Planar posets have dimension at most linear in their height
  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 $(4/3)h-2$. 
Gwena\"el Joret and Piotr Micek and Veit Wiechert
math.CO cs.DM