A Family of Partially Ordered Sets with Small Balance Constant
Evan Chen

TL;DR
This paper constructs a sequence of finite partially ordered sets with balance constants approaching approximately 0.349, providing new examples that challenge existing bounds related to the 1/3-2/3 conjecture.
Contribution
It introduces a new family of posets with smaller balance constants than previously known, advancing understanding of the 1/3-2/3 conjecture.
Findings
Sequence of posets with balance constants near 0.349
First known examples with balance constants below 1/3
Progress towards understanding the minimal possible balance constant
Abstract
Given a finite poset and two distinct elements and , we let denote the fraction of linear extensions of in which precedes . The balance constant of is then defined by \[ \delta(\mathcal P) = \max_{x \neq y \in \mathcal P} \min \left\{ \operatorname{pr}_{\mathcal P}(x \prec y), \operatorname{pr}_{\mathcal P}(y \prec x) \right\}. \] The - conjecture asserts that whenever is not a chain, but except from certain trivial examples it is not known when equality occurs, or even if balance constants can approach . In this paper we make some progress on the conjecture by exhibiting a sequence of posets with balance constants approaching , answering a question of…
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Taxonomy
TopicsAdvanced Graph Theory Research · Advanced Combinatorial Mathematics · Chemistry and Stereochemistry Studies
