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.