Problem Statement
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 $[a,b]$ denotes the least common multiple?
Categories:
Number Theory
Progress
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]).This was also asked by Erdős at the 1991 problem session of West Coast Number Theory.
In the comments Weisenberg sketches a construction of a set $A\subseteq [1,N]$ without this property such that\[\lvert A\rvert \gg (\log\log N)^{f(N)}\frac{N}{\log N}\]for some $f(N)\to \infty$. Weisenberg also sketches a proof of the main problem when $\epsilon>\frac{221}{225}$.
See also [535], [537], and [856]. A related combinatorial problem is asked at [857].
Source: erdosproblems.com/536 | Last verified: January 15, 2026