I am interested in a family of sets $C$ with the following property:

- if you take any $k$ distinct subsets $S_1, S_2, \dotsc, S_k \in C$, then $S_1$ is not a subset of $S_2 \cup S_3 \cup \dotso \cup S_k$.

If we have $k = 2$, these are Sperner families.

Has the case $k > 2$ been discussed in the literature?

**Edit:** A colleague just pointed out that Jukna (2011): *"Extremal Combinatorics"*, Section 8.6 briefly discusses these families and gives some references. The book uses the term **union-free families**.

However, it seems that none of the results mentioned in the book address the following corner case that I am mainly interested in:

- The size of the universe $U$ is $O(k)$.
- The size of the family is "non-trivial" (say, $\Omega(k^2)$).

I would like to know more about *positive* results for this case (I do not need an explicit construction; an existential result is fine).

It seems that in the literature, the main focus has been on families that are exponentially large in $|U|$, and to construct such families we need to have a fairly large $|U|$ in comparison with $k$. However, I would like to understand the case of a small $|U|$ (and not necessarily exponentially large $|C|$).

equalto a set in a family. You ask about containment, not equality. What are you really interested in? $\endgroup$