  Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to
be $k$-Sperner if it does not contain a chain of length $k+1$ under set
inclusion and it is saturated if it is maximal with respect to this property.
Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to
$k$, then the minimum size of a saturated $k$-Sperner system
$\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture
by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X|
\geq n_0(k)$ there exists a saturated $k$-Sperner system
$\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most
$2^{(1-\varepsilon)k}$.
  A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an
oversaturated $k$-Sperner system if, for every
$S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more
chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if
$|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and
$O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$,
then the lower bound is best possible, up to a polynomial factor.
