Dimension and Height for Posets with Planar Cover Graphs

We show that for each integer h ≥ 2, there exists a least positive integer ch so that if P is a poset having a planar cover graph and the height of P is h, then the dimension of P is at most ch. Trivially, c1 = 2. Also, Felsner, Li and Trotter showed that c2 exists and is 4, but their proof techniques do not seem to apply when h ≥ 3. We focus on establishing the existence of ch, although we suspect that the upper bound provided by our proof is far from best possible. From below, a construction of Kelly is easily modified to show that ch must be at least h + 2. © 2013 Elsevier Ltd. All rights reserved.