0\}\},\end{align*} $$

that is, $\mathcal {S}=\{Y\subseteq [n]: C_Y$ is syndetic $\}$ . For each $Y\in \mathcal {S}$ , choose an $F_Y$ , such that ${\mathbb {Q}}\setminus \{0\}=\bigcup _{f\in F_Y} C_Y/ f$ . Since $\mathcal {S}$ is a finite collection, we can choose a finite set $F\subseteq {\mathbb {Q}}\setminus \{0\}$ which is a superset of all the finite sets $F_Y$ .

For $x\in {\mathbb {Q}}\setminus \{0\}$ , write

$$ \begin{align*}A_x=\{m\in[n]: \exists f\in F\ x\in C_m/ f\}.\end{align*} $$

Note that by the definition of $\mathcal {S}$ , for every $x\in {\mathbb {Q}}\setminus \{0\}$ , we have

(3.1) $$ \begin{align} A_x\cap Y\not=\emptyset\quad\text{for every }Y\in\mathcal{S}. \end{align} $$

Claim 3.4. For every $x\in {\mathbb {Q}}\setminus \{0\}$ , there exists $Y_x\in \mathcal {T}$ , such that $Y_x\subseteq A_x$ .

Proof. For the sake of contradiction, suppose that no such set $Y_x$ exists. Then $[n]\setminus A_x$ belongs to $\mathcal {S}$ , which contradicts (3.1).

Finally, there are finitely many choices for $Y_x$ as in Claim 3.4, and so these choices correspond to the desired $Y_1,\ldots ,Y_k$ .

4 Finding the pattern $\{x,y,xy,x+y\}$

Before giving the proof of Theorem 1.1, we discuss two special cases as a warm-up. Namely, we first present the proof in the case when each color class is syndetic, and next, we present the proof in the case when the color classes are all multiplicatively thick. In the proof of Theorem 1.1, we will carry out the arguments of these special cases simultaneously, using the ideas of Section 3.

4.1 Two special cases

We begin with the proof in the case that each color class is multiplicatively syndetic. Below, we use Moreira’s theorem [Mor17, Theorem 7.5] in the general form that holds for any field (cf. the remarks after [Mor17, Definition 7.1]). We also mention that this proof also works in $\mathbb {N}$ and can be adapted to show that any subset of $\mathbb {N}$ which is both multiplicatively syndetic and central contains a subset of the form $\{x,y,xy,x+y\}.$

Claim 4.1. Suppose that $n\in \mathbb {N}$ , ${\mathbb {Q}}\setminus \{0\}$ is colored as $\bigcup _{i=1}^n C_i$ , where each $C_i$ is multiplicatively syndetic. Then there are monochromatic sets of the form $\{x,y,xy,x+y\}$ .

Proof. By definition, for each $i\in [n]$ , there is a finite set $F_i\subseteq {\mathbb {Q}}\setminus \{0\}$ so that ${\mathbb {Q}}\setminus \{0\}=\bigcup _{t\in F_i} (C_i\setminus \{0\})/ t$ . Let $F=\bigcup _{i=1}^n F_i.$ Then for each $x\in {\mathbb {Q}}\setminus \{0\}$ , we may pick $(f_{x,1},\ldots ,f_{x,n})\in F^n$ , such that for each $i\in [n]$ , we have $x\in C_i/ f_{x,i}$ . For $\vec {f}\in F^n$ , let $D_{\vec {f}}=\{x\in {\mathbb {Q}}\setminus \{0\}: (f_{x,1},\ldots ,f_{x,n})=\vec {f}\}$ . Then ${\mathbb {Q}}=\{0\}\cup \bigcup _{\vec {f}\in F^n} D_{\vec {f}}$ . For $f\in F$ , let $g_f\in {\mathbb {Q}}[x]$ be defined by $g_f(x)=x/ f$ , and let $G=\{0\}\cup \{g_f:f\in F\}$ .

Pick by [Mor17, Theorem 7.5] some $\vec {f}=(f_1,\ldots ,f_n)\in F^n$ and $x_0,x_1\in {\mathbb {Q}}\setminus \{0\}$ , such that $\{x_0,x_1\}\cup \{x_0+g(x_1) : g \in G\}\subseteq D_{\vec {f}}$ . Then $\{x_0 x_1, x_0\} \cup \{x_0 + x_1/ f : f \in F\}\subseteq D_{\vec {f}}$ . Pick j, such that $x_1 \in C_j$ . Let $y=x_1$ and $x=x_0 f_j$ . Then $\{x,y,xy,x+y\}\subseteq C_j$ .

