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import FormalConjecturesUtilErdős Problem 645
[BrLa99] Brown, Tom C. and Landman, Bruce M., Monochromatic arithmetic progressions with large differences. Bull. Austral. Math. Soc. (1999), 21--35.
namespace Erdos645
If ℕ is $2$-coloured then there must exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$.
This was first proved by Brown and Landman [BrLa99], who in fact show that this is always possible with $d>f(x)$ for any increasing function $f$.
This was formalized in Lean by Alexeev using Aristotle and ChatGPT.
@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos645.lean"]
theorem erdos_645 (c : ℕ → Bool) : ∃ x d, 0 < x ∧ x < d ∧
(∃ C, c x = C ∧ c (x + d) = C ∧ c (x + 2 * d) = C) := c:ℕ → Bool⊢ ∃ x d, 0 < x ∧ x < d ∧ ∃ C, c x = C ∧ c (x + d) = C ∧ c (x + 2 * d) = C
All goals completed! 🐙
end Erdos645