A subset $A$ of $\{1,...,n\}$ is called primitive if it has the property that no element of $A$ is a multiple of any other element in $A$. Behrend's theorem states that the logarithmic density of any primitive subset must be $O(1/\sqrt{\log \log n})$ (https://en.wikipedia.org/wiki/Behrend%27s_theorem).
I'm looking for a similar result for what I call an anti-primitive subset of $\{1,...,n\}$: A subset which contains all divisors of any of its elements. That is, a subset with the property that $m \in A, \ i | m \implies i \in A$.
Any reference to papers in this direction would be useful.