The other extreme case is when the colors are all multiplicatively thick. For the sake of simplicity, we only present the two color case here; the proof extends naturally to more colors, albeit with more complicated notation.

Claim 4.2. Suppose that ${\mathbb {Q}}\setminus \{0\}=C_1\cup C_2$ , where both $C_i$ are multiplicatively thick in $({\mathbb {Q}}\setminus \{0\},\cdot )$ . There are monochromatic sets of the form $\{x,y,xy,x+y\}.$

Proof. Let d be a left-invariant mean on $({\mathbb {Q}},+)$ . Without loss of generality, we may assume that $d(C_1)>0$ . Let $\alpha =d(C_1)$ . Apply Theorem 2.3 with $\alpha =d(C_1)$ and $s=1$ , to get $r_1$ and $\alpha '$ , and apply it again with $\alpha '$ and $s=2$ to get $r_2$ .

Since $C_2$ is multiplicatively thick, we can find an additively IP $_{r_2}$ set $S_2$ contained in $C_2$ . Using the fact that $C_1$ is multiplicatively thick, we can find an additively IP $_{r_1}$ set $S_1$ contained in $C_1$ , such that

(4.1) $$ \begin{align} S_1 S_2\subseteq C_1. \end{align} $$

Now, since $S_1$ is additively IP $_{r_1}$ , applying Theorem 2.3 to the homomorphism $x\mapsto x$ , we find $C_1'\subseteq C_1$ with $d(C_1')=\alpha '$ and $y_1\in S_1$ , such that

(4.2) $$ \begin{align} C_1'+y_1\subseteq C_1. \end{align} $$

If $y_1C_1' \cap C_1\not =\emptyset $ , then pick any $x\in C_1'$ , such that $xy_1\in C_1$ , and put $y=y_1$ . Note that (4.2) implies $x+y\in C_1$ , so $\{x,y,xy,x+y\}\subseteq C_1$ .

Thus, we can assume that $y_1C_1'\subseteq C_2$ . Since $S_2$ is additively IP $_{r_2}$ , applying Theorem 2.3 to the two homomorphisms $x\mapsto y_1 x$ and $x\mapsto \frac {1}{y_1}x$ , we find $C_1"\subseteq C_1'$ and $y_2\in S_2$ , such that

(4.3) $$ \begin{align} C_1"+y_1 y_2\subseteq C_1'. \end{align} $$

(4.4) $$ \begin{align} C_1"+\frac{y_2}{y_1}\subseteq C_1'. \end{align} $$

Note that $y_1 y_2\in C_1$ by (4.1).

If $y_1 y_2 C_1"\cap C_1\not =\emptyset $ , then put $y=y_1 y_2$ , and choose $x\in C_1"$ , such that $xy\in C_1$ . Note that (4.3) implies that $x+y\in C_1$ . Thus, $\{x,y,xy,x+y\}\subseteq C_1$ .

Otherwise, $y_1y_2 C_1"\cap C_2\not =\emptyset $ . Put $y=y_2,$ (which is an element of $S_2\subset C_2$ ), and let $x\in y_1 C_1"$ be such that $xy\in C_2$ . Since $y_1 C_1"\subseteq y_1 C_1'\subseteq C_2$ , we have $x\in C_2$ . Note that (4.4) implies that $x+y\in C_2$ . Thus, in this case, $\{x,y,xy,x+y\}\subseteq C_2$ .

To end this subsection, we mention that Claims 4.1 and 4.2 alone can be used to show that any $2$ -coloring of ${\mathbb {Q}}$ contains a monochromatic set of the form $\{x,y,xy,x+y\}$ . This was the original motivation of the proof of Theorem 1.1 given below.

4.2 The general case

In this subsection, we prove our main result.

Theorem 4.3. For any finite coloring of ${\mathbb {Q}}$ , there exist nonzero y and infinitely many $x\in {\mathbb {Q}}$ , such that the tuples $\{x,y,xy,x+y\}$ are monochromatic.

