On the Minimum Width of a Cutset in the Truncated Boolean Lattice

For integers $0\le m\le l\le n-m$, the truncated Boolean lattice ${\cal B}n(m,l)$istheposetofallsubsetsof$[n] = {1, 2, \ldots, n}$ which have size at least $m$ and at most $l$. $\mathcal{C}\subseteq {\mathcal{B}}_n(m,l)$ is a {\em cutset} if it meets every chain of length $l-m$ in ${\mathcal{B}}_n(m,l)$, and the {\em width} of $\mathcal{C}$ is the size of the largest antichain in $\mathcal{C}$. We conjecture that for $n>>m$ the minimum width $h_n(m,l)$ of a cutset in ${\mathcal{B}}_n(m,l)$ 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 ${\mathcal{B}}_n(m,l)$ and $\Delta_n(k)={n \choose k}- {n \choose k-1}$\mathrm{.}Weestablishourconjectureforthecasesof\mathrm{"}shortlattices\mathrm{"}($l=m$,$l=m+1$,and$l=m+2$)\mathrm{.}For\mathrm{"}tallerlattices\mathrm{"}($l \geq 2m$)ourconjecturegives${n \choose m} - {n \choose m-1}$,independentlyof$l$. Our main result is that $h_n(m,l)\le \left(\genfrac{}{}{0px}{}{n}{m}\right)-\left(\genfrac{}{}{0px}{}{n}{m-1}\right)$ if $l\ge 2m$.