/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
import FormalConjecturesUtilGreen's Open Problem 22
[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, xy} $." 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 $\mathbb {Q} $." arXiv preprint arXiv:2307.08901 (2023).
open Finset Filter
namespace Green22
The monochromatic sum-product property: a colouring $c$ of ${1, \ldots, N}$ has a pair $(x, y)$ with $x, y \geq 3$ such that $x + y$ and $xy$ are both in ${1, \ldots, N}$ and receive the same colour.
def HasMonochromaticSumProduct (N : ℕ) (r : ℕ) (coloring : Icc 1 N → Fin r) : Prop :=
∃ x y : ℕ, 3 ≤ x ∧ 3 ≤ y ∧
∃ h_sum : x + y ∈ Icc 1 N, ∃ h_prod : x * y ∈ Icc 1 N,
coloring ⟨x + y, h_sum⟩ = coloring ⟨x * y, h_prod⟩
$N_0(r)$ is the smallest $N$ such that every $r$-colouring of ${1, \ldots, N}$ has the monochromatic sum-product property.
noncomputable def N₀ (r : ℕ) : ℕ :=
sInf {N | ∀ c : Icc 1 N → Fin r, HasMonochromaticSumProduct N r c}
open scoped AsymptoticsThe upper bound function from [GrSa25].
noncomputable def GreenSawhneyBound (r : ℕ) : ℝ := Real.exp (Real.exp (r ^ 50))
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.
@[category research open, AMS 5 11]
theorem green_22 :
let ans := (answer(sorry) : ℕ → ℝ)
∀ᶠ r in atTop, N₀ r ≤ ans r ∧
ans =o[atTop] GreenSawhneyBound := ⊢ let ans := sorry;
∀ᶠ (r : ℕ) in atTop, ↑(N₀ r) ≤ ans r ∧ ans =o[atTop] GreenSawhneyBound
All goals completed! 🐙
[GrSa25, Theorem 1.1] found a permissible upper bound.
@[category research solved, AMS 5 11]
theorem green_22.variants.green_sawhney_bound :
∀ᶠ r in atTop, N₀ r ≤ GreenSawhneyBound r := ⊢ ∀ᶠ (r : ℕ) in atTop, ↑(N₀ r) ≤ GreenSawhneyBound r
All goals completed! 🐙
[Mo17, Corollary 1.5] For any finite coloring of $\mathbb{N}$ there exist (infinitely many) $x, y \in \mathbb{N}$ such that ${xy, x + y}$ is monochromatic.
This guarantees that $N_0(r)$ is well-defined.
Note: [Mo17] also establishes that $x$ is of the same colour.
@[category research solved, AMS 5 11]
theorem green_22.variants.moreira_infinite :
∀ (r : ℕ) (coloring : ℕ → Fin r),
Set.Infinite { p : ℕ × ℕ |
0 < p.1 ∧ 0 < p.2 ∧
coloring p.1 = coloring (p.1 * p.2) ∧
coloring p.1 = coloring (p.1 + p.2) } := ⊢ ∀ (r : ℕ) (coloring : ℕ → Fin r),
{p | 0 < p.1 ∧ 0 < p.2 ∧ coloring p.1 = coloring (p.1 * p.2) ∧ coloring p.1 = coloring (p.1 + p.2)}.Infinite
All goals completed! 🐙
Since $x, y \geq 3$, we must have $xy \geq 9$, so $N_0(r) \geq 9$ (assuming $N_0(r)$ is well-defined, which follows from [Mo17]).
@[category research solved, AMS 5 11]
theorem green_22.variants.lower_nine : ∀ r : ℕ, r ≠ 0 → 9 ≤ N₀ r := ⊢ ∀ (r : ℕ), r ≠ 0 → 9 ≤ N₀ r
All goals completed! 🐙
-- TODO(jeangud) Add Hindman's problem versions mentioned in [Gr26].
end Green22