Note that the above statement, in particular, implies that all elements in the quadruple can be chosen to be distinct. First, we can assure that $y\not =1$ by modifying the coloring by giving $1$ a separate color, and then, using the fact that for a fixed $y\not =1$ , the equation $xy=x+y$ has one solution, we can choose x so that all numbers $x,y,xy,x+y$ are distinct. In fact, experts may notice from the proof that we may even ensure that we obtain a positive multiplicative density set of y that works for a set of positive multiplicative and additive density $x,$ although we will not check this in detail during the proof.

In the proof, we will use Lemma 3.3 to carry out the proofs of Claims 4.1 and 4.2 simultaneously.

Proof of Theorem 4.3.

Let d be an additive invariant mean on ${\mathbb {Q}}$ . Suppose ${\mathbb {Q}}$ is colored into n colors, and write ${\mathbb {Q}}\setminus \{0\}= C_1\cup \ldots \cup C_n$ . Using Lemma 3.3 for the colors $C_i$ , find $k\in \mathbb {N}$ and finite sets $F\subseteq {\mathbb {Q}}\setminus \{0\}$ and $Y_1,\ldots ,Y_k\subseteq [n]$ , such that

1. (i) for each $l \leq k$ , the set $\bigcup _{m\in Y_l} C_m$ is multiplicatively thick,

2. (ii) for each $x\in {\mathbb {Q}}\setminus \{0\}$ , there exists $l\leq k$ , such that for each $m\in Y_l$ , we have $x\in \bigcup _{f\in F} C_m/ f$ .

This means that for each $x\in {\mathbb {Q}}\setminus \{0\}$ , there is $l\leq k$ and a tuple $f_1,\ldots ,f_n\in F$ , so that for each $m\in Y_l$ , we have $x\in C_m/ f_m$ .

Note that there are finitely many tuples $(l,f_1,\ldots ,f_n)$ with $l\leq k$ and $f_1,\ldots ,f_n\in F$ . This allows us to define a new finite coloring of ${\mathbb {Q}}\setminus \{0\}$ , where for each tuple $(l,f_1,\ldots ,f_n)$ as above, we have a color $D_{(l,f_1, \ldots ,f_n)}$ , and we put

$$ \begin{align*}x\in D_{(l,f_1,\ldots,f_n)}\quad\text{ if }\quad x\in C_m/ f_m\text{ for each } m\in Y_l.\end{align*} $$

Write K for the number of colors in the above coloring. These colors do not need to be disjoint, but we can always disjointify them.

Pick via the pigeonhole principle $N\in \mathbb {N}$ large enough so that given any sequence $\vec {v}_1,\ldots \vec {v}_N$ in $[k]\times F^n$ , there exist i and j with $i<j-1<N-1$ , such that $\vec {v}_i=\vec {v}_j$ . Let $s={N\choose 2}\cdot |F|$ . We define a sequence of positive numbers $\alpha _1,\alpha _1',\ldots ,\alpha _N,\alpha _N'$ with $\alpha _1=\frac {1}{K}$ and a sequence $r_1,\ldots ,r_N$ in $\mathbb {N}$ . At the j-th step, given $\alpha _j$ , we apply Theorem 2.3 and obtain $\alpha _j'$ and $r_j$ , such that for any set $A\subseteq {\mathbb {Q}}$ with $d(A)\geq \alpha _j$ and any $q_1,\ldots ,q_s$ , the set

(4.5) $$ \begin{align} \{y\in {\mathbb{Q}}: d(\{x\in A: x+q_1 y,\ldots, x+q_s y\in A \})\geq\alpha_j'\}\text{ is additively IP}_r^*. \end{align} $$

We put $\alpha _{j+1}=\frac {\alpha _j'}{K}$ . When the construction is complete, we write $r=\max \{r_1,\ldots ,r_N\}$ so that any set which is additively IP $_{r_j}^*$ for some $j\in [N]$ is additively IP $_{r}^*$ .

Since the sets $\bigcup _{m\in Y_1} C_m,\ldots , \bigcup _{m\in Y_k} C_m$ are multiplicatively thick, by Lemma 3.2, there are additively IP $_r$ sets

$$ \begin{align*}S_{1,j}\subseteq \bigcup_{m\in Y_1} C_m,\ldots,S_{k,j}\subseteq \bigcup_{m\in Y_k} C_m\end{align*} $$

for each $j<N$ , such that

(4.6) $$ \begin{align} S_{l_i,i}\cdot S_{l_{i+1},i+1}\cdot\ldots\cdot S_{l_j,j}\subseteq \bigcup_{m\in Y_{l_i}}C_m \end{align} $$

for any $i\leq j\leq N$ and any choice $l_i,l_{i+1},\ldots ,l_j\leq k$ .

