Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

TL;DR

This paper compares Dushnik-Miller, Boolean, and local dimensions of posets, revealing their relationships and bounds with respect to graph parameters like tree-width and path-width, using combinatorial and Ramsey theoretic methods.

Contribution

It establishes bounds and contrasts among the three dimension parameters, especially highlighting the limitations of local dimension in terms of tree-width.

Findings

Boolean dimension is bounded by tree-width, independent of height.

Local dimension cannot be bounded solely by tree-width, but is bounded by path-width.

Ramsey theoretic methods are used to derive several results.

Abstract

The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and local dimension. For a poset $P$, the Boolean dimension of $P$ and the local dimension of $P$ are both bounded from above by the dimension of $P$ and can be considerably less. Our primary goal will be to study analogies and contrasts among these three parameters. As one example, it is known that the dimension of a poset is bounded as a function of its height and the tree-width of its cover graph. The Boolean dimension of a poset is bounded in terms of the tree-width of its cover graph, independent of its height. We show that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of height. We also prove that the local dimension of a poset is bounded in terms of the path-width of its cover graph. In several of our results, Ramsey theoretic methods will be applied.