OPEN
Let $C>0$ be some constant and $n$ be large. If $A\subseteq\{1,\ldots,n\}$ has $\sum_{n\in A}\frac{1}{n}\leq C$ then is there some $c$ (which may depend on $C$) such that
\[\{ m\leq n : a\nmid m\textrm{ for all }a\in A\}\]
has size $\geq n/(\log n)^{c}$?
An example of Schinzel and Szekeres
[ScSz59] shows that this would be best possible (up to the value of $c$).
See also [542].
