Local Dimension is Unbounded for Planar Posets

TL;DR

This paper proves that the local dimension of planar posets can be arbitrarily large, answering longstanding questions about the boundedness of local and Boolean dimensions in these structures.

Contribution

It demonstrates that local dimension is unbounded for planar posets and provides new insights into its relationship with block structure and tree-width.

Findings

Local dimension is unbounded for planar posets.

Local dimension cannot be bounded by the maximum local dimension of blocks.

Local dimension is unbounded in terms of the cover graph's tree-width, regardless of height.

Abstract

In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Ne\v{s}et\v{r}il and Pudl\'ak in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width, and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height.