Let ⊲ be a relation between graphs. We say a graph G is ⊲-ubiquitous if whenever Γ is a graph with $nG⊲\mathrm{\Gamma }$ for all $n\in \mathbb{N}$, then one also has ${\mathrm{\aleph }}_{0}G⊲\mathrm{\Gamma }$, where αG is the disjoint union of α many copies of G.

The Ubiquity Conjecture of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation.

In this paper we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.

中文翻译：

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树的拓扑普遍性

设⊲是图之间的关系。我们说一个图G是 ⊲ -ubiquitous如果每当 Γ 是一个图$nG⊲\mathrm{\Gamma }$对所有人$n\in \mathbb{ñ}$, 那么一个也有${\mathrm{\aleph }}_{0}G⊲\mathrm{\Gamma }$，其中αG是α的多个G副本的不相交并集。

Andreae的普遍存在猜想是无限图论中的一个众所周知的开放问题，它断言每个局部有限连通图对于次要关系都是普遍存在的。

在本文中，我们展示了所有树在拓扑次要关系方面都是普遍存在的，而与它们的基数无关。这回答了 1979 年 Andreae 的一个问题。