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 $\epsilon (A)$ be the distance from $1$ to the largest reciprocal subsum of $A$ that does not exceed $1$. Erdős and Graham proved that $\epsilon (A)\ll K^{-2}$ whenever $\mathcal{R}(A)>K$, and asked whether one always has $\epsilon (A)\le \exp (-cK)$ for an absolute constant $c>0$. We prove the stretched-exponential estimate $$\epsilon (A)\le \exp (-c\sqrt{K\log K})$$ for all sufficiently large $K$.