Let $h(n)$ be such that, for any $m\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\leq i\leq \pi(n)$ such that $p_i\mid a_i$, where $p_i$ denotes the $i$th prime.
A problem of Erdős and Pomerance [ErPo80], who proved that
\[h(n) \ll \frac{n^{3/2}}{(\log n)^{1/2}}.\]
Erdős and Selfridge proved $h(n)>(3-o(1))n$, and Ruzsa proved $h(n)/n\to \infty$.
