SOLVED - $100
Let $A\subseteq \{1,\ldots,N\}$ be such that there are no $a,b,c\in A$ such that $a\mid(b+c)$ and $a<\min(b,c)$. Is it true that $\lvert A\rvert\leq N/3+O(1)$?
Asked by Erdős and Sárközy, who observed that $(2N/3,N]\cap \mathbb{N}$ is such a set. The answer is yes, as proved by Bedert
[Be23].
For the infinite version see [12].
In [Er92c] Erdős asks about the general version where $a\nmid (b_1+\cdots+b_r)$ for $a<\min(b_1,\ldots,b_r)$, and whether $\lvert A\rvert \leq N/(r-1)+O(1)$.
