Problem 321

Erdős Problem #321: declared status 'open'. Formalized: yes. What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$? Current best: Results of Bleicher and Erd\H{o}s from [BlEr75] and [BlEr76b] imply that$$\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq R(N)\leq \frac{1}{\log 2}\log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),$$valid for any $k\geq 4$ with $\log_kN\geq k$ and any $r\geq 1$ with $\log_{2r}N\geq 1$. Prize: no. OEIS: A384927, A391592. Tags: number theory, unit fractions.