We inductively define

• • a sequence of subsets $A_1\supseteq \ldots \supseteq A_N$ of ${\mathbb {Q}}$ ,

• • finite sets $Q_1,\ldots ,Q_N\subseteq {\mathbb {Q}}$ ,

• • tuples $(l_1,f_{1,1},\ldots ,f_{n,1}),\ldots , (l_N,f_{1,N},\ldots ,f_{n,N})$ , such that $l_j\leq k$ and $f_{1,j},\ldots ,f_{n,j}\in F$ for every $j\leq N$ ,

• • and elements $y_1,\ldots ,y_{N-1}\in {\mathbb {Q}}\setminus \{0\}$ ,

such that for every $j< N$ , we have

1. 1. $A_{j+1}\subseteq A_j\cap \bigcap _{q\in Q_j}(A_j-qy_j)$ ,

2. 2. $A_{j+1}\subseteq \{x\in A_j: xy_1\cdot \ldots \cdot y_j\in D_{(l_{j+1},f_{1,j+1},\ldots ,f_{n,j+1})}\}$ ,

3. 3. $y_j\in S_{l_{j},j}$ .

and $d(A_j)\geq \alpha _j$ for every $j<N$ with

$$ \begin{align*}Q_j=\left\{\frac{y_i\cdot\ldots\cdot y_{j-1}}{fy_1\cdot\ldots\cdot y_{i-1}}: f\in F, 1\leq i<j\right\}\end{align*} $$

for every $j< N$ , where by convention, we write $y_1\cdot \ldots \cdot y_{0}=1$ and $Q_1=\{\frac {1}{f}:f\in F\}$ . Note that $|Q_j|\leq |F|\cdot {j-1\choose 2}<s$ .

To start off our inductive construction, we choose $(l_1,f_{1,1},\ldots ,f_{n,1})$ , such that $d(D_{(l_1,f_{1,1},\ldots ,f_{n,1})})>\alpha _1$ and put $A_1= D_{(l_1,f_{1,1},\ldots ,f_{n,1})}$ . Such tuple $(l_1,f_{1,1},\ldots ,f_{n,1})$ can be found by our choice of $\alpha _1$ .

Now we describe the induction. Suppose that $A_j$ , $Q_j$ , $(l_j,f_{1,j}\ldots ,f_{n,j})$ as well as $y_{j-1}$ have been constructed. We proceed to find $y_j$ as well as $A_{j+1}$ , $Q_{j+1}$ and $(l_{j+1},f_{1,j+1}\ldots ,f_{n,j+1})$ .

First, since we can assume that s is chosen big enough so that $|Q_j|\leq s$ , and since the set $S_{l_j,j}$ is additively IP $_r$ , by (4.5), we can find $y_j\in S_{l_{j},j}$ and $A_j'\subseteq A_j$ , such that $d(A_j')=\alpha _j'$ and for every $x\in A_j'$ , we have

$$ \begin{align*}x+qy_{j}\in A_j \quad\text{for all }q\in Q_j.\end{align*} $$

Second, look at $y_1\cdot \ldots \cdot y_j A_j'$ , and note that for some $(l_{j+1},f_{1,j+1},\ldots ,f_{n,j+1})$ , the set $\{x\in A_j': xy_1\cdot \ldots \cdot y_j\in D_{(l_{j+1},f_{1,j+1},\ldots ,f_{n,j+1})}\}$ has density at least $\alpha _{j+1}$ , by our choice of $\alpha _{j+1}$ . Put

$$ \begin{align*}A_{j+1}=\{x\in A_j': xy_1\cdot\ldots\cdot y_j\in D_{(l_{j+1},f_{1,j+1},\ldots,f_{n,j+1})} \}.\end{align*} $$

We find $i < j-1 < N-1$ so that for some $(l, f_1,. .. , f_n)$ , we have

(4.7) $$ \begin{align} (l_i,f_{1,i},\ldots,f_{n,i})=(l_j,f_{1,j},\ldots,f_{n,j})=(l,f_1,\ldots,f_n). \end{align} $$

Let $y=y_i\cdot \ldots \cdot y_{j-1}$ . By the property (3) and (4.6), there exists $m\in Y_{l}$ , such that $y\in C_m$ . Let $x'\in A_{j}$ , and put $x=f_m x' y_1\cdot \ldots \cdot y_{i-1}$ .

