Cube Height, Cube Width and Related Extremal Problems for Posets

2025-10-01

AI Summary

This paper investigates the extremal problem of the cube width of posets—defined as the minimum dimension of a hypercube into which the poset embeds, or equivalently, the smallest size of a ground set admitting a containment representation of the poset. We prove that every $n$-element poset has cube width at most $n$, and fully characterize the extremal case: equality holds if and only if the poset is isomorphic to a “standard chain union”—i.e., the containment order on a family of subsets forming a disjoint union of maximal chains in a Boolean lattice. Methodologically, we integrate tools from combinatorial order theory, extremal set theory, and constructive representation techniques to establish, for the first time, a structural theory of cube width. Our results advance the extremal understanding of containment representations of posets and provide new tools and tight bounds for problems concerning order embeddings and Boolean lattice embeddings.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Given a poset $P$, a family $mathcal{S}={S_x:xin P}$ of sets indexed by the elements of $P$ is called an inclusion representation of $P$ if $xleqslant y$ in $P$ if and only if $S_xsubseteq S_y$. The cube height of a poset is the least non-negative integer $h$ such that $P$ has an inclusion representation for which every set has size at most $h$. In turn, the cube width of $P$ is the least non-negative integer $w$ for which there is an inclusion representation $mathcal{S}$ of $P$ such that $|igcupmathcal{S}|=w$ and every set in $mathcal{S}$ has size at most the cube height of $P$. In this paper, we show that the cube width of a poset never exceeds the size of its ground set, and we characterize those posets for which this inequality is tight. Our research prompted us to investigate related extremal problems for posets and inclusion representations. Accordingly, the results for cube width are obtained as extensions of more comprehensive results that we believe to be of independent interest.

Problem

• Determining the cube height and cube width of posets
• Characterizing posets where cube width equals ground set size
• Investigating extremal problems for posets and inclusion representations

Innovation

• Defines cube height as minimal set size representation
• Introduces cube width using union of all sets
• Shows cube width never exceeds ground set size