Dimension and Height for Posets with Planar Cover Graphs

Posets of height 2 can have arbitrarily large dimension. Also, planar posets can have arbitrarily large dimension. However, we show that the dimension of a planar poset is bounded as a function of its height. In fact this statement holds for all posets with planar cover graphs. More precisely, 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.