Erdős asks in [Er75b] if for every 2-coloring of ℝ, there is an uncountable set $A ⊆ ℝ$ such that
all sums $a + b$ for $a, b ∈ A, a ≠ b$ have the same colour.
In [Ko16] Péter Komjáth constructed a counterexample.
The same result was proven independently in [SWCol] by Sokoup and Weiss.
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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)
