Dimension is polynomial in height for posets with planar cover graphs

Published 30 Jun 2019 in math.CO and cs.DM | (1907.00380v3)

Abstract: We show that height $h$ posets that have planar cover graphs have dimension $\mathcal{O}(h^6)$. Previously, the best upper bound was $2^{{\mathcal{O}(h^3)}$.} 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.

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