This says "solved", but I don't see a solution here... I was interested to see if this was correct:
Let's let . Suppose we wanted to attempt to pick elements such that no one is a multiple of another. Since , at least one choice must come from . Let's pick . This eliminates at least one possible choice from ; namely, . Now there are only possible choices left in (at most) and choices left to make. It follows that we need to choose another , which eliminates at least one more choice from . Iterating this process, we see that we need to pick every element from and can't pick anything from , which is of course bad, since we have only chosen elements. Of course, we could probably come to a contradiction a bunch of different ways before exhausting all of , but this seems to work.
I think this is right. If someone could make sure, that would be great; and if there are better solutions, I'd be interested in seeing those too.