Is it true that, for any $2$-colouring of $\mathbb{R}$, there is a set $A\subseteq \mathbb{R}$ of cardinality ${\aleph }_1$ such that all sums $a+b$ with $a\ne b$ and $a,b\in A$ are the same colour?

AMS 3 5 · solved (literature)

theorem erdos_965 :
    answer(False) ↔ ∀ f : ℝ → Fin 2, ∃ A : Set ℝ, ¬ A.Countable ∧
      ∀ᵉ (a ∈ A) (b ∈ A) (c ∈ A) (d ∈ A), a ≠ b → c ≠ d → f (a + b) = f (c + d)