Problem 12

Erdős Problem #12: declared status 'open'. Formalized: yes. Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with$$\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0?$$Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with$$\lvert A\cap\{1,\ldots,N\}\rvert<N^{1-c}?$$Is it true that$$\sum_{n\in A}\frac{1}{n}<\infty?$$ Current best: Elsholtz and Planitzer [ElPl17] have constructed such an $A$ with$$\lvert A\cap\{1,\ldots,N\}\rvert\gg \frac{N^{1/2}}{(\log N)^{1/2}(\log\log N)^2(\log\log\log N)^2}.$$Schoen [Sc01] proved that if all elements in $A$ are pairwise coprime then$$\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}$$for infinitely many $N$. Prize: no. Tags: number theory.