Local Dimension is Unbounded for Planar Posets

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 DushnikMiller 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. 1. Notation, Terminology and Background Discussion In this paper, we investigate combinatorial problems for finite posets. As has become standard in the literature, we use the terms elements and points interchangeably in referring to the members of the ground set of a poset. We will write x ‖ y in P when x and y are incomparable in a poset P , and we let Inc(P ) denote the set of all ordered pairs (x, y) with x ‖ y in P . As a binary relation, Inc(P ) is symmetric. Recall that a non-empty family R of linear extensions of P is called a realizer of P when x < y in P if and only if x < y in L for each L ∈ R. Clearly, a non-empty family R of linear extensions of P is a realizer of P if and only if for each (x, y) ∈ Inc(P ), there is some L ∈ R for which x > y in L. The dimension of a poset P , as defined by Dushnik and Miller in their seminal paper [3], is the least positive integer d for which P has a realizer R with |R| = d. 2010 Mathematics Subject Classification. 06A07, 05C35.