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Claim. Let $R(N)$ be the largest size of a set $A\subseteq\{1,\ldots,N\}$ whose subset sums $\sum_{n\in S}1/n$, $S\subseteq A$, are pairwise distinct, let $S(N)$ count the distinct reciprocal subset sums of $\{1,\ldots,N\}$, and let $\log_j$ be the $j$-fold iterated natural logarithm. Theorem 3 of the paper (p. 610) gives $\log S(N)\le\frac{N\log_rN}{\log N}\prod_{j=3}^r\log_jN$ for $r\ge1$ and $\log_{2r}N\ge1$. The $2^{R(N)}$ subset sums of an extremal set are distinct values among those $S(N)$ counts, so $2^{R(N)}\le S(N)$, and hence

$$R(N)\ \le\ \frac1{\log2}\,\frac{N\log_rN}{\log N}\prod_{j=3}^{r}\log_jN \qquad(r\ge1,\ \log_{2r}N\ge1),$$

the upper bound the site prints for Problem 321.

Covers. An upper bound for $R(N)$. The extra factor $\log_rN$ makes it exceed the order of magnitude by an unbounded factor; the upper bound of that order is Young, Zhu and Luo's accepted claim.

Depends on. Theorem 3, the library page that states the bound for $\log S(N)$.

Acceptance. Refereed: M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions II, Illinois J. Math. 20 (1976), no. 4, 598--613, DOI 10.1215/ijm/1256049650; the issue is dated 1 December 1976 in the Crossref record, the date this page carries. The proof is not verified by this corpus.