TITLE: Alternating sum over collections closed under containment
QUESTION [9 upvotes]: Let $\mathscr{C}$ be a collection of subsets of a finite set $P$. Assume $\mathscr{C}$ is closed under containment: if $S\subset P$ is in $\mathscr{C}$, then every set $S'\subset P$ containing $S$ is in $\mathscr{C}$.
What can we say about
$$\sum_{S\in \mathscr{C}} (-1)^{|S|},$$ where $|S|$ is the number of elements of $S$? In particular, is the absolute value of this sum bounded
by the number of minimal elements of $\mathscr{C}$, i.e.,
$$\left|\sum_{S\in \mathscr{C}} (-1)^{|S|}\right| \leq |\{S\in \mathscr{C}:\not \exists S'\subsetneq S \;\text{s.t.}\; S'\in \mathscr{C}\}|?$$

REPLY [8 votes]: There is an interpretation of your inequality using simplicial complex and $h$-vector. Namely, let $\Delta$ be the set of all $P-S$ with $S$ in your set. Then $\Delta$ is a collection closed under inclusion, hence a simplicial complex.
The invariant you are interested in is, up to sign, the alternating sum of $f_i$: the number of $i$-dim faces of $\Delta$. Then, by a well-known formula, it is equal to $|h_d|$, where $h_i$ forms the $h$-vector of $\Delta$, and $d-1$ is the dimension of $\Delta$.
The right hand side of the inequality you are interested in is the number of facets (maximal elements) of $\Delta$. As people have pointed out, in general the inequality you want does not hold.
However, I would like to point out that it is likely to hold under certain extra topological/homological  assumptions. For instance, if $\Delta$ is Cohen-Macaulay, then all the $h_i$ are non-negative, and the number of facets is $f_{d-1}=\sum_{i\geq 0} h_i\geq h_d$, which is what you need. In fact, as $h_0=1$ and $h_1=n-d$ where $n=|P|$, you get something a little stronger.
One can prove other inequalities for $h_i$ under weaker conditions. For example, if $\Delta$ satisfies Serre's conditions $(S_{r})$,  one still have non-negativity of $h_{\leq r}$, a result first proved by Murai-Terai.  I discussed some of them in a recent talk (but it is perhaps a bit algebraic).