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

TL;DR

This paper investigates the minimal size of uniform cutsets within a truncated Boolean lattice, providing bounds and exact values for these structures using combinatorial theorems.

Contribution

It introduces the function g_n(m,l) to measure minimal uniform cutsets and applies the Kruskal-Katona Theorem to characterize and bound these cutsets.

Findings

Derived bounds for g_n(m,l)

Provided exact values for specific cases

Characterized cutsets using Kruskal-Katona Theorem

Abstract

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 $g_n(m,l)$.