Erdős problem 536

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 distinct $a,b,c\in A$ such that $$[a, b]=[b, c]=[a, c],$$ where $[\cdot, \cdot]$ denotes the least common multiple?