Fractional Local Dimension

TL;DR

This paper introduces fractional local dimension, a new parameter that generalizes fractional and local dimensions of posets, and determines its asymptotic behavior for specific posets, providing bounds related to the maximum degree of comparability graphs.

Contribution

The paper defines fractional local dimension, analyzes its asymptotic value for certain posets, and establishes bounds linking it to the maximum degree in comparability graphs.

Findings

Fractional local dimension tends to a specific value FLD(d) as n increases.

For all d ≥ 2, FLD(d) is less than d+1.

As d grows large, FLD(d) behaves like d divided by (log d - log log d).

Abstract

The original notion of dimension for posets was introduced by Dushnik and Miller in 1941 and has been studied extensively in the literature. In 1992, Brightwell and Scheinerman developed the notion of fractional dimension as the natural linear programming relaxation of the Dushnik-Miller concept. In 2016, Ueckerdt introduced the concept of local dimension, and in just three years, several research papers studying this new parameter have been published. In this paper, we introduce and study fractional local dimension. As suggested by the terminology, our parameter is a common generalization of fractional dimension and local dimension. For a pair $(n,d)$ with $2\le d<n$, we consider the poset $P(1,d;n)$ consisting of all $1$-element and $d$-element subsets of $\{1,\dots,n\}$ partially ordered by inclusion. This poset has fractional dimension $d+1$, but for fixed $d\ge2$, its local dimension goes to infinity with $n$. On the other hand, we show that as $n$ tends to infinity, the fractional local dimension of $P(1,d;n)$ tends to a value $\text{FLD}(d)$ which we will be able to determine exactly. For all $d\ge2$, $\text{FLD}(d)$ is strictly less than $d+1$, and for large $d$, $\text{FLD}(d)\sim d/(\log d-\log\log d-o(1))$. As an immediate corollary, we show that if $P$ is a poset, and $d$ is the maximum degree of a vertex in the comparability graph of $P$, then the fractional local dimension of $P$, is at most $2+\text{FLD}(d).$ Our arguments use both discrete and continuous methods.