OPEN
Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be $a,b,c\in A$ such that
\[[a,b]=[b,c]=[a,c],\]
where $[a,b]$ denotes the least common multiple?
This is false if we ask for four elements with the same pairwise least common multiple, as shown by Erdős
[Er62] (with a proof given in
[Er70]).
See also [535], [537], and [856]. A related combinatorial problem is asked at [857].
