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.

theorem Erdos1199.erdos_1199 : True ↔ ∀ (color : ℕ → Fin 2), ∃ (A : Set ℕ), A.Infinite ∧ ∀ n ∈ A + A, ∀ m ∈ A + A, color n = color m

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].

theorem Erdos1199.erdos_1199.variants.three : ∃ (color : ℕ → Fin 3), ∀ (A : Set ℕ), A.Infinite → ∃ n ∈ A + A, ∃ m ∈ A + A, color n ≠ color m

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