erdos_949

Let $S \subseteq \mathbb{R}$ be a set containing no solutions to $a + b = c$. Must there be a set $A \subseteq \mathbb{R} \setminus S$ of cardinality continuum such that $A + A \subseteq \mathbb{R}\setminus S$?

theorem erdos_949 : answer(sorry) ↔
    ∀ S : Set ℝ, (∀ a ∈ S, ∀ b ∈ S, a + b ∉ S) → ∃ A ⊆ Sᶜ, #A = 𝔠 ∧ A + A ⊆ Sᶜ

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