Since $A_{j}\subseteq A_{i}$ , by the property (2) at step i of the construction, and we know that

$$ \begin{align*}x' y_1\cdot\ldots\cdot y_{i-1}\in D_{(l,f_1,\ldots,f_n)},\end{align*} $$

so $x=f_m x'y_1\cdot \ldots \cdot y_{i-1}\in C_m$ by (4.7).

By property (2) at step j, we know that

$$ \begin{align*}x' y_1\cdot\ldots\cdot y_{j-1}\in D_{(l,f_1,\ldots,f_n)},\end{align*} $$

so $xy=f_m x' y_1\cdot \ldots \cdot y_{j-1}\in C_m$ .

Finally, by property (1), we have $A_{j}\subseteq A_{j-1}\cap \bigcap _{q\in Q_{j-1}}(A_{j-1}-qy_{j-1})\subseteq A_{i}\cap \bigcap _{q\in Q_{j-1}}(A_i-qy_{j-1})$ , so

$$ \begin{align*}x'+q y_{j-1}\in A_i\end{align*} $$

for every $q\in Q_{j-1}$ . Put

$$ \begin{align*}q=\frac{y_{i}\cdot\ldots\cdot y_{j-2}}{f_m y_1\cdot\ldots\cdot y_{i-1}}.\end{align*} $$

Then by (2) again, since $x'+q y_{j-1}\in A_i$ , we have $y_1\cdot \ldots \cdot y_{i-1}(x'+q y_{j-1})\in D_{(l,f_1,\ldots ,f_n)}$ , that is,

$$ \begin{align*}x'y_1\cdot\ldots\cdot y_{i-1}+\frac{1}{f_m}y_i\ldots y_{j-1}=\frac{1}{f_m}x+\frac{1}{f_m}y\in D_{(l,f_1,\ldots,f_n)},\end{align*} $$

which means that $x+y\in C_m$ .

4.3 Other fields

Now, we give the short proof of Corollary 1.2 using a standard compactness argument.

Proof of Corollary 1.2.

Write $T_F$ for the theory of fields. Fix n, and let L be the language of the theory of fields together with n unary predicates $C_1,\ldots , C_n$ . By Theorem 4.3, the sentence

$$ \begin{align*}\sigma=\left(\forall x\bigvee_{i=1}^n C_i(x)\right)\to\left(\exists x,y\not=0 \bigvee_{i=1}^n C_i(x)\wedge C_i(y)\wedge C_i(xy)\wedge C_i(x+y)\right)\end{align*} $$

is satisfied in $\mathbb {Q}$ and hence in any field of characteristic zero. Thus, $\sigma $ is provable in the theory of fields of characteristic zero

$$ \begin{align*}T_0=T_F\cup\{\underbrace{1+\ldots+1}_{p}\not=0:p\text{ prime}\}.\end{align*} $$

By the first-order logic compactness theorem [TZ12, Theorem 2.2.1], $\sigma $ is provable from a finite subset of $T_0$ , hence there exists $p_0$ , such that any field of characteristic greater than $p_0$ satisfies $\sigma $ .

5 Generalizations

Here, we prove a generalization of Theorem 1.1 to the setting of more variables and more complicated monochromatic patterns.

Theorem 5.1. Let $H\subseteq \{h:{\mathbb {Q}}^i\to {\mathbb {Q}}: i\in \mathbb {N}\}$ be a finite set of functions, and let $t\in \mathbb {N}$ . In any finite coloring of ${\mathbb {Q}}$ , there are $x_0,\ldots ,x_t\in {\mathbb {Q}}$ , such that the following numbers have the same color:

$$ \begin{align*}x_i\cdot\ldots\cdot x_j,\quad x_0\cdot\ldots\cdot x_i+ h_{i+1}(x_1,\ldots,x_{i})x_{i+1}+ \ldots + h_t(x_1,\ldots,x_{t-1})x_{t}\end{align*} $$

for all $0\leq i\leq j\leq t$ and any $h_{i+1},\ldots ,h_t\in H$ (of appropriate arity).

Note in the above that the functions from H do not take $x_0$ as an input.

