Erdős Problem #792

declared status 'open'. Formalized: no. Let $f(n)$ be maximal such that in any $A\subset \mathbb{Z}$ with $\lvert A\rvert=n$ there exists some sum-free subset $B\subseteq A$ with $\lvert B\rvert \geq f(n)$, so that there are no solutions to$$a+b=c$$with $a,b,c\in B$. Estimate $f(n)$. Current best: The best lower bound known is$$f(n)\geq \frac{n}{3}+c\log\log n$$for some constant $c>0$, due to Bedert [Be25b]. The best upper bound known is$$f(n) \leq \frac{n}{3}+o(n),$$due to Eberhard, Green, and Manners [EGM14]. Prize: no. Tags: additive combinatorics.

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