# Monochromatic sums and products

@article{Green2016MonochromaticSA, title={Monochromatic sums and products}, author={Ben Green and Tom Sanders}, journal={arXiv: Number Theory}, year={2016}, pages={613} }

Monochromatic sums and products, Discrete Analysis 2016:5, 48pp.
An old and still unsolved problem in Ramsey theory asks whether if the positive integers are coloured with finitely many colours, then there are positive integers $x$ and $y$ such that $x, y, x+y$ and $xy$ all have the same colour. In fact, it is not even known whether it is always possible to find $x$ and $y$ such that $x+y$ and $xy$ have the same colour.
This paper is about the corresponding question when $\mathbb{N}$ is…

## 18 Citations

### Monochromatic sums and products in $\mathbb{N}$

- Mathematics
- 2016

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by…

### Exponential Patterns in Arithmetic Ramsey Theory

- Mathematics
- 2016

We show that for every finite colouring of the natural numbers there exists $a,b >1$ such that the triple $\{a,b,a^b\}$ is monochromatic. We go on to show the partition regularity of a much richer…

### Monochromatic products and sums in $2$-colorings of $\mathbb{N}$

- Mathematics
- 2022

We show that any 2-coloring of N contains inﬁnitely many monochromatic sets of the form { x, y, xy, x + y } , and more generally monochromatic sets of the form { x i , (cid:81) x i , (cid:80) x i : i…

### Monochromatic sums and products in N

- Mathematics
- 2017

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair {x+y, xy}. We answer this question affirmatively in a strong sense by exhibiting a…

### A Diophantine Ramsey Theorem

- MathematicsComb.
- 2021

It is shown that for a wide class of coefficients c 1, …, c s in every finite coloring there is a monochromatic solution to the equation c_1x_1^k + \cdots + {c_s}x_s^k = \text{p}(y).

### Bootstrapping partition regularity of linear systems

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2020

Abstract Suppose that A is a k × d matrix of integers and write $\Re _A:{\mathbb N}\to {\mathbb N}\cup \{ \infty \} $ for the function taking r to the largest N such that there is an r-colouring…

### Partition Regularity of Generalised Fermat Equations

- MathematicsComb.
- 2018

It is proved that given an $r-colouring of $\mathbb{F}_p$ with $p$ prime, there are more than c_{r,\alpha,\beta,\gamma} p^2 solutions to the equation $x+y=z^2$ with all of $x,y,z$ of the same colour.

### Measure preserving actions of affine semigroups and $\{x+y,xy\}$ patterns

- MathematicsErgodic Theory and Dynamical Systems
- 2016

Ergodic and combinatorial results obtained in Bergelson and Moreira [Ergodic theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns. Ergod. Th. & Dynam. Sys. to…

### Partition regular polynomial patterns in commutative semigroups

- Mathematics
- 2016

In 1933 Rado characterized all systems of linear equations with rational coefficients which have a monochromatic solution whenever one finitely colors the natural numbers. A natural follow-up problem…

## References

SHOWING 1-10 OF 29 REFERENCES

### Ergodic theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns

- MathematicsErgodic Theory and Dynamical Systems
- 2015

We establish a ‘diagonal’ ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg’s correspondence principle,…

### A Szemerédi-type regularity lemma in abelian groups, with applications

- Mathematics
- 2003

Abstract.Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the…

### On monochromatic solutions of some nonlinear equations in ℤ/pℤ

- Mathematics
- 2009

Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are x, y ∈ ℤ such that x + y, xy, and x have the…

### Linear Forms and Higher-Degree Uniformity for Functions On $${\mathbb{F}^{n}_{p}}$$

- Mathematics
- 2010

In [GW1] we began an investigation of the following general question. Let L1, . . . , Lm be a system of linear forms in d variables on $${F^n_p}$$, and let A be a subset of $${F^n_p}$$ of positive…

### On a question of Erdős and Moser

- Mathematics
- 2005

For two finite sets of real numbers A and B, one says that B is sum-free with respect to A if the sum set {b + b | b, b ∈ B, b 6= b} is disjoint from A. Forty years ago, Erdős and Moser posed the…

### On sets of integers containing k elements in arithmetic progression

- Mathematics
- 1975

In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic…

### New polynomial and multidimensional extensions of classical partition results

- MathematicsJ. Comb. Theory, Ser. A
- 2017

### Linear forms and quadratic uniformity for functions on ℤN

- Mathematics
- 2011

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom…

### Tight bounds on additive Ramsey‐type numbers

- MathematicsJ. Lond. Math. Soc.
- 2017

A classic result of Rado states that for every homogenous regular equation with integer coefficients there is the least natural number R(n) such that if the elements of [N]={1,…,N} are colored into n…

### Dense Subsets of Pseudorandom Sets

- Mathematics, Computer Science2008 49th Annual IEEE Symposium on Foundations of Computer Science
- 2008

A new proof inspired by Nisan's proof of Impagliazzo's hardcore set theorem via iterative partitioning and an energy increment is presented and provides a new characterization of the notion of "pseudoentropy" of a distribution.