Example 5.2. Considering the constant $0$ and constant $1$ function together with $h_1(y,z)=y$ , $h_2(y,z)=z$ and $h_3(y,z)=yz$ , we get the following monochromatic pattern:

$$ \begin{align*} \{x,y,z,xy,yz,xyz, \\ x+y,x+z,x+y+z,x + y + yz, x+yz,\\ xy+z,xy+yz\}. \end{align*} $$

Example 5.3. For a given k, considering the constant functions $1,\ldots , k$ , we get the following monochromatic pattern:

$$ \begin{align*}\{x,y,xy,x+iy: i\leq k\}.\end{align*} $$

The proof is essentially the same as that of Theorem 1.1, but now we will need to consider more complicated sets $Q_i$ in the proof and use Ramsey’s theorem together with the pigeonhole principle. We repeat the first part of the proof for the sake of completeness.

Proof. Suppose ${\mathbb {Q}}$ is colored into n colors, and write ${\mathbb {Q}}\setminus \{0\}= C_1\cup \ldots \cup C_n$ . Let d be an invariant mean on $({\mathbb {Q}},+)$ . By Lemma 3.3, we get $k\in \mathbb {N}$ and finite sets $F\subseteq {\mathbb {Q}}$ and $Y_1,\ldots ,Y_k\subseteq [n],$ such that

1. (i) for each $l \leq k$ , the set $\bigcup _{m\in Y_l} C_m$ is multiplicatively thick,

2. (ii) for each $x\in {\mathbb {Q}}\setminus \{0\}$ , there exists $l\leq k$ , such that for each $m\in Y_l$ , we have $x\in \bigcup _{f\in F} C_m/ f$ .

This gives the coloring of ${\mathbb {Q}}\setminus \{0\}$ with colors $D_{(l,f_1,\ldots ,f_n)}$ for each tuple $(l,f_1,\ldots ,f_n)$ where

$$ \begin{align*}x\in D_{(l,f_1,\ldots,f_n)}\quad\text{ if }\quad x\in C_m/ f_m\text{ for each } m\in Y_l.\end{align*} $$

These colors do not need to be disjoint, but we can always disjointify them. Write K for the number of colors in the above coloring.

Similarly, as in the proof of Theorem 4.3, we find large numbers $N,s\in \mathbb {N}$ with s depending on N, and $r\in \mathbb {N}$ obtained by iteratively applying Theorem 2.3 N many times.

We define a sequence of positive numbers $\alpha _1,\alpha _1'\ldots ,\alpha _N,\alpha _N'$ with $\alpha _1=\frac {1}{K}$ . At the j-th step, given $\alpha _j$ and s, we apply Theorem 2.3 and obtain $\alpha _j'$ , such that for any set $A\subseteq {\mathbb {Q}}$ with $d(A)\geq \alpha _j$ and for any $q_1,\ldots ,q_s\in {\mathbb {Q}}$ , the set

(5.1) $$ \begin{align} \{y\in {\mathbb{Q}}: d(\{x\in A: x+q_1 y,\ldots, x+q_s y\in A \})\geq\alpha_j'\}\text{ is additively IP}_r^*. \end{align} $$

We put $\alpha _{j+1}=\frac {\alpha _j'}{K}$ .

Lemma 3.2 gives us additively IP $_r$ sets $S_{1,j}\subseteq \bigcup _{m\in Y_1} C_m,\ldots ,S_{k,j}\subseteq \bigcup _{m\in Y_k} C_m$ for each $j<N$ , such that

(5.2) $$ \begin{align} S_{l_i,i}\cdot S_{l_{i+1},i+1}\cdot\ldots\cdot S_{l_j,j}\subseteq \bigcup_{m\in Y_{l_i}}C_m \end{align} $$

for any $i\leq j\leq N$ and any choice of $l_i,l_{i+1},\ldots ,l_j\leq k$

Inductively define a sequence of subsets $A_1\supseteq \ldots \supseteq A_N$ of ${\mathbb {Q}}$ as well as finite sets $Q_1,\ldots ,Q_N\subseteq {\mathbb {Q}}$ , tuples $(l_1,f_{1,1},\ldots ,f_{n,1}),\ldots , (l_N,f_{1,N},\ldots ,f_{n,N})$ with $l_j\leq k$ and $f_{1,j},\ldots ,f_{n,j}\in F$ for every $j\leq N$ , as well as elements $y_1,\ldots ,y_{N-1}\in {\mathbb {Q}}\setminus \{0\}$ , such that for every $j\leq N$ , we have

