Planar Posets Have Dimension at Most Linear in Their Height
Gwenaël Joret, Piotr Micek, Veit Wiechert · SIAM Journal on Discrete Mathematics · 2017
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 $(4/3)h-2$.