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?

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 $\liminf \frac{|A \cap \{1, \dotsc, N\}|}{N^{1/2}} > 0$ ?

The DeepMind prover agent has found a formal proof of this statement.