On Saturated $k$-Sperner Systems

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 $\mathrm{\mid }X\mathrm{\mid }$ 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 $\epsilon >0$ such that for every $k$ and $|X| \geq n_0(k)$thereexistsasaturated$k$-Spernersystem$$\mathcal{F}$\subseteq$\mathcal{P}$(X)$withcardinalityatmost$2^{(1-\varepsilon)k}$. 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 $\mathrm{\mid }X\mathrm{\mid }\ge k$, then the smallest such collection contains between $2^{k\mathrm{/}2-1}$ and $O\left(\frac{\log k}{k}{2}^k\right)$ elements. We show that if $\mathrm{\mid }X\mathrm{\mid }\ge k^2+k$, then the lower bound is best possible, up to a polynomial factor.