On a problem involving unit fractions

Abstract

Erd\H{o}s 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 $\sum_{s \in S} 1/s \leq 1$ is smaller than $2^{0.93n}$.