Erdős Problem #487

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.