TITLE: What is the maximal number of subsets of a finite set, such that no one is the unions of some other ones?
QUESTION [10 upvotes]: Suppose $S$ is a set with $n$ elements.  What is the maximal value of $m$ such that there exists a collection $\{A_k\}_{k=1,\dots m}$ of subsets of $S$ such that each $A_i$ is not the union of some other elements of $\{A_k\}_{k=1,\dots m}$?
Is there a elegant formula for this number?

REPLY [6 votes]: This was stated as an open problem on page 161 of Mitchell, Turan's graph theorem and maximum independent sets in Brandt semigroups, which is pages 151-162 of Isabel M Araujo et al., eds., Proceedings of the Workshop Semigroups and Languages, 2004: "Let $X$ denote the set of subsets of an $n$ element set. What is the maximum cardinality of a subset $Y$ of $X$ with the property that no set in $Y$ is the union of other sets in $Y$?"
There is no further discussion of the problem in the paper. 
A pdf of the paper is available here.