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Bounds for monochromatic solutions to $\{x+y,xy\}$

Ben Green, Mehtaab Sawhney

TL;DR

This work establishes an explicit finite bound for monochromatic occurrences of the pattern ${x+y,xy}$ in $r$-colourings of large initial segments of the integers. The authors develop a soft-analytic framework based on logarithmic averages, a hierarchical multiscale decomposition, and a quantitative inverse theorem that detects biased structure along progressions, while carefully majorizing primes with a Fourier-decomposed majorant. They combine Diophantine properties of almost-primes with averaging projections to transfer combinatorial configurations into arithmetic structure, ultimately producing a monochromatic ${x+y,xy}$ for $x>y>2$ once $N \geq \exp\exp(r^{50})$ and $r$ is large. The result advances finitary Ramsey-type bounds for nonlinear patterns and leverages a rich blend of harmonic analysis, Diophantine approximation, and additive combinatorics. The approach yields effective bounds subject to current method limitations and highlights several challenging directions for extending to stronger patterns such as ${x,y,x+y,xy}$.

Abstract

Let $r$ be a sufficiently large positive integer, and let $N \ge \exp\exp(r^{50})$. Then any $r$-colouring of $[N]$ contains a monochromatic copy of $\{x+y,xy\}$ with $x > y > 2$.

Bounds for monochromatic solutions to $\{x+y,xy\}$

TL;DR

This work establishes an explicit finite bound for monochromatic occurrences of the pattern in -colourings of large initial segments of the integers. The authors develop a soft-analytic framework based on logarithmic averages, a hierarchical multiscale decomposition, and a quantitative inverse theorem that detects biased structure along progressions, while carefully majorizing primes with a Fourier-decomposed majorant. They combine Diophantine properties of almost-primes with averaging projections to transfer combinatorial configurations into arithmetic structure, ultimately producing a monochromatic for once and is large. The result advances finitary Ramsey-type bounds for nonlinear patterns and leverages a rich blend of harmonic analysis, Diophantine approximation, and additive combinatorics. The approach yields effective bounds subject to current method limitations and highlights several challenging directions for extending to stronger patterns such as .

Abstract

Let be a sufficiently large positive integer, and let . Then any -colouring of contains a monochromatic copy of with .

Paper Structure

This paper contains 20 sections, 22 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Proof outline
  4. Acknowledgments
  5. Notation
  6. Diophantine sets and averages
  7. Diophantine properties of almost primes
  8. Fourier decomposition of a majorant for the primes
  9. An inverse theorem
  10. Setting up the proof of the inverse theorem
  11. Proof of the inverse theorem
  12. Averaging projections and orthogonality
  13. Proof of the main theorem
  14. Setting up parameters.
  15. Positivity for $x, xy$
  16. ...and 5 more sections

Key Result

Theorem 1.1

There is a constant $r_0$ such that the following holds. Let $r \geqslant r_0$ be an integer and let $N \geqslant \exp\exp(r^{50})$. Then any $r$-colouring of $[N] :=\{1,\ldots,N\}$ contains a monochromatic copy of $\{x+y,xy\}$ with $x > y > 2$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Remark
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 38 more