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.

翻译：我们显示,有平面覆盖图的高度($h) 方块图有维度$\ mathcal{O}(h) 6) 。 以前, 最好的上限是$2\ mathcal{O}(h) }(h) 美元。 Planity在我们的论点中起着关键作用, 因为有这样的方块:(1) 方块在高度上是指数的, (2) 封面图将5美元作为次要值排除在外 。