Long Intervals Without Distinct Multiples of the First $n$ Positive Integers

For positive integers $n$ and $m$, let $f(n,m)$ be the least integer $h\ge 0$ such that $(m,m+h]$ contains distinct integers $a_1,\dots ,a_n$ satisfying $i\mid a_i$ for $1\le i\le n$, and put $F(n)={\max }_{m\in \mathbb{N}}f(n,m)$. A recent theorem of van Doorn [INTEGERS, 2026; arXiv:2601.16972] gives $F(n)-f(n,n)>0.36n\log n\mathrm{/}\log \log n$ for sufficiently large $n$. We prove $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{F(n)-f(n,n)}{n\log n}\ge \frac{1}{\mathrm{e}}\mathrm{.}$$ Thus, for every fixed $c<1\mathrm{/}\mathrm{e}$ and all sufficiently large $n$, some interval of length $cn\log n$ contains no system of pairwise distinct multiples of $1,2,\dots ,n$. The proof applies an Erdős–Pomerance smooth-number obstruction at starting points $m\asymp n\log n$, using local saddle-point estimates of Hildebrand and Tenenbaum.