Erdős Problem #47 — Unit-fraction representations of 1 in dense sequences

Perhaps this is really a Turán type problem and not a Ramsey problem. In other words, if $m$ is sufficiently large and $1 < a_1 < a_2 < \cdots < a_\ell \leq m$ is a sequence of integers for which $\sum_\ell 1/a_\ell > \delta \log m$ then

$$1 = \sum \frac{\varepsilon_i}{a_i} \qquad (\varepsilon_i = 0 \text{ or } 1)$$

is always solvable. I offer 100 dollars for a proof or disproof. Perhaps it suffices to assume that

$$\sum_{a_i < m} \frac{1}{a_i} > C(\log\log m)^2$$

for some large enough $C$.