Let $A \subset \mathbb{N}$ be infinite with no distinct $a, b, c \in A$ such that $a \mid (b + c)$ with $b, c > a$. Can $|A \cap [1, N]|/\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?

Given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$, such that there are infinitely many $N$ such that $$\lvert A\cap\{1,\ldots,N\}\rvert > \frac{N}{f(N)}.