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.

中文翻译：

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对于具有平面覆盖图的偏序集，维度是高度的多项式

我们证明具有平面覆盖图的高度h偏序集具有维度$\mathcal{氧}（H^6）$。以前，最佳上限是$2^{\mathcal{氧}（H^3）}$。平面性在我们的论证中起着关键作用，因为存在这样的偏序集：(1) 维度在高度上呈指数关系，并且 (2) 覆盖图不包括$K_5$作为未成年人。