Planar Posets Have Dimension at Most Linear in Their Height

Abstract

Abstract. We prove that every planar poset P of height h has dimension at most 192h + 96. This improves on previous exponential bounds and is best possible up to a constant factor. We complement this result with a construction of planar posets of height h and dimension at least (4/3)h − 2.

Nowhere Dense Graph Classes and Dimension

“…Let us remark that it is rather remarkable that linear or polynomial bounds can be obtained when assuming that the poset has a planar diagram or a planar cover graph, respectively. Indeed, for the slightly larger class of posets with K 5 -minor-free cover graphs, constructions show that the dimension can already be exponential in the height, as shown in [11]. (This also follows from Theorem 14 below, applied with t = 3.…”

“…It is in fact suspected that posets with planar cover graphs have dimension at most linear in their height. This was recently proved [11] for posets whose diagrams can be drawn in a planar way; these posets form a strict subclass of posets with planar cover graphs.…”

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

“…Joret, Micek and Wiechert [17] have recently shown that for fixed t ≥ 3, d(t, h) grows exponentially with h. The best bound to date in the general case is due to Joret, Micek, Ossona de Mendez and Wiechert [16], where they prove:…”

“…Joret, Micek, Ossona de Mendez and Wiechert have shown how their results in [16] can be extended to obtain this conclusion. Meanwhile, Kozik, Krawczyk, Micek and Trotter [24] have a much more complicated argument which [17] showed that the c h ≥ 2h − 2.…”

“…For planar posets, Joret, Micek and Wiechert [17] have a linear upper bound, i.e., they show that a planar poset of height h has dimension at most 192h + 96. They have also given 4h/3 − 2 as a lower bound.…”