Let $A\subseteq \mathbb{N}$ have positive density. Must there exist distinct $a,b,c\in A$ such that $[a,b]=c$ (where $[a,b]$ is the least common multiple of $a$ and $b$)?

Let $A\subseteq \mathbb{N}$ have positive density. Must there exist distinct $a,b,c\in A$ such that $[a,b]=c$ (where $[a,b]$ is the least common multiple of $a$ and $b$)?

This is true, a consequence of the positive solution to [447] by Kleitman [Kl71].

Davenport and Erdős [DaEr36] showed that there must exist an infinite sequence $a_1<a_2\cdots$ in $A$ such that $a_i\mid a_j$ for all $i\leq j$, under the assumption that the upper logarithmic density of $A$ is positive.