Dimension and height for posets with planar cover graphs

We show that for each integer h ≥ 2 , there exists a least positive integer c h 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 c h . Trivially, c 1 = 2 . Also, Felsner, Li and Trotter showed that c 2 exists and is 4 , but their proof techniques do not seem to apply when h ≥ 3 . We focus on establishing the existence of c h , 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 c h must be at least h + 2 .