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Claim. P. Erdős and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples in another interval, Section 6, display (20) (p. 159). With $h_{\mathcal P}(n)=\max_mf_{\mathcal P}(n,m)$, where $f_{\mathcal P}(n,m)$ is the least length $L$ for which $(m,m+L]$ holds distinct integers, one divisible by each prime up to $n$, the paper proves

$$h_{\mathcal P}(n)/n\ll\sqrt{n/\log n},$$

from its uniform bound (17) taken with $k=\pi(n)$. Since $h_{\mathcal P}(n)$ is the $h(n)$ of Problem 860 less one (the site's open interval holds $h(n)-1$ integers), this is $h(n)\ll n^{3/2}/(\log n)^{1/2}$. The paper sets it against the Erdős--Selfridge lower bound, its display (19), and writes that the authors do not know how to narrow the gap between the two. The paper is compiled at Erdős and Pomerance (1980).

Covers. The upper bound $h(n)\ll n^{3/2}/(\log n)^{1/2}$. The order of magnitude of $h(n)$, which the problem asks to estimate, stays open.

Acceptance. The result is refereed: Indag. Math. (Proc.) 83 (1980), 147--161, in the issue headed 13 June 1980, which dates this page. The site's commentary records the bound on a problem it labels open, which is not acceptance.

Depends on. Nothing on this wiki; the argument is the paper's own.