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 $A + A \subseteq \mathbb{R}\setminus S$?

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1True ↔ ∀ (S : Set ℝ), (∀ a ∈ S, ∀ b ∈ S, a + b ∉ S) → ∃ A ⊆ Sᶜ, Cardinal.mk ↑A = Cardinal.continuum ∧ A + A ⊆ Sᶜ

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