Source labels openChecked July 26, 2026

Green's Open Problem 22

If $\{1, \ldots, N\}$ 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.

Combinatorics AMS 5 AMS 11

Mathematical statement

If $\{1, \ldots, N\}$ 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.

Statement source: Green's Open Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

green_22

Canonical source

Complete statement target, proof intentionally absentLean 4

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

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Ben Green's Open Problemsbackground