Problem #949: Let $S\subset \mathbb{R}$ be a set containing no solutions...

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Progress

Erdős suggests that if the answer is no one could consider the variant where we assume that $S$ is Sidon (i.e. all $a+b$ with $a,b\in S$ are distinct, aside from the trivial coincidences).

In the comments Dillies gives a positive proof of this, found by AlphaProof: in other words, if $S\subset \mathbb{R}$ is a Sidon set then there exists $A\subseteq \mathbb{R}\backslash S$ of cardinality continuum such that $A+A\subseteq \mathbb{R}\backslash S$.