FormalConjectures.ErdosProblems.«1199»

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import FormalConjecturesUtil

Erdős Problem 1199

References:

erdosproblems.com/1199

[Hi79] Hindman, Neil, Partitions and sums of integers with repetition. J. Combin. Theory Ser. A (1979), 19--32.

[Ow74] J. Owings, E2494. Amer. Math. Monthly (1974), 902.

open Pointwisenamespace Erdos1199

Is it true that in any 2-colouring of $\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?

A conjecture of Owings [Ow74].

@[category research open, AMS 5]
theorem erdos_1199 :
    answer(sorry) ↔
    ∀ (color : ℕ → Fin 2), ∃ (A : Set ℕ),
    A.Infinite ∧ ∀ n ∈ (A+A), ∀ m ∈ (A+A), color n = color m := by⊢ True ↔ ∀ (color : ℕ → Fin 2), ∃ A, A.Infinite ∧ ∀ n ∈ A + A, ∀ m ∈ A + A, color n = color m
  sorryAll goals completed! 🐙

Hindman [Hi79] has shown that this is false for 3-colourings.

@[category research solved, AMS 5,
  formal_proof using lean4 at "https://github.com/HowieHwong/lean-erdos-proofs/blob/b8b641ba2d00dc4d1fe205a078a4159372672459/Erdos/P1199.lean#L85"]
theorem erdos_1199.variants.three :
    ∃ (color : ℕ → Fin 3), ∀ (A : Set ℕ),
    A.Infinite → ∃ n ∈ (A+A), ∃ m ∈ (A+A), color n ≠ color m := by⊢ ∃ color, ∀ (A : Set ℕ), A.Infinite → ∃ n ∈ A + A, ∃ m ∈ A + A, color n ≠ color m
  sorryAll goals completed! 🐙

end Erdos1199