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

erdos_949 (general sum-free version) is OPEN; the Sidon variant in the Lean file is already proven. State of the art covers #S<𝔠 (unconditional), S Sidon, and S with the Baire property (Chojecki). My

erdos_949 (general sum-free version) is OPEN; the Sidon variant in the Lean file is already proven. State of the art covers #S<𝔠 (unconditional), S Sidon, and S with the Baire property (Chojecki). My new rigorous contribution: the Lebesgue-MEASURABLE case is also YES (null S ⟹ bad relation null ⟹ Mycielski perfect set; |S|>0 ⟹ 0∉cl(S) via a density lemma ⟹ interval near 0). This reduces any counterexample to a doubly-pathological set that is BOTH non-measurable AND lacks the Baire property; the residual core is genuinely set-theoretic. Full problem not settled — honest partial progress + reduction.