  We call a family $\mathcal{F}$ of subsets of $[n]$ $s$-saturated if it
contains no $s$ pairwise disjoint sets, and moreover no set can be added to
$\mathcal{F}$ while preserving this property (here $[n] = \{1,\ldots,n\}$).
  More than 40 years ago, Erd\H{o}s and Kleitman conjectured that an
$s$-saturated family of subsets of $[n]$ has size at least $(1 -
2^{-(s-1)})2^n$. It is easy to show that every $s$-saturated family has size at
least $\frac{1}{2}\cdot 2^n$, but, as was mentioned by Frankl and Tokushige,
even obtaining a slightly better bound of $(1/2 + \varepsilon)2^n$, for some
fixed $\varepsilon > 0$, seems difficult. In this note, we prove such a result,
showing that every $s$-saturated family of subsets of $[n]$ has size at least
$(1 - 1/s)2^n$.
  This lower bound is a consequence of a multipartite version of the problem,
in which we seek a lower bound on $|\mathcal{F}_1| + \ldots + |\mathcal{F}_s|$
where $\mathcal{F}_1, \ldots, \mathcal{F}_s$ are families of subsets of $[n]$,
such that there are no $s$ pairwise disjoint sets, one from each family
$\mathcal{F}_i$, and furthermore no set can be added to any of the families
while preserving this property. We show that $|\mathcal{F}_1| + \ldots +
|\mathcal{F}_s| \ge (s-1)\cdot 2^n$, which is tight e.g.\ by taking
$\mathcal{F}_1$ to be empty, and letting the remaining families be the families
of all subsets of $[n]$.
