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All Random Solved Random Open
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].