On a problem involving unit fractions

Abstract

Erdős and Graham proposed to determine the number of subsets $S\subseteq \left\{1,2,\dots ,n\right\}$ with ${\sum }_{s\in S}1/s=1$ and asked, among other things, whether that number could be as large as $2^{n-o(n)}$. We show that the number of subsets $S\subseteq \left\{1,2,\dots ,n\right\}$ with ${\sum }_{s\in S}1/s\le 1$ is smaller than $2^{0.93n}$.