Dimension is polynomial in height for posets with planar cover graphs

06/30/2019

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by   Jakub Kozik, et al.

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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.