In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Ne\v{s}et\v{r}il and Pudl\'ak in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width, and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height.

中文翻译：

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平面位姿的局部维度是无界的

1981 年，Kelly 表明平面偏序集可以具有任意大的维度。然而，Kelly 示例中的偏序集具有有界的布尔维数和有界的局部维数，这自然导致了关于平面偏序集类是布尔维数还是局部维数有界的问题。布尔维数的问题首先由 Ne\v{s}et\v{r}il 和 Pudl\'ak 在 1989 年提出，至今仍未得到解答。局部维度的概念很新，由 Ueckerdt 于 2016 年引入。从那时起，研究人员获得了许多关于布尔维数和局部维数的有趣结果，将这些参数与经典的 Dushnik-Miller 维数概念进行对比，并在参数与结构图论、路径宽度和树宽之间建立联系。特定。在这里，我们表明局部维度不受平面偏序集类的限制。我们的证明还表明，偏序集的局部维度不受其块的最大局部维度的限制，并且它提供了一个替代证明，即偏序集的局部维度不能在树方面有界——其覆盖图的宽度，与其高度无关。