Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that
$$
\frac{N}{\log N} \prod_{i=3}^{k} \log_i N \le R(N),
$$
valid for any $k \ge 4$ with $\log_k N \ge k$ and any $r \ge 1$ with $\log_{2r} N \ge 1$. (In these bounds $\log_i n$ denotes the $i$-fold iterated logarithm.)
[BlEr75] Bleicher, M. N. and Erdős, P., _The number of distinct subsums of $\sum \sb{1}\spN\,1/i$_. Math. Comp. (1975), 29-42.
[BlEr76b] Bleicher, Michael N. and Erdős, Paul, _Denominators of Egyptian fractions. II_. Illinois J. Math. (1976), 598-613.
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