1. 1. $A_{j+1}\subseteq A_j\cap \bigcap _{q\in Q_j}(A_j-qy_j)$ ,

2. 2. $A_{j+1}\subseteq \{x\in A_j: xy_1\cdot \ldots \cdot y_j\in D_{(l_{j+1},f_{1,j+1},\ldots ,f_{n,j+1})}\}$ ,

3. 3. $y_j\in S_{l_{j},j}$ .

and $d(A_j)\geq \alpha _j$ for every $j\leq N$ and

$$ \begin{align*} Q_j=\left\{\frac{h(y_1\cdot\ldots\cdot y_{i_1-1},,\ldots,y_{i_{s-1}}\cdot\ldots\cdot y_{i_s-1})y_{i_s}\cdot\ldots\cdot y_{j-1}}{fy_1\cdot\ldots\cdot y_{i_q-1}}\right.: \\ \left.\vphantom{\frac{A^A}{A^A}}f\in F, h\in H, s\leq j, 0\leq i_1<\ldots<i_s\leq j, q\leq s\right\}, \end{align*} $$

where by convention, we write $y_1\cdot \ldots \cdot y_{0}=1$ and $Q_1=\{\frac {1}{f}:f\in F\}$ .

Now, we describe the induction. Suppose that $A_j$ , $Q_j$ , $(l_j,f_{1,j}\ldots ,f_{n,j})$ as well as $y_{j-1}$ have been constructed. We proceed to find $y_j$ as well as $A_{j+1}$ , $Q_{j+1}$ and $(l_{j+1},f_{1,j+1}\ldots ,f_{n,j+1})$ .

First, since we can assume that s is chosen big enough so that $|Q_j|\leq s$ , and since the set $S_{l_j,j}$ is additively IP $_r$ , by (4.5), we can find $y_j\in S_{l_{j},j}$ and $A_j'\subseteq A_j$ , such that $d(A_j')=\alpha _j'$ and for every $x\in A_j'$ , we have

$$ \begin{align*}x+qy_{j}\in A_j \quad\text{for all }q\in Q_j.\end{align*} $$

Second, look at $y_1\ldots y_j A_j'$ , and note that for some $(l_{j+1},f_{1,j+1},\ldots ,f_{n,j+1})$ , the set $\{x\in A_j': xy_1\cdot \ldots \cdot y_j\in D_{(l_{j+1},f_{1,j+1},\ldots ,f_{n,j+1})}\}$ has density at least $\alpha _{j+1}$ , by our choice of $\alpha _{j+1}$ . Put

$$ \begin{align*}A_{j+1}=\{x\in A_j': xy_1\cdot\ldots\cdot y_j\in D_{(l_{j+1},f_{1,j+1},\ldots,f_{n,j+1})} \}.\end{align*} $$

So long as N was chosen large enough, by the pigeonhole principle, we find $j_1,\ldots ,j_M$ so that for some $(l,f_1,\ldots ,f_n)$ , we have

(5.3) $$ \begin{align} (l_{j_1},f_{1,j_1},\ldots,f_{n,j_1})=(l_{j_i},f_{1,j_i},\ldots,f_{n,j_i})=(l,f_1,\ldots,f_n) \end{align} $$

for each $i\leq M$ . For simplicity of notation, suppose $j_1=1,\ldots , j_M=M$ .

Note that if $i<j\leq M$ , then by (3) and (5.2), there exists $m\in Y_l$ , such that $y_i\cdot \ldots \cdot y_{j-1}\in C_m$ . Now, consider the coloring of $[M]\times [M]$ , where we give $(i,j)$ with $i<j$ the color m if

$$ \begin{align*}y_{i}\cdot\ldots\cdot y_{j-1} \in C_m.\end{align*} $$

So long as M is large enough, by Ramsey’s theorem, there is a color m and subsequence $j_1,\ldots ,j_{t+1}$ so that all pairs from the subsequence have the same color.

Put $x_1=y_{j_1}\cdot \ldots \cdot y_{j_2-1}, \ \ldots ,\ x_t=y_{j_{t}}\cdot \ldots \cdot y_{j_{t+1}-1}$ .

