Denote by $\mathcal{P}(S)$ the set of non-empty subsets of a finite $S$.
Suppose that $A\subset \mathcal{P}(S)$ is a downset, i.e., every subset $Q$ of any $P\in A$ is also contained in $A$. We can also assume that the atoms are in $A$, i.e. $\{x\}\in A$ for each $x\in S$.
Now, take $\overline{A}=\mathcal{P}(S)\setminus A$. From the definition of a downset, it follows that $\overline{A}$ is an upset, i.e. every subset $Q$ containing any $P\in \overline{A}$ is also contained in $\overline{A}$.
Let $M=\operatorname{max}\{\left|P\right|:P\in A\}$ and $\overline{m}=\operatorname{\min}\{\left|P\right|:P\in \overline{A}\}$.
It seems like these two things should be related. We can say one obvious thing, and that is that $M\geq \overline{m} - 1$. ($\overline{m}$ is the smallest size of a set not contained in $A$, so, all the $k$-subsets of $S$ are contained in $A$ for each $k=1,\ldots ,m-1$. Obviously $M\geq k$ for each of these.)
What can be said about the relationship between $M$ and $\overline{m}$? Is there a nontrivial bound on $M$ in terms of $\overline{m}$ and $|S|$? Is there a good reference which talks about this (and/or other relations involving the members of the sets$^\star$ $\{\left|P\right|:P\in A\}$ and $\{\left|P\right|:P\in \overline{A}\}$)?
$^\star$ Note. It may help to note that these sets could be replaced by the sets $\{\left|P\right|:P\text{ is a maximal element of }A\}$ and $\{\left|P\right|:P\text{ is a minimal element of }\overline{A}\}$, respectively, with no loss of information.
You really can’t say much. Fix $s\in S$, and let $A=\{s\}\cup\wp(S\setminus\{s\})$; $A$ is a downset containing all of the atoms, $M=|S|-1$, and $\overline m=2$. On the other hand, it’s easy to get $M=\overline m-1$ by taking $A=\{a\subseteq A:|a|\le k\}$ for some $k$.