Dimension is polynomial in height for posets with planar cover graphs

Jakub Kozik, Piotr Micek, William T. Trotter

Published 2019-06-30Version 1

We show that height~$h$ posets that have planar cover graphs have dimension $O(h^6)$. Previously, this upper bound was $2^{O(h^3)}$. Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes $K_5$ as a minor.