Erdős Problem 949

Reference: erdosproblems.com/949

theorem Erdos949.erdos_949 : sorry ↔ ∀ (S : Set ℝ), (∀ a ∈ S, ∀ b ∈ S, a + b ∉ S) → ∃ A ⊆ Sᶜ, Cardinal.mk ↑A = Cardinal.continuum ∧ A + A ⊆ Sᶜ

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 Erdos949.erdos_949.variants.sidon : True ↔ ∀ (S : Set ℝ), IsSidon S → ∃ A ⊆ Sᶜ, Cardinal.mk ↑A = Cardinal.continuum ∧ A + A ⊆ Sᶜ

Let $S\sub \mathbb{R}$ be a Sidon set. Must there be a set $A\sub \mathbb{R}∖S$ of cardinality continuum such that $A + A \sub \mathbb{R}∖S$?