Maximum size of antichain-like collection Lets say we have a set $X$ of $n$ elements and a family $S\subseteq\mathcal{P}(X)$ of subsets such that no set $A\in S$ is the union of some of the others, that is,
$$\forall S'\subseteq S\setminus\{A\}\quad\text{we have}\quad A\ne\bigcup_{B\in S'}B$$
what is the maximum size of such a collection $S$ in terms of $n$? Are there estimates/proofs similar to that of Sperner's theorem?
Thanks!
 A: Andy Loo (Union-Free Families of Subsets, arXiv 2015) has studied this question.
For $n=1,2,3,4$ Loo gives exact maximum values ($1,2,4,7$ nonempty sets, respectively), and for $n=5,\ldots,30$ he gives lower and upper bounds; lower bounds are constructive (explicit set families are given).
For $n=5$ the bounds given are $13$ and $15$. (In fact, my brute-force search, using simple Julia code and 300 seconds of computing, says $13$ is the correct value.)
Note that a Sperner family is always a valid solution (if a set is not a superset of any other set, then it cannot be a union of other sets either). This suggests a strategy of taking a maximal Sperner family and then throwing in some extra small sets.
Examples of maximal families:
$$
\begin{array}{lll}
1 & 1 & \{1\} \\
2 & 2 & \{1\},\{2\} \\
3 & 4 & \{1\}, \{1,2\}, \{1,3\}, \{2,3\} \\
4 & 7 & \{1\}, \{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\} \\
5 & 13 &  \{1, 5\},
 \{2, 3\},
 \{2, 4\},
 \{3, 4\},\\
&& \{1, 2, 3\},
 \{1, 2, 4\},
 \{1, 2, 5\},
 \{1, 3, 4\},
 \{1, 3, 5\},
 \{1, 4, 5\},\\
&& \{2, 3, 5\},
 \{2, 4, 5\},
 \{3, 4, 5\}
\end{array}
$$
Caveat: it seems that "union-free family" more often refers to a family where no set is the union of two other sets. This version was apparently proposed by Erdős.
