Are there any known upper bounds on the number of maximal independent sets in a hypergraph? I'm aware that simple graphs have an upper bound of $O(3^{n/3})$. How about on the number of independent sets?



Number of maximal independent sets in a hypergraph
The family of maximal independent sets of a hypergraph has the property that no member of the family is contained in another. Such a family of sets is called an *antichain* or a *clutter* or a *Sperner family*. By [Sperner's theorem](https://en.wikipedia.org/wiki/Sperner%27s_theorem), a Sperner family of subsets of an $n$-element set has at most $\binom n{\lfloor n/2\rfloor}$ members. In particular, a hypergraph $H=(V,E)$ of order $|V|=n$ has at most $\binom n{\lfloor n/2\rfloor}$ maximal independent sets. This bound is attained when $E$ is the set of all $\left(\lfloor n/2\rfloor+1\right)$-element subsets of $V$.


