Take the tour ×  Let $n$ be a natural number, and consider the
discrete cube $2^{[n]} := \{ A: A \subset \{1,\ldots,n\}\}$
consisting of all subsets of the $n$-element set $[n] :=
\{1,\ldots,n\}$. Define a downset in $2^{[n]}$ to be a
collection ${\mathcal D}$ of elements $A$ in $2^{[n]}$ with the
property that if $A \in {\mathcal D}$ and $B \subset A$, then $B
\in {\mathcal D}$.  My question is: what are the largest and
smallest possible values for the alternating sum $\sum_{A \in
{\mathcal D}} (-1)^{|A|}$, as ${\mathcal D}$ ranges over
downsets in $2^{[n]}$, as a function of $n$? (Here $|A|$ denotes
the cardinality of $A$.)  The trivial bounds here are $\pm
2^{n-1}$, by taking only the positive or negative values of
${\mathcal D}$, but of course these values are attained on a
"checkerboard" set which is very far from being a downset, and
this suggests that significant improvement is possible.  By
taking ${\mathcal D}$ to be the set of all subsets $A$ of
$2^{[n]}$ of cardinality at most $r$ for some $1 \leq