Problem 787

Erdős Problem #787: declared status 'open'. Formalized: no. Let $g(n)$ be maximal such that given any set $A\subset \mathbb{R}$ with $\lvert A\rvert=n$ there exists some $B\subseteq A$ of size $\lvert B\rvert\geq g(n)$ such that $b_1+b_2\not\in A$ for all $b_1\neq b_2\in B$. Estimate $g(n)$. Current best: The current best bounds known are$$(\log n)^{1+c} \ll g(n) \ll \exp(\sqrt{\log n})$$for some constant $c>0$, the lower bound due to Sanders [Sa21] and the upper bound due to Ruzsa [Ru05]. Prize: no. Tags: additive combinatorics.