  For integers $0 \leq m \leq l \leq n-m$, the truncated Boolean lattice ${\cal
B}_n(m,l)$ is the poset of all subsets of $[n] = \{1, 2, \ldots, n\}$ which
have size at least $m$ and at most $l$. ${\cal C} \subseteq {\cal B}_n(m,l)$ is
a {\em cutset} if it meets every chain of length $l-m$ in ${\cal B}_n(m,l)$,
and the {\em width} of ${\cal C}$ is the size of the largest antichain in
${\cal C}$. We conjecture that for $n >> m$ the minimum width $h_n(m,l)$ of a
cutset in ${\cal B}_n(m,l)$ is $\Sigma_{j \geq 0} \Delta_n(m-jc) =
\Delta_n(m)+\Delta_n(m-c)+\Delta_n(m-2c)+ \dots$, where $c=l-m+1$ is the number
of level sets in ${\cal B}_n(m,l)$ and $\Delta_n(k)={n \choose k}- {n \choose
k-1}$. We establish our conjecture for the cases of "short lattices" ($l=m$,
$l=m+1$, and $l=m+2$). For "taller lattices" ($l \geq 2m$) our conjecture gives
${n \choose m} - {n \choose m-1}$, independently of $l$. Our main result is
that $h_n(m,l) \leq {n \choose m} - {n \choose m-1}$ if $l \geq 2m$.
