Local Dimension is Unbounded for Planar Posets

12/17/2017
by   Bartłomiej Bosek, et al.
0

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 Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. In just the last year, 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/14/2022

Boolean dimension and dim-boundedness: Planar cover graph with a zero

In 1989, Nešetřil and Pudlák posed the following challenging question: D...
research
08/23/2023

Cliquewidth and dimension

We prove that every poset with bounded cliquewidth and with sufficiently...
research
10/06/2018

Local Boxicity, Local Dimension, and Maximum Degree

In this short note we focus on two recently introduced parameters in the...
research
03/23/2018

On difference graphs and the local dimension of posets

The dimension of a partially-ordered set (poset), introduced by Dushnik ...
research
06/12/2018

The queue-number of posets of bounded width or height

Heath and Pemmaraju conjectured that the queue-number of a poset is boun...
research
09/23/2022

Planar graph with twin-width seven

We construct a planar graph with twin-width equal to seven....
research
02/09/2023

t-sails and sparse hereditary classes of unbounded tree-width

It has long been known that the following basic objects are obstructions...

Please sign up or login with your details

Forgot password? Click here to reset