Green's Open Problem 22

If 1, …, N is r-coloured then, for N geqslant N_0(r), there are integers x, y geqslant 3 such that x + y, xy have the same colour. Find reasonable bounds for N_0(r). The goal is to improve upon the Green-Sawhney bound.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-green-22,
  title        = {Green's Open Problem 22},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-22}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

If ${1, \dots, N}$ is $r$ -coloured then, for $N ⩾ N_{0} (r)$ , there are integers $x, y ⩾ 3$ such that $x + y, x y$ have the same colour.

Find reasonable bounds for $N_{0} (r)$ . The goal is to improve upon the Green-Sawhney bound.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«22». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_22 :
    let ans := (answer(sorry) : ℕ → ℝ)
    ∀ᶠ r in atTop, N₀ r ≤ ans r ∧
    ans =o[atTop] GreenSawhneyBound

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[Gr26] Ben Green's Open Problems
[Mo17] Moreira, Joel. "Monochromatic sums and products in N." Annals of Mathematics 185.3 (2017): 1069-1090.
[GrSa25] Green, Ben, and Mehtaab Sawhney. "Bounds for monochromatic solutions to ${x + y, x y}$ ." arXiv preprint arXiv:2511.09365 (2025).
[Ri25] Richter, Florian K. "Sums and products in sets of positive density." arXiv preprint arXiv:2507.00515 (2025).
[BoSa24] Bowen, Matt, and Marcin Sabok. "Monochromatic products and sums in the rationals." Forum of Mathematics, Pi. Vol. 12. Cambridge University Press, 2024.
[Bo25] Bowen, Matt. "Monochromatic products and sums in 2-colorings of N." Advances in Mathematics 462 (2025): 110095.
[Al23] Alweiss, Ryan. "Monochromatic Sums and Products over $Q$ ." arXiv preprint arXiv:2307.08901 (2023).

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.