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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 lowest common multiple of $a$ and $b$)?
#487
:
[Er61]
number theory
Davenport and Erdős
[DaEr37]
showed that there must exist an infinite sequence $a_1<a_2\cdots$ in $A$ such that $a_i\mid a_j$ for all $i\leq j$.
This is true, a consequence of the positive solution to
[447]
by Kleitman
[Kl71]
.
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