Dimension of unicycle posets

TL;DR

This work resolves Bollobás and Brightwell's conjecture by giving a self-contained, constructive proof that every unicycle poset has dimension at most $3$. The authors develop explicit realizers for crowns $\mathscr{C}_n$ and for rooted-tree posets, then orchestrate grafting operations on cycle posets to extend these realizers to all unicycle posets. The core strategy combines three realizers to realize both crown and tree substructures and then stitches them together through grafting on cycle posets, ensuring all necessary incomparabilities are maintained. The result has implications for the study of dimension in random posets and demonstrates a general method to bound dimension via structured decompositions and explicit linear extensions.

Abstract

Motivated by the study of the dimension of random posets, it was conjectured by Bollobás and Brightwell in 1997 that if $P$ is a finite poset whose cover graph contains at most one cycle then its order dimension is at most $3$. In this paper we prove this conjecture by giving a constructive proof with explicit triplets of linear extensions realizing such posets.

Figures (9)

• Figure 1: A tree poset $\textbf{T}$ with $a\in \textbf{T}$.
• Figure 2:
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Theorems & Definitions (44)

• Theorem 1.1: bollobas1997dimension, Conjecture 1.1
• Corollary 1.2: bollobas1997dimension
• proof
• Lemma 2.1
• Lemma 2.2
• Remark 2.3
• Definition 3.1
• Lemma 3.2
• proof
• Definition 3.3