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import FormalConjecturesUtilErdős Problem 303
[BrRo91] Brown, Tom C. and Rödl, Voijtech, Monochromatic solutions to equations with unit fractions. Bull. Austral. Math. Soc. (1991), 387-392.
namespace Erdos303
Is it true that in any finite colouring of the integers there exists a monochromatic solution to $\frac 1 a = \frac 1 b + \frac 1 c$ with distinct $a, b, c$?
This is true, as proved by Brown and Rödl [BrRo91].
This was formalized in Lean by Yuan using Seed-Prover.
@[category research solved, AMS 5 11, formal_proof using lean4 at "https://www.erdosproblems.com/forum/thread/303"]
theorem erdos_303 :
answer(True) ↔
-- For any finite colouring of the integers
∀ (𝓒 : ℤ → ℤ), (Set.range 𝓒).Finite →
-- There exists integers `a, b, c`
∃ (a b c : ℤ),
-- that are non-zero and distinct.
[a, b, c, 0].Nodup ∧
-- `a, b, c` satisfy the equation
(1/a : ℝ) = 1/b + 1/c ∧
-- `a, b, c` have the same color
(𝓒 '' {a, b, c}).Subsingleton := ⊢ True ↔
∀ (𝓒 : ℤ → ℤ),
(Set.range 𝓒).Finite → ∃ a b c, [a, b, c, 0].Nodup ∧ 1 / ↑a = 1 / ↑b + 1 / ↑c ∧ (𝓒 '' {a, b, c}).Subsingleton
All goals completed! 🐙
end Erdos303