Nowhere dense graph classes provide one of the least restrictive notions of sparsity for graphs. Several equivalent characterizations of nowhere dense classes have been obtained over the years, using a wide range of combinatorial objects. In this paper we establish a new characterization of nowhere dense classes, in terms of poset dimension: A monotone graph class is nowhere dense if and only if for every $h \geq 1$ and every $\epsilon > 0$, posets of height at most $h$ with $n$ elements and whose cover graphs are in the class have dimension $\mathcal{O}(n^{\epsilon})$.

中文翻译：

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无处密集图类和维度

无处密集图类为图提供了限制最少的稀疏概念之一。多年来，使用广泛的组合对象已经获得了无处密集类的几个等效特征。在本文中，我们根据偏序集维度建立了无处稠密类的新特征：当且仅当对于每个 $h \geq 1$ 和每个 $\epsilon > 0$，高度偏序集，单调图类是无处稠密的至多 $h$ 具有 $n$ 个元素并且其覆盖图在类中具有维度 $\mathcal{O}(n^{\epsilon})$。