On Uniform f-vectors of Cutsets in the Truncated Boolean Lattice

Let $[n] = \{1, 2, \ldots, n\}$ and let $2^{[n]}$ be the collection of all subsets of $[n]$ ordered by inclusion. ${\cal C} \subseteq 2^{[n]}$ is a {\em cutset} if it meets every maximal chain in $2^{[n]}$, and the {\em width} of ${\cal C} \subseteq 2^{[n]}$ is the minimum number of chains in a chain decomposition of ${\cal C}$. Fix $0 \leq m \leq l \leq n$. What is the smallest value of $k$ such that there exists a cutset that consists only of subsets of sizes between $m$ and $l$, and such that it contains exactly $k$ subsets of size $i$ for each $m \leq i \leq l$? The answer, which we denote by $g_n(m,l)$, gives a lower estimate for the width of a cutset between levels $m$ and $l$ in $2^{[n]}$. After using the Kruskal-Katona Theorem to give a general characterization of cutsets in terms of the number and sizes of their elements, we find lower and upper bounds (as well as some exact values) for $g_n(m,l)$.Comment: 12 page

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

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 $l \geq 2m$

On Uniform f-vectors of Cutsets in the Truncated Boolean Lattice

Let [n] = {1, 2, . . . , n} and let 2 [n] be the collection of all subsets of [n] ordered by inclusion. C # 2 [n] is a cutset if it meets every maximal chain in 2 [n] , and the width of C # 2 [n] is the minimum number of chains in a chain decomposition of C. Fix 0 # m # l # n. What is the smallest value of k such that there exists a cutset that consists only of subsets of sizes between m and l, and such that it contains exactly k subsets of size i for each m # i # l? The answer, which we denote by gn (m, l), gives a lower estimate for the width of a cutset between levels m and l in 2 [n] . After using the Kruskal-Katona Theorem to give a general characterization of cutsets in terms of the number and sizes of their elements, we find lower and upper bounds (as well as some exact values) for gn (m, l). 1 Introduction Let 2 [n] be the Boolean lattice of order n, that is the lattice of all subsets (often called nodes) of [n] = {1, 2, . . . , n} ordered by i..