OPEN
Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that
\[[a_i,a_{i+1},\ldots,a_{i+k-1}] < X,\]
where the left-hand side is the least common multiple. Is it true that, for every $\epsilon >0$, there exists some $k$ such that
\[F(A,X,k)<X^\epsilon?\]
A problem of Erdős and Szemerédi, who proved that for every $A$
\[F(A,X,3) \ll X^{1/3}\log X,\]
and there is an $A$ such that
\[F(A,X,3) \gg X^{1/3}\log X\]
for infinitely many $X$. There may be a sequence for which this holds for every $X$.