First, note that by the choice of the subsequence $j_1,\ldots ,j_t$ , we have

$$ \begin{align*}x_i\cdot\ldots\cdot x_j\in C_m\end{align*} $$

for every $1\leq i\leq j$ .

Choose $x_0'\in A_{j_t}$ , and put $x_0= f_m x_0' y_1\cdot \ldots \cdot y_{j_1-1}$ . Since $A_{j}\subseteq A_{i}$ for $i<j$ , by property (1), we know that by (5.3), we have $x_0' y_1\cdot \ldots \cdot y_{j_i-1}\in D_{(l,f_1,\ldots ,f_n)}$ , so

$$ \begin{align*}x_0 x_1\cdot\ldots\cdot x_i= f_m x_0' y_1\cdot\ldots\cdot y_{j_i-1}\in C_m\end{align*} $$

for each $0\leq i\leq n$ .

Now, fix i and $h_{i+1},\ldots ,h_t\in H$ . By property (1), we know that if $i<j$ , then $A_{j}\subseteq A_{j-1}\cap \bigcap _{q\in Q_{j-1}}(A_{j-1}-qy_{j-1})\subseteq A_{i}\cap \bigcap _{q\in Q_{j-1}}(A_i-qy_{j-1})$ , so

(5.4) $$ \begin{align} x_0'+q y_{j-1}\in A_i \end{align} $$

for every $q\in Q_{j-1}$ .

Put

$$ \begin{align*}q_{s}=\frac{h_s(x_1,\ldots,x_{s-1})y_{j_{s}}\cdot\ldots\cdot y_{j_{s+1}-2}}{f_m y_1\cdot\ldots\cdot y_{j_i-1}}\in Q_{j_{s+1}-1}\end{align*} $$

for every s, such that $i<s\leq t$ . Then, by iterating (5.4), we have

$$ \begin{align*}x_0'+ q_{j_{i}-1}y_{j_{i+2}-1}+\ldots+q_{j_{s}}y_{j_{s+1}-1} \in A_{j_s}\end{align*} $$

for every $s\leq t$ , so $x_0'+ q_{i+1}y_{j_{i+2}-1}+\ldots +q_{t}y_{j_{t+1}-1} \in A_{j_t}$ . Since $A_{j_t}\subseteq A_{j_i}$ , we get $y_1\cdot \ldots \cdot y_{j_i-1}(x_0'+ q_{i+1}y_{j_{i+2}-1}+\ldots +q_{t}y_{j_{t+1}-1} )\in D_{(l,,f_1\ldots ,f_n)}$ , which means that $f_m y_1\cdot \ldots \cdot y_{j_i-1}(x_0'+ q_{i+1}y_{j_{i+2}-1}+\ldots +q_{t}y_{j_{t+1}-1} )\in C_m$ , and so

$$ \begin{align*}x_0\cdot\ldots\cdot x_i+ h_{i+1}(x_1,\ldots,x_{i})x_{i+1}+ \ldots + h_t(x_1,\ldots,x_{t-1})x_{t}\in C_m\end{align*} $$

as needed.

6 Open problems

Our main result is a common extension of Hindman’s conjecture and the van der Waerden theorem [Wae27], so it is natural to wonder if our approach might extend to generalize the polynomial van der Waerden theorem [BL96] as well.

Question 6.1. Let P be a finite set of integral polynomials. Does any finite coloring of $\mathbb {Q}$ contain a monochromatic set of the form $\{x,y,xy,x+p(y):p\in P\}?$

This is known to hold even for $\mathbb {N}$ if the y term is dropped [Mor17]. Our approach would immediately extend to prove this if the IP $^*_r$ polynomial theorem were known to be true, but perhaps finding the right quantitative version of the polynomial van der Waerden theorem [BL96] will suffice.

Another potential direction is to extend our results to generalize the geometric theorem as well. Currently, even the following is open.

Question 6.2. Does every finite coloring of $\mathbb {Q}$ contain a monochromatic set of the form $\{xy,xy^2,x+y\}?$

Finally, let us mention that Hindman also conjectured stronger statements, namely, that any finite coloring of $\mathbb {N}$ contains monochromatic sets of the form $FS(A)\cup FP(A)$ for $A\subseteq \mathbb {N}$ arbitrarily large. Our approach can deal with some subsets of these configurations in $\mathbb {Q}$ (see Theorem 5.1 and Example 5.2), but the problem still seems difficult even for $|A|=3$ in $\mathbb {Q}.$