SOLVED
This has been resolved in some other way than a proof or disproof.
Let $A(N)$ denote the maximal cardinality of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in S}\frac{1}{n}\neq 1$ for all $S\subseteq A$. Estimate $A(N)$.
Erdős and Graham believe the answer is $A(N)=(1+o(1))N$. Croot
[Cr03] disproved this, showing the existence of some constant $c<1$ such that $A(N)<cN$ for all large $N$. It is trivial that $A(N)\geq (1-\frac{1}{e}+o(1))N$.
Liu and Sawhney
[LiSa24] have proved that $A(N)=(1-1/e+o(1))N$.
View the LaTeX source
Additional thanks to: Zachary Hunter
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #300, https://www.erdosproblems.com/300, accessed 2026-01-16
