A Stretched-Exponential Bound for an Erdos--Graham Unit-Fraction Problem

For a finite multiset $A$ of positive integers, write $\mathcal{R}(A)=\sum_{a\in A}a^{-1}$ and let $\varepsilon(A)$ be the distance from $1$ to the largest reciprocal subsum of $A$ that does not exceed $1$. Erd\H{o}s and Graham proved that $\varepsilon(A)\ll K^{-2}$ whenever $\mathcal{R}(A)>K$, and asked whether one always has $\varepsilon(A)\leq \exp(-cK)$ for an absolute constant $c>0$. We prove the stretched-exponential estimate $$\varepsilon(A)\leq \exp\bigl(-c\sqrt{K\log K}\bigr)$$ for all sufficiently large $K$.