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Local Dimension is Unbounded for Planar Posets. (arXiv:1712.06099v1 [math.CO])
来源于:arXiv
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. In just the
last year, 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 o 